Path Integrals and Hamiltonians
Baaquie, Belal E.
2014-03-01
1. Synopsis; Part I. Fundamental Principles: 2. The mathematical structure of quantum mechanics; 3. Operators; 4. The Feynman path integral; 5. Hamiltonian mechanics; 6. Path integral quantization; Part II. Stochastic Processes: 7. Stochastic systems; Part III. Discrete Degrees of Freedom: 8. Ising model; 9. Ising model: magnetic field; 10. Fermions; Part IV. Quadratic Path Integrals: 11. Simple harmonic oscillators; 12. Gaussian path integrals; Part V. Action with Acceleration: 13. Acceleration Lagrangian; 14. Pseudo-Hermitian Euclidean Hamiltonian; 15. Non-Hermitian Hamiltonian: Jordan blocks; 16. The quartic potential: instantons; 17. Compact degrees of freedom; Index.
Barth, A. M.; Vagov, A.; Axt, V. M.
2016-09-01
We present a numerical path-integral iteration scheme for the low-dimensional reduced density matrix of a time-dependent quantum dissipative system. Our approach simultaneously accounts for the combined action of a microscopically modeled pure-dephasing-type coupling to a continuum of harmonic oscillators representing, e.g., phonons, and further environmental interactions inducing non-Hamiltonian dynamics in the inner system represented, e.g., by Lindblad-type dissipation or relaxation. Our formulation of the path-integral method allows for a numerically exact treatment of the coupling to the oscillator modes and moreover is general enough to provide a natural way to include Markovian processes that are sufficiently described by rate equations. We apply this new formalism to a model of a single semiconductor quantum dot which includes the coupling to longitudinal acoustic phonons for two cases: (a) external laser excitation taking into account a phenomenological radiative decay of the excited dot state and (b) a coupling of the quantum dot to a single mode of an optical cavity taking into account cavity photon losses.
Chromatic roots and hamiltonian paths
Thomassen, Carsten
2000-01-01
We present a new connection between colorings and hamiltonian paths: If the chromatic polynomial of a graph has a noninteger root less than or equal to t(n) = 2/3 + 1/3 (3)root (26 + 6 root (33)) + 1/3 (3)root (26 - 6 root (33)) = 1.29559.... then the graph has no hamiltonian path. This result...
On the Reaction Path Hamiltonian
孙家钟; 李泽生
1994-01-01
A vector-fiber bundle structure of the reaction path Hamiltonian, which has been introduced by Miller, Handy and Adams, is explored with respect to molecular vibrations orthogonal to the reaction path. The symmetry of the fiber bundle is characterized by the real orthogonal group O(3N- 7) for the dynamical system with N atoms. Under the action of group O(3N- 7). the kinetic energy of the reaction path Hamiltonian is left invariant. Furthermore , the invariant behaviour of the Hamiltonian vector fields is investigated.
New sufficient conditions for Hamiltonian paths.
Rahman, M Sohel; Kaykobad, M; Firoz, Jesun Sahariar
2014-01-01
A Hamiltonian path in a graph is a path involving all the vertices of the graph. In this paper, we revisit the famous Hamiltonian path problem and present new sufficient conditions for the existence of a Hamiltonian path in a graph.
Path integrals for awkward actions
Amdahl, David
2016-01-01
Time derivatives of scalar fields occur quadratically in textbook actions. A simple Legendre transformation turns the lagrangian into a hamiltonian that is quadratic in the momenta. The path integral over the momenta is gaussian. Mean values of operators are euclidian path integrals of their classical counterparts with positive weight functions. Monte Carlo simulations can estimate such mean values. This familiar framework falls apart when the time derivatives do not occur quadratically. The Legendre transformation becomes difficult or so intractable that one can't find the hamiltonian. Even if one finds the hamiltonian, it usually is so complicated that one can't path-integrate over the momenta and get a euclidian path integral with a positive weight function. Monte Carlo simulations don't work when the weight function assumes negative or complex values. This paper solves both problems. It shows how to make path integrals without knowing the hamiltonian. It also shows how to estimate complex path integrals b...
Quantization of noncommutative completely integrable Hamiltonian systems
Giachetta, G; Sardanashvily, G
2007-01-01
Integrals of motion of a Hamiltonian system need not be commutative. The classical Mishchenko-Fomenko theorem enables one to quantize a noncommutative completely integrable Hamiltonian system around its invariant submanifold as an abelian completely integrable Hamiltonian system.
Hamiltonian formalism and path entropy maximization
Davis, Sergio; González, Diego
2015-10-01
Maximization of the path information entropy is a clear prescription for constructing models in non-equilibrium statistical mechanics. Here it is shown that, following this prescription under the assumption of arbitrary instantaneous constraints on position and velocity, a Lagrangian emerges which determines the most probable trajectory. Deviations from the probability maximum can be consistently described as slices in time by a Hamiltonian, according to a nonlinear Langevin equation and its associated Fokker-Planck equation. The connections unveiled between the maximization of path entropy and the Langevin/Fokker-Planck equations imply that missing information about the phase space coordinate never decreases in time, a purely information-theoretical version of the second law of thermodynamics. All of these results are independent of any physical assumptions, and thus valid for any generalized coordinate as a function of time, or any other parameter. This reinforces the view that the second law is a fundamental property of plausible inference.
Quantization of noncommutative completely integrable Hamiltonian systems
Giachetta, G. [Department of Mathematics and Informatics, University of Camerino, 62032 Camerino (Italy); Mangiarotti, L. [Department of Mathematics and Informatics, University of Camerino, 62032 Camerino (Italy); Sardanashvily, G. [Department of Theoretical Physics, Moscow State University, 117234 Moscow (Russian Federation)]. E-mail: gennadi.sardanashvily@unicam.it
2007-02-26
Integrals of motion of a Hamiltonian system need not commute. The classical Mishchenko-Fomenko theorem enables one to quantize a noncommutative completely integrable Hamiltonian system around its invariant submanifold as the Abelian one.
Kulshreshtha, Usha, E-mail: ushakulsh@gmail.com [Department of Physics and Astronomy, Iowa State University, 50011, Ames, IA (United States); Department of Physics, Kirori Mal College, University of Delhi, 110007, Delhi (India); Kulshreshtha, Daya Shankar, E-mail: dskulsh@gmail.com [Department of Physics and Astronomy, Iowa State University, 50011, Ames, IA (United States); Department of Physics and Astrophysics, University of Delhi, 110007, Delhi (India); Vary, James P., E-mail: jvary@iastate.edu [Department of Physics and Astronomy, Iowa State University, 50011, Ames, IA (United States)
2015-04-28
Recently Grinstein, Jora, and Polosa have studied a theory of large-N scalar quantum chromodynamics in one space and one time dimension. This theory admits a Bethe–Salpeter equation describing the discrete spectrum of quark–antiquark bound states. They consider gauge fields in the adjoint representation of SU(N) and scalar fields in the fundamental representation. The theory is asymptotically free and linearly confining. The theory could possibly provide a good field theoretic framework for the description of a large class of diquark–antidiquark (tetra-quark) states. Recently we have studied the light-front quantization of this theory without a Higgs potential. In the present work, we study the light-front Hamiltonian, path integral, and BRST formulations of the theory in the presence of a Higgs potential. The light-front theory is seen to be gauge invariant, possessing a set of first-class constraints. The explicit occurrence of spontaneous symmetry breaking in the theory is shown in unitary gauge as well as in the light-front ’t Hooft gauge.
Kulshreshtha, Usha [Iowa State University, Department of Physics and Astronomy, Ames, IA (United States); University of Delhi, Department of Physics, Kirori Mal College, Delhi (India); Kulshreshtha, Daya Shankar [Iowa State University, Department of Physics and Astronomy, Ames, IA (United States); University of Delhi, Department of Physics and Astrophysics, Delhi (India); Vary, James P. [Iowa State University, Department of Physics and Astronomy, Ames, IA (United States)
2015-04-01
Recently Grinstein, Jora, and Polosa have studied a theory of large- N scalar quantum chromodynamics in one space and one time dimension. This theory admits a Bethe-Salpeter equation describing the discrete spectrum of quark-antiquark bound states. They consider gauge fields in the adjoint representation of SU(N) and scalar fields in the fundamental representation. The theory is asymptotically free and linearly confining. The theory could possibly provide a good field theoretic framework for the description of a large class of diquark-antidiquark (tetra-quark) states. Recently we have studied the light-front quantization of this theory without a Higgs potential. In the present work, we study the light-front Hamiltonian, path integral, and BRST formulations of the theory in the presence of a Higgs potential. The light-front theory is seen to be gauge invariant, possessing a set of first-class constraints. The explicit occurrence of spontaneous symmetry breaking in the theory is shown in unitary gauge as well as in the light-front 't Hooft gauge. (orig.)
Kulshreshtha, Usha; Vary, James P
2015-01-01
Recently Grinstein, Jora, and Polosa have studied a theory of large-$N$ scalar quantum chromodynamics in one-space one-time dimension. This theory admits a Bethe-Salpeter equation describing the discrete spectrum of quark-antiquark bound states. They consider gauge fields in the adjoint representation of $SU(N)$ and scalar fields in the fundamental representation. The theory is asymptotically free and linearly confining. The theory could possibly provide a good field theoretic framework for the description of a large class of diquark-antidiquark (tetra-quark) states. Recently we have studied the light-front quantization of this theory without a Higgs potential. In the present work, we study the light-front Hamiltonian, path integral and BRST formulations of the theory in the presence of a Higgs potential. The light-front theory is seen to be gauge-invariant, possessing a set of first-class constraints. The explicit occurrence of spontaneous symmetry breaking in the theory is shown in unitary gauge as well as ...
On third order integrable vector Hamiltonian equations
Meshkov, A. G.; Sokolov, V. V.
2017-03-01
A complete list of third order vector Hamiltonian equations with the Hamiltonian operator Dx having an infinite series of higher conservation laws is presented. A new vector integrable equation on the sphere is found.
Jacobi fields of completely integrable Hamiltonian systems
Giachetta, G.; Mangiarotti, L.; Sardanashvily, G
2003-03-31
We show that Jacobi fields of a completely integrable Hamiltonian system of m degrees of freedom make up an extended completely integrable system of 2m degrees of freedom, where m additional first integrals characterize a relative motion.
Solving a Hamiltonian Path Problem with a bacterial computer
Treece Jessica
2009-07-01
Full Text Available Abstract Background The Hamiltonian Path Problem asks whether there is a route in a directed graph from a beginning node to an ending node, visiting each node exactly once. The Hamiltonian Path Problem is NP complete, achieving surprising computational complexity with modest increases in size. This challenge has inspired researchers to broaden the definition of a computer. DNA computers have been developed that solve NP complete problems. Bacterial computers can be programmed by constructing genetic circuits to execute an algorithm that is responsive to the environment and whose result can be observed. Each bacterium can examine a solution to a mathematical problem and billions of them can explore billions of possible solutions. Bacterial computers can be automated, made responsive to selection, and reproduce themselves so that more processing capacity is applied to problems over time. Results We programmed bacteria with a genetic circuit that enables them to evaluate all possible paths in a directed graph in order to find a Hamiltonian path. We encoded a three node directed graph as DNA segments that were autonomously shuffled randomly inside bacteria by a Hin/hixC recombination system we previously adapted from Salmonella typhimurium for use in Escherichia coli. We represented nodes in the graph as linked halves of two different genes encoding red or green fluorescent proteins. Bacterial populations displayed phenotypes that reflected random ordering of edges in the graph. Individual bacterial clones that found a Hamiltonian path reported their success by fluorescing both red and green, resulting in yellow colonies. We used DNA sequencing to verify that the yellow phenotype resulted from genotypes that represented Hamiltonian path solutions, demonstrating that our bacterial computer functioned as expected. Conclusion We successfully designed, constructed, and tested a bacterial computer capable of finding a Hamiltonian path in a three node
Integrable Hamiltonian systems and spectral theory
Moser, J
1981-01-01
Classical integrable Hamiltonian systems and isospectral deformations ; geodesics on an ellipsoid and the mechanical system of C. Neumann ; the Schrödinger equation for almost periodic potentials ; finite band potentials ; limit cases, Bargmann potentials.
Integrability and Non-integrability of Hamiltonian Normal Forms
Verhulst, Ferdinand
2015-01-01
This paper summarizes the present state of integrability of Hamiltonian normal forms and it aims at characterizing non-integrable behaviour in higher-dimensional systems. Non-generic behaviour in Hamiltonian systems can be a sign of integrability, but it is not a conclusive indication. We will discu
Discrete Coherent State Path Integrals
Marchioro, Thomas L., II
1990-01-01
The quantum theory provides a fundamental understanding of the physical world; however, as the number of degrees of freedom rises, the information required to specify quantum wavefunctions grows geometrically. Because basis set expansions mirror this geometric growth, a strict practical limit on quantum mechanics as a numerical tool arises, specifically, three degrees of freedom or fewer. Recent progress has been made utilizing Feynman's Path Integral formalism to bypass this geometric growth and instead calculate time -dependent correlation functions directly. The solution of the Schrodinger equation is converted into a large dimensional (formally infinite) integration, which can then be attacked with Monte Carlo techniques. To date, work in this area has concentrated on developing sophisticated mathematical algorithms for evaluating the highly oscillatory integrands occurring in Feynman Path Integrals. In an alternative approach, this work demonstrates two formulations of quantum dynamics for which the number of mathematical operations does not scale geometrically. Both methods utilize the Coherent State basis of quantum mechanics. First, a localized coherent state basis set expansion and an approximate short time propagator are developed. Iterations of the short time propagator lead to the full quantum dynamics if the coherent state basis is sufficiently dense along the classical phase space path of the system. Second, the coherent state path integral is examined in detail. For a common class of Hamiltonians, H = p^2/2 + V( x) the path integral is reformulated from a phase space-like expression into one depending on (q,dot q). It is demonstrated that this new path integral expression contains localized damping terms which can serve as a statistical weight for Monte Carlo evaluation of the integral--a process which scales approximately linearly with the number of degrees of freedom. Corrections to the traditional coherent state path integral, inspired by a
Global Properties of Integrable Hamiltonian Systems
Lukina, O.V.; Takens, F.; Broer, H.W.
2008-01-01
This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our
Global Properties of Integrable Hamiltonian Systems
Lukina, O.V.; Takens, F.; Broer, H.W.
2008-01-01
This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approa
Shakeri, Nadim; Jalili, Saeed; Ahmadi, Vahid; Rasoulzadeh Zali, Aref; Goliaei, Sama
2015-01-01
The problem of finding the Hamiltonian path in a graph, or deciding whether a graph has a Hamiltonian path or not, is an NP-complete problem. No exact solution has been found yet, to solve this problem using polynomial amount of time and space. In this paper, we propose a two dimensional (2-D) optical architecture based on optical electronic devices such as micro ring resonators, optical circulators and MEMS based mirror (MEMS-M) to solve the Hamiltonian Path Problem, for undirected graphs in linear time. It uses a heuristic algorithm and employs n+1 different wavelengths of a light ray, to check whether a Hamiltonian path exists or not on a graph with n vertices. Then if a Hamiltonian path exists, it reports the path. The device complexity of the proposed architecture is O(n2).
Smooth prime integrals for quasi-integrable Hamiltonian systems
Chierchia, L.; Gallavotti, G. (Rome Univ. (Italy). Ist. di Matematica)
1982-02-11
A Hamiltonian with N degrees of freedom, analytic perturbation of a canonically integrable strictly nonisochronous analytic Hamiltonian, is considered. We show the existence of N functions on phase space and of class Csup(infinity) which are prime integrals for the perturbed motions on a suitable region whose Lebesgue measure tends to fill locally the phase space as the perturbation's magnitude approaches zero. An application to the perturbations of isochronous nonresonant linear oscillators is given.
Integrable Coupling of KN Hierarchy and Its Hamiltonian Structure
GUO Fu-Kui; ZHANG Yu-Feng
2006-01-01
The Hamiltonian structure of the integrable couplings obtained by our method has not been solved. In this paper, the Hamiltonian structure of the KN hierarchy is obtained by making use of the quadratic-form identity.
Lax operator algebras and Hamiltonian integrable hierarchies
Sheinman, Oleg K
2009-01-01
We consider the theory of Lax equations in complex simple and reductive classical Lie algebras with the spectral parameter on a Riemann surface of finite genus. Our approach is based on the new objects -- the Lax operator algebras, and develops the approach of I.Krichever treating the $\\gl(n)$ case. For every Lax operator considered as the mapping sending a point of the cotangent bundle on the space of extended Tyrin data to an element of the corresponding Lax operator algebra we construct the hierarchy of mutually commuting flows given by Lax equations and prove that those are Hamiltonian with respect to the Krichever-Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example we derive elliptic $A_n$, $C_n$, $D_n$ Calogero-Moser systems in frame of our approach.
Lax operator algebras and Hamiltonian integrable hierarchies
Sheinman, Oleg K [Steklov Mathematical Institute, Russian Academy of Sciences, Moscow (Russian Federation)
2011-02-28
This paper considers the theory of Lax equations with a spectral parameter on a Riemann surface, proposed by Krichever in 2001. The approach here is based on new objects, the Lax operator algebras, taking into consideration an arbitrary complex simple or reductive classical Lie algebra. For every Lax operator, regarded as a map sending a point of the cotangent bundle on the space of extended Tyurin data to an element of the corresponding Lax operator algebra, a hierarchy of mutually commuting flows given by the Lax equations is constructed, and it is proved that they are Hamiltonian with respect to the Krichever-Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example, elliptic A{sub n}, C{sub n}, and D{sub n} Calogero-Moser systems are derived in the framework of our approach. Bibliography: 13 titles.
Path Integral Approach to Atomic Collisions
Harris, Allison
2016-09-01
The Path Integral technique is an alternative formulation of quantum mechanics that is based on a Lagrangian approach. In its exact form, it is completely equivalent to the Hamiltonian-based Schrödinger equation approach. Developed by Feynman in the 1940's, following inspiration from Dirac, the path integral approach has been widely used in high energy physics, quantum field theory, and statistical mechanics. However, only in limited cases has the path integral approach been applied to quantum mechanical few-body scattering. We present a theoretical and computational development of the path integral method for use in the study of atomic collisions. Preliminary results are presented for some simple systems. Ultimately, this approach will be applied to few-body ion-atom collisions. Work supported by NSF grant PHY-1505217.
An alternative path integral for quantum gravity
Krishnan, Chethan; Kumar, K. V. Pavan; Raju, Avinash
2016-10-01
We define a (semi-classical) path integral for gravity with Neumann boundary conditions in D dimensions, and show how to relate this new partition function to the usual picture of Euclidean quantum gravity. We also write down the action in ADM Hamiltonian formulation and use it to reproduce the entropy of black holes and cosmological horizons. A comparison between the (background-subtracted) covariant and Hamiltonian ways of semi-classically evaluating this path integral in flat space reproduces the generalized Smarr formula and the first law. This "Neumann ensemble" perspective on gravitational thermodynamics is parallel to the canonical (Dirichlet) ensemble of Gibbons-Hawking and the microcanonical approach of Brown-York.
An Alternate Path Integral for Quantum Gravity
Krishnan, Chethan; Raju, Avinash
2016-01-01
We define a (semi-classical) path integral for gravity with Neumann boundary conditions in $D$ dimensions, and show how to relate this new partition function to the usual picture of Euclidean quantum gravity. We also write down the action in ADM Hamiltonian formulation and use it to reproduce the entropy of black holes and cosmological horizons. A comparison between the (background-subtracted) covariant and Hamiltonian ways of semi-classically evaluating this path integral in flat space reproduces the generalized Smarr formula and the first law. This "Neumann ensemble" perspective on gravitational thermodynamics is parallel to the canonical (Dirichlet) ensemble of Gibbons-Hawking and the microcanonical approach of Brown-York.
An alternative path integral for quantum gravity
Krishnan, Chethan; Kumar, K.V. Pavan; Raju, Avinash [Center for High Energy Physics, Indian Institute of Science,Bangalore 560012 (India)
2016-10-10
We define a (semi-classical) path integral for gravity with Neumann boundary conditions in D dimensions, and show how to relate this new partition function to the usual picture of Euclidean quantum gravity. We also write down the action in ADM Hamiltonian formulation and use it to reproduce the entropy of black holes and cosmological horizons. A comparison between the (background-subtracted) covariant and Hamiltonian ways of semi-classically evaluating this path integral in flat space reproduces the generalized Smarr formula and the first law. This “Neumann ensemble” perspective on gravitational thermodynamics is parallel to the canonical (Dirichlet) ensemble of Gibbons-Hawking and the microcanonical approach of Brown-York.
An integrable Hamiltonian hierarchy and associated integrable couplings system
Chen Xiao-Hong; Xia Tie-Cheng; Zhu Lian-Cheng
2007-01-01
This paper establishes a new isospectral problem. By making use of the Tu scheme, a new integrable system is obtained. It gives integrable couplings of the system obtained. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.
Path integrals for actions that are not quadratic in their time derivatives
Cahill, Kevin
2015-01-01
The standard way to construct a path integral is to use a Legendre transformation to find the hamiltonian, to repeatedly insert complete sets of states into the time-evolution operator, and then to integrate over the momenta. This procedure is simple when the action is quadratic in its time derivatives, but in most other cases Legendre's transformation is intractable, and the hamiltonian is unknown. This paper shows how to make path integrals without using the hamiltonian.
Modeling DNA Dynamics by Path Integrals
Zoli, Marco
2013-01-01
Complementary strands in DNA double helix show temporary fluctuational openings which are essential to biological functions such as transcription and replication of the genetic information. Such large amplitude fluctuations, known as the breathing of DNA, are generally localized and, microscopically, are due to the breaking of the hydrogen bonds linking the base pairs (\\emph{bps}). I apply imaginary time path integral techniques to a mesoscopic Hamiltonian which accounts for the helicoidal geometry of a short circular DNA molecule. The \\emph{bps} displacements with respect to the ground state are interpreted as time dependent paths whose amplitudes are consistent with the model potential for the hydrogen bonds. The portion of the paths configuration space contributing to the partition function is determined by selecting the ensemble of paths which fulfill the second law of thermodynamics. Computations of the thermodynamics in the denaturation range show the energetic advantage for the equilibrium helicoidal g...
Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems
XU Xi-Xiang; YANG Hong-Xiang; LU Rong-Wu
2008-01-01
A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-fore matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.
Propagators and path integrals
Holten, J.W. van
1995-08-22
Path-integral expressions for one-particle propagators in scalar and fermionic field theories are derived, for arbitrary mass. This establishes a direct connection between field theory and specific classical point-particle models. The role of world-line reparametrization invariance of the classical action and the implementation of the corresponding BRST-symmetry in the quantum theory are discussed. The presence of classical world-line supersymmetry is shown to lead to an unwanted doubling of states for massive spin-1/2 particles. The origin of this phenomenon is traced to a `hidden` topological fermionic excitation. A different formulation of the pseudo-classical mechanics using a bosonic representation of {gamma}{sub 5} is shown to remove these extra states at the expense of losing manifest supersymmetry. (orig.).
A New Scheme of Integrability for (bi)Hamiltonian PDE
De Sole, Alberto; Kac, Victor G.; Valeri, Daniele
2016-10-01
We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method is based on the notion of an Adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant. We also introduce the notion of a dispersionless Adler type series, which is applied to the study of dispersionless Hamiltonian equations. Non-commutative Hamiltonian equations are discussed in this framework as well.
Mignemi, S., E-mail: smignemi@unica.it [Dipartimento di Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09123 Cagliari (Italy); INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato (Italy); Štrajn, R. [Dipartimento di Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09123 Cagliari (Italy); INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato (Italy)
2016-04-29
The definition of path integrals in one- and two-dimensional Snyder space is discussed in detail both in the traditional setting and in the first-order formalism of Faddeev and Jackiw. - Highlights: • The definition of the path integral in Snyder space is discussed using phase space methods. • The same result is obtained in the first-order formalism of Faddeev and Jackiw. • The path integral formulation of the two-dimensional Snyder harmonic oscillator is outlined.
Comparing Maps to Symplectic Integrators in a Galactic Type Hamiltonian
N. D. Caranicolas; N. J. Papadopoulos
2003-09-01
We obtain the - Poincare phase plane for a two dimensional, resonant, galactic type Hamiltonian using conventional numerical integration, a second order symplectic integrator and a map based on the averaged Hamiltonian. It is found that all three methods give good results, for small values of the perturbation parameter, while the symplectic integrator does a better job than the mapping, for large perturbations. The dynamical spectra are used to distinguish between regular and chaotic motion.
Path Integrals in Quantum Physics
Rosenfelder, R
2012-01-01
These lectures aim at giving graduate students an introduction to and a working knowledge of path integral methods in a wide variety of fields in physics. Consequently, the lecture notes are organized in three main parts dealing with non-relativistic quantum mechanics, many-body physics and field theory. In the first part the basic concepts of path integrals are developed in the usual heuristic, non-mathematical way followed by standard examples and special applications including numerical evaluation of (euclidean) path integrals by Monte-Carlo methods with a program for the anharmonic oscillator. The second part deals with the application of path integrals in statistical mechanics and many-body problems treating the polaron problem, dissipative quantum systems, path integrals over ordinary and Grassmannian coherent states and perturbation theory for both bosons and fermions. Again a simple Fortran program is included for illustrating the use of strong-coupling methods. Finally, in the third part path integra...
Path Integral Quantization of Generalized Quantum Electrodynamics
Bufalo, Rodrigo; Zambrano, German Enrique Ramos
2010-01-01
It is shown in this paper a complete covariant quantization of Generalized Electrodynamics by path integral approach. To this goal we first studied the hamiltonian structure of system following Dirac's methodology, and then we follow the Faddeev-Senjanovic procedure to attain the amplitude transition. The complete propagators (Schwinger-Dyson-Fradkin equations) on correct gauge fixation and the generalized Ward-Fradkin-Takahashi identities are also obtained. Afterwards, an explicit calculation on one-loop approximation of all Green's functions and a discussion about the obtained results are presented.
An Introduction into the Feynman Path Integral
Grosche, C
1993-01-01
In this lecture a short introduction is given into the theory of the Feynman path integral in quantum mechanics. The general formulation in Riemann spaces will be given based on the Weyl- ordering prescription, respectively product ordering prescription, in the quantum Hamiltonian. Also, the theory of space-time transformations and separation of variables will be outlined. As elementary examples I discuss the usual harmonic oscillator, the radial harmonic oscillator, and the Coulomb potential. Lecture given at the graduate college ''Quantenfeldtheorie und deren Anwendung in der Elementarteilchen- und Festk\\"orperphysik'', Universit\\"at Leipzig, 16-26 November 1992.
Boundary conditions: The path integral approach
Asorey, M [Departamento de Fisica Teorica, Universidad de Zaragoza 50009 Zaragoza (Spain); Clemente-Gallardo, J [BIFI, Universidad de Zaragoza, 50009 Zaragoza (Spain); Munoz-Castaneda, J M [Departamento de Fisica Teorica, Universidad de Zaragoza 50009 Zaragoza (Spain)
2007-11-15
The path integral approach to quantum mechanics requires a substantial generalisation to describe the dynamics of systems confined to bounded domains. Nonlocal boundary conditions can be introduced in Feynman's approach by means of boundary amplitude distributions and complex phases to describe the quantum dynamics in terms of the classical trajectories. The different prescriptions involve only trajectories reaching the boundary and correspond to different choices of boundary conditions of selfadjoint extensions of the Hamiltonian. One dimensional particle dynamics is analysed in detail.
INTEGRABLE COUPLINGS OF THE TB HIERARCHY AND ITS HAMILTONIAN STRUCTURE
无
2008-01-01
In this paper,we obtain integrable couplings of the TB hierarchy using the new subalgebra of the loop algebra A_3.Then the Hamiltonian structure of the above system is given by the quadratic-form identity.
Path integrals, hyperbolic spaces and Selberg trace formulae
Grosche, Christian
2013-01-01
In this second edition, a comprehensive review is given for path integration in two- and three-dimensional (homogeneous) spaces of constant and non-constant curvature, including an enumeration of all the corresponding coordinate systems which allow separation of variables in the Hamiltonian and in the path integral. The corresponding path integral solutions are presented as a tabulation. Proposals concerning interbasis expansions for spheroidal coordinate systems are also given. In particular, the cases of non-constant curvature Darboux spaces are new in this edition.The volume also contains r
Path Integration in Conical Space
Inomata, Akira; Junker, Georg
2011-01-01
Quantum mechanics in conical space is studied by the path integral method. It is shown that the curvature effect gives rise to an effective potential in the radial path integral. It is further shown that the radial path integral in conical space can be reduced to a form identical with that in flat space when the discrete angular momentum of each partial wave is replaced by a specific non-integral angular momentum. The effective potential is found proportional to the squared mean curvature of ...
Equivariant Localization of Path Integrals
Szabo, Richard J.
1996-01-01
We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background -- equivariant cohomology and the Duistermaat-Heckman theorem, we demonstrate how the localization ideas are relat...
Path integrals and quantum processes
Swanson, Marc S
1992-01-01
In a clearly written and systematic presentation, Path Integrals and Quantum Processes covers all concepts necessary to understand the path integral approach to calculating transition elements, partition functions, and source functionals. The book, which assumes only a familiarity with quantum mechanics, is ideal for use as a supplemental textbook in quantum mechanics and quantum field theory courses. Graduate and post-graduate students who are unfamiliar with the path integral will also benefit from this contemporary text. Exercise sets are interspersed throughout the text to facilitate self-
Scattering theory with path integrals
Rosenfelder, R. [Particle Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI (Switzerland)
2014-03-15
Starting from well-known expressions for the T-matrix and its derivative in standard nonrelativistic potential scattering, I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.
Scattering Theory with Path Integrals
Rosenfelder, R
2013-01-01
Starting from well-known expressions for the $T$-matrix and its derivative in standard nonrelativistic potential scattering I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.
Path Integrals in Quantum Physics
2012-01-01
These lectures aim at giving graduate students an introduction to and a working knowledge of path integral methods in a wide variety of fields in physics. Consequently, the lecture notes are organized in three main parts dealing with non-relativistic quantum mechanics, many-body physics and field theory. In the first part the basic concepts of path integrals are developed in the usual heuristic, non-mathematical way followed by standard examples and special applications including numerical ev...
A systematic construction of completely integrable Hamiltonians from coalgebras
Ballesteros, A; Ballesteros, Angel; Ragnisco, Orlando
1998-01-01
A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of Hamiltonian systems with coalgebra symmetry. In order to illustrate this general method, the $so(2,1)$ algebra and the oscillator algebra $h_4$ are used to derive new classical integrable systems including a generalization of Gaudin-Calogero systems and oscillator chains. Quantum deformations are then used to obtain some explicit integrable deformations of the previous long-range interacting systems and a (non-coboundary) deformation of the $(1+1)$ Poincaré algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.
Geometry of KAM tori for nearly integrable Hamiltonian systems
Broer, Hendrik; Cushman, Richard; Fassò, Francesco; Takens, Floris
2007-01-01
We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangian tori by gluing together local KAM conjugacies with the help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly integrable system and an integrable one. This leads to the pr
A Path Integral Approach to Inclusive Processes
Nachtmann, O
2000-01-01
The single-particle inclusive differential cross-section for a reaction$a+b\\to c+X$ is written as the imaginary part of a correlation function in afor ward scattering amplitude for $a+b\\to a+b$ in a modified effective theory.In this modified theory the interaction Hamiltonian $\\tilde H_I$ equals $H_I$in the original theory up to a certain time. Then there is a sign change and$\\tilde H_I$ becomes nonlocal. This is worked out in detail for scalar fieldmodels and for QED plus the abelian gluon model. A suitable path integral fordirect calculations of inclusive cross sections is presented.
Spin Observables and Path Integrals
López, J A
2000-01-01
We discuss the formulation of spin observables associated to a non-relativistic spinning particles in terms of grassmanian differential operators. We use as configuration space variables for the pseudo-classical description of this system the positions $x$ and a Grassmanian vector quantum amplitudes as path integrals in this superspace. We compute the quantum action necessary for this description including an explicit expression for the boundary terms. Finally we shown how for simple examples, the path integral may be performed in the semi-classical approximation, leading to the correct quantum propagator.
The discrete variational principle in Hamiltonian formalism and first integrals
Zhang Hong-Bin; Chen Li-Qun; Liu Rong-Wan
2005-01-01
The aim of this paper is to show that first integrals of discrete equation of motion for Hamiltonian systems can be determined explicitly by investigating the invariance properties of the discrete Lagrangian in phase space. The result obtained is a discrete analog of the theorem of Noether in the calculus of variations.
Path integral approach to two-dimensional QCD in the light-front frame
Gaete, P. (Instituto de Fisica, Universidade Federal do Rio de Janeiro, C.P. 68528, BR-21945, Rio de Janeiro (Brazil)); Gamboa, J. (Fachbereich 7 Physik, Universitaet Siegen, Siegen, D-57068 (Germany)); Schmidt, I. (Departamento de Fisica, Universidad Tecnica Federico Santa Maria, Casilla 110-V, Valparaiso (Chile))
1994-05-15
Two-dimensional quantum chromodynamics in the light-front frame is studied following Hamiltonian methods. The theory is quantized using the path integral formalism and an effective theory similar to the Nambu--Jona-Lasinio model is obtained. Confinement in two dimensions is derived by analyzing directly the constraints in the path integral.
The path integral representation kernel of evolution operator in Merton-Garman model
Blazhyevskyi, L F; 10.5488/CMP.14.23001
2011-01-01
In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.
Path integral for inflationary perturbations
Prokopec, T.; Rigopoulos, G.
2010-01-01
The quantum theory of cosmological perturbations in single-field inflation is formulated in terms of a path integral. Starting from a canonical formulation, we show how the free propagators can be obtained from the well-known gauge-invariant quadratic action for scalar and tensor perturbations, and
Searching for integrable Hamiltonian systems with Platonic symmetries
Rastelli, Giovanni
2010-01-01
In this paper we try to find examples of integrable natural Hamiltonian systems on the sphere $S^2$ with the symmetries of each Platonic polyhedra. Although some of these systems are known, their expression is extremely complicated; we try here to find the simplest possible expressions for this kind of dynamical systems. Even in the simplest cases it is not easy to prove their integrability by direct computation of the first integrals, therefore, we make use of numerical methods to provide evidences of integrability; namely, by analyzing their Poincar\\'e sections (surface sections). In this way we find three systems with platonic symmetries, one for each class of equivalent Platonic polyhedra: tetrahedral, exahedral-octahedral, dodecahedral-icosahedral, showing evidences of integrability. The proof of integrability and the construction of the first integrals are left for further works. As an outline of the possible developments if the integrability of these systems will be proved, we show how to build from th...
Note on integrability of certain homogeneous Hamiltonian systems
Szumiński, Wojciech [Institute of Physics, University of Zielona Góra, Licealna 9, PL-65-407, Zielona Góra (Poland); Maciejewski, Andrzej J. [Institute of Astronomy, University of Zielona Góra, Licealna 9, PL-65-407, Zielona Góra (Poland); Przybylska, Maria, E-mail: M.Przybylska@if.uz.zgora.pl [Institute of Physics, University of Zielona Góra, Licealna 9, PL-65-407, Zielona Góra (Poland)
2015-12-04
In this paper we investigate a class of natural Hamiltonian systems with two degrees of freedom. The kinetic energy depends on coordinates but the system is homogeneous. Thanks to this property it admits, in a general case, a particular solution. Using this solution we derive necessary conditions for the integrability of such systems investigating differential Galois group of variational equations. - Highlights: • Necessary integrability conditions for some 2D homogeneous Hamilton systems are given. • Conditions are obtained analysing differential Galois group of variational equations. • New integrable and superintegrable systems are identified.
A Few Expanding Integrable Models, Hamiltonian Structures and Constrained Flows
ZHANG Yu-Feng
2011-01-01
Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained.From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively.Especially, the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation.The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variational identity.Finally,we decompose the BK hierarchy of evolution equations into x-constrained flows and tn-constrained flows whose adjoint representations and the Lax pairs are given.
Non-Hamiltonian perturbations of integrable systems and resonance trapping
Ghil, M.; Wolansky, G.
1992-01-01
This paper studies general, non-Hamiltonian perturbations of integrable systems with two degrees of freedom and derives conditions for temporary and permanent resonance trapping. The analysis involves a noncanonical transformation of variables near the resonant manifold and averaging with respect to the fast phase to investigate oscillatory behavior on the intermediate timescale. The resulting reduced system is Hamiltonian to leading order and permits, after averaging on the intermediate, or libration, timescale, a canonical transformation to action-angle variables in the oscillation zone. The final system so obtained reveals the possible existence of two- and three-dimensional invariant tori in the vicinity of the resonant manifold. An explicit divergence condition for general perturbations to be dissipative on the slow timescale follows from the analysis. An application of this approach to the problem of resonant trapping and escape is outlined for the restricted problem of three bodies subject to dissipative perturbations with a radial symmetry.
Path Integral for Lattice Staggered Fermions in the Loop Representation
Aroca, J M; Gambini, R
1998-01-01
The path integral formulation in terms of loop variables is introduced for lattice gauge theories with dynamical fermions. The path integral of lattice compact QED with staggered fermions is expressed as a sum over surfaces with border on self-avoiding fermionic paths. Each surface is weighted with a classical action -- written in terms of integer gauge invariant variables -- which gives via transfer matrix method the Hamiltonian of the loop or P-representation. The surfaces correspond to the world sheets of loop-like pure electric flux excitations and meson-like configurations (open electric flux tubes carrying matter fields at their ends). The gauge non-redundancy and the geometric transparency are two appealing features of this description. From the computational point of view, it involves fewer degrees of freedom than the Kogut-Susskind formulation and offers the possibility of alternative numerical methods for dynamical fermions.
The symmetry groups of bifurcations of integrable Hamiltonian systems
Orlova, E I [M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow (Russian Federation)
2014-11-30
Two-dimensional atoms are investigated; these are used to code bifurcations of the Liouville foliations of nondegenerate integrable Hamiltonian systems. To be precise, the symmetry groups of atoms with complexity at most 3 are under study. Atoms with symmetry group Z{sub p}⊕Z{sub q} are considered. It is proved that Z{sub p}⊕Z{sub q} is the symmetry group of a toric atom. The symmetry groups of all nonorientable atoms with complexity at most 3 are calculated. The concept of a geodesic atom is introduced. Bibliography: 9 titles.
Third derivatives of the integrable part of an asteroid Hamiltonian
Pavlović R.
2007-01-01
Full Text Available To apply the theorem of Nekhoroshev (1977 to asteroids, one first has to check whether a necessary geometrical condition is fulfilled: either convexity, or quasi-convexity, or only a 3-jet non-degeneracy. This requires computation of the derivatives of the integrable part of the corresponding Hamiltonian up to the third order over actions and a thorough analysis of their properties. In this paper we describe in detail the procedure of derivation and we give explicit expressions for the obtained derivatives. .
Integrated assignment and path planning
Murphey, Robert A.
2005-11-01
A surge of interest in unmanned systems has exposed many new and challenging research problems across many fields of engineering and mathematics. These systems have the potential of transforming our society by replacing dangerous and dirty jobs with networks of moving machines. This vision is fundamentally separate from the modern view of robotics in that sophisticated behavior is realizable not by increasing individual vehicle complexity, but instead through collaborative teaming that relies on collective perception, abstraction, decision making, and manipulation. Obvious examples where collective robotics will make an impact include planetary exploration, space structure assembly, remote and undersea mining, hazardous material handling and clean-up, and search and rescue. Nonetheless, the phenomenon driving this technology trend is the increasing reliance of the US military on unmanned vehicles, specifically, aircraft. Only a few years ago, following years of resistance to the use of unmanned systems, the military and civilian leadership in the United States reversed itself and have recently demonstrated surprisingly broad acceptance of increasingly pervasive use of unmanned platforms in defense surveillance, and even attack. However, as rapidly as unmanned systems have gained acceptance, the defense research community has discovered the technical pitfalls that lie ahead, especially for operating collective groups of unmanned platforms. A great deal of talent and energy has been devoted to solving these technical problems, which tend to fall into two categories: resource allocation of vehicles to objectives, and path planning of vehicle trajectories. An extensive amount of research has been conducted in each direction, yet, surprisingly, very little work has considered the integrated problem of assignment and path planning. This dissertation presents a framework for studying integrated assignment and path planning and then moves on to suggest an exact
Continuous-Discrete Path Integral Filtering
Bhashyam Balaji
2009-08-01
Full Text Available A summary of the relationship between the Langevin equation, Fokker-Planck-Kolmogorov forward equation (FPKfe and the Feynman path integral descriptions of stochastic processes relevant for the solution of the continuous-discrete filtering problem is provided in this paper. The practical utility of the path integral formula is demonstrated via some nontrivial examples. Specifically, it is shown that the simplest approximation of the path integral formula for the fundamental solution of the FPKfe can be applied to solve nonlinear continuous-discrete filtering problems quite accurately. The Dirac-Feynman path integral filtering algorithm is quite simple, and is suitable for real-time implementation.
Techniques and applications of path integration
Schulman, L S
2005-01-01
A book of techniques and applications, this text defines the path integral and illustrates its uses by example. It is suitable for advanced undergraduates and graduate students in physics; its sole prerequisite is a first course in quantum mechanics. For applications requiring specialized knowledge, the author supplies background material.The first part of the book develops the techniques of path integration. Topics include probability amplitudes for paths and the correspondence limit for the path integral; vector potentials; the Ito integral and gauge transformations; free particle and quadra
STOCHASTIC HOPF BIFURCATION IN QUASI-INTEGRABLE-HAMILTONIAN SYSTEMS
GAN Chunbiao
2004-01-01
A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrableHamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.
Critical Review of Path Integral Formulation
Fujita, Takehisa
2008-01-01
The path integral formulation in quantum mechanics corresponds to the first quantization since it is just to rewrite the quantum mechanical amplitude into many dimensional integrations over discretized coordinates $x_n$. However, the path integral expression cannot be connected to the dynamics of classical mechanics, even though, superficially, there is some similarity between them. Further, the field theory path integral in terms of many dimensional integrations over fields does not correspond to the field quantization. We clarify the essential difference between Feynman's original formulation of path integral in QED and the modern version of the path integral method prevailing in lattice field theory calculations, and show that the former can make a correct second quantization while the latter cannot quantize fields at all and its physical meaning is unknown.
The Path Integral Quantization corresponding to the Deformed Heisenberg Algebra
Pramanik, Souvik; Moussa, Mohamed; Ali, Ahmed Farag
2014-01-01
In this paper, we analyze a deformation of the Heisenberg algebra consistent with both the generalized uncertainty principle and doubly special relativity. We observe that this algebra can give rise to fractional derivatives terms in the corresponding quantum mechanical Hamiltonian. However, a formal meaning can be given to such fractional derivative terms, using the theory of harmonic extensions of functions. Thus we obtain the expression of the propagator of path integral corresponding to this deformed Heisenberg algebra. In fact, we explicitly evaluate this expression for a free particle in one dimension and check its consistency.
A mollified numerical integrator of ring polymer Hamiltonian dynamics with constraints
Xiong, Yunfeng
2014-01-01
In this paper, a symplectic and time-reversible integrator is proposed of simulating the Hamiltonian dynamics with constraints in path integral molecular dynamics. The constraints are tackled by Matrix Inverted Linearized Constraint algorithm (MILC), while a slight modification is requested under normal mode representation, and the slow potential is mollified by Equilibrium method (Equilibrium MOLLY) to ameliorate the numerical resonance. It is demonstrated that the slow force impulse can be evaluated only at the centroid of beads, instead of being evaluated at the positions of each bead independently. Therefore, it not only allows longer time step but also reduces the complexity of computation. The numerical experiment is performed using SPC/E model in 298K with eight beads. Further discussion will involve the application of Equilibrium MOLLY in flexible bond model.
White Noise Path Integrals in Stochastic Neurodynamics
Carpio-Bernido, M. Victoria; Bernido, Christopher C.
2008-06-01
The white noise path integral approach is used in stochastic modeling of neural activity, where the primary dynamical variables are the relative membrane potentials, while information on transmembrane ionic currents is contained in the drift coefficient. The white noise path integral allows a natural framework and can be evaluated explicitly to yield a closed form for the conditional probability density.
Efficient fourth order symplectic integrators for near-harmonic separable Hamiltonian systems
Nielsen, Kristian Mads Egeris
2015-01-01
Efficient fourth order symplectic integrators are proposed for numerical integration of separable Hamiltonian systems H(p,q)=T(p)+V(q). Symmetric splitting coefficients with five to nine stages are obtained by higher order decomposition of the simple harmonic oscillator. The performance of the methods is evaluated for various Hamiltonian systems: Integration errors are compared to those of acclaimed integrators composed by S. Blanes et al. (2013), W. Kahan et al. (1999) and H. Yoshida (1990). Numerical tests indicate that the integrators obtained in this paper perform significantly better than previous integrators for common Hamiltonian systems.
Completely Integrable Hamiltonian Systems Generated by Poisson Structures in R3
LEI De-Chao; ZHANG Xiang
2005-01-01
@@ The completely integrable Hamiltonian systems have been applied to physics and mechanics intensively. We generate a family of completely integrable Hamiltonian systems from some kinds of exact Poisson structures in R3 by the realization of the Poisson algebra. Moreover, we prove that there is a Poisson algebra which cannot be realized by an exact Poisson structure.
Action-angle coordinates for time-dependent completely integrable Hamiltonian systems
Giachetta, Giovanni; Mangiarotti, Luigi [Department of Mathematics and Physics, University of Camerino, Camerino (Italy)]. E-mails: giovanni.giachetta@unicam.it; luigi.mangiarotti@unicam.it; Sardanashvily, Gennadi [Department of Theoretical Physics, Physics Faculty, Moscow State University, Moscow (Russian Federation)]. E-mail: sard@grav.phys.msu.su
2002-07-26
A time-dependent completely integrable Hamiltonian system is proved to admit the action-angle coordinates around any instantly compact regular invariant manifold. Written relative to these coordinates, its Hamiltonian and first integrals are functions only of action coordinates. (author). Letter-to-the-editor.
Abedi-Fardad, J., E-mail: j.abedifardad@bonabu.ac.ir [Department of Mathematics, Bonab University, Tabriz (Iran, Islamic Republic of); Rezaei-Aghdam, A., E-mail: rezaei-a@azaruniv.edu [Department of Physics, Azarbaijan Shahid Madani University, 53714-161 Tabriz (Iran, Islamic Republic of); Haghighatdoost, Gh., E-mail: gorbanali@azaruniv.edu [Department of Mathematics, Bonab University, Tabriz (Iran, Islamic Republic of); Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz (Iran, Islamic Republic of)
2014-05-15
We construct integrable and superintegrable Hamiltonian systems using the realizations of four dimensional real Lie algebras as a symmetry of the system with the phase space R{sup 4} and R{sup 6}. Furthermore, we construct some integrable and superintegrable Hamiltonian systems for which the symmetry Lie group is also the phase space of the system.
Square-root actions, metric signature, and the path-integral of quantum gravity
Carlini, A; Carlini, A; Greensite, J
1995-01-01
We consider quantization of the Baierlein-Sharp-Wheeler form of the gravitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argue that path-integral quantization of the gravitational action should be based on a path integrand \\exp[ \\sqrt{i} S ] rather than the familiar Feynman expression \\exp[ i S ], and that unitarity requires integration over manifolds of both Euclidean and Lorentzian signature. We discuss the relation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions.
Square-root actions, metric signature, and the path integral of quantum gravity
Carlini, A.; Greensite, J.
1995-12-01
We consider quantization of the Baierlein-Sharp-Wheeler form of the gravitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argue that path-integral quantization of the gravitational action should be based on a path integrand exp[ √i S] rather than the familiar Feynman expression exp[iS], and that unitarity requires integration over manifolds of both Euclidean and Lorentzian signature. We discuss the relation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions.
VON NEUMANN STABILITY ANALYSIS OF SYMPLECTIC INTEGRATORS APPLIED TO HAMILTONIAN PDEs
Helen M. Regan
2002-01-01
Symplectic integration of separable Hamiltonian ordinary and partial differential equations is discussed. Avon Neumann analysis is performed to achieve general linear stability criteria for symplectic methods applied to a restricted class of Hamiltonian PDEs. In this treatment, the symplectic step is performed prior to the spatial step, as opposed to the standard approach of spatially discretising the PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied. In this way stability criteria are achieved by considering the spectra of linearised Hamiltonian PDEs rather thanspatial step size.
Coherent-state path integrals in the continuum: The SU(2) case
Kordas, G.; Kalantzis, D.; Karanikas, A. I.
2016-09-01
We define the time-continuous spin coherent-state path integral in a way that is free from inconsistencies. The proposed definition is used to reproduce known exact results. Such a formalism opens new possibilities for applying approximations with improved accuracy and can be proven useful in a great variety of problems where spin Hamiltonians are used.
Langevin equation path integral ground state.
Constable, Steve; Schmidt, Matthew; Ing, Christopher; Zeng, Tao; Roy, Pierre-Nicholas
2013-08-15
We propose a Langevin equation path integral ground state (LePIGS) approach for the calculation of ground state (zero temperature) properties of molecular systems. The approach is based on a modification of the finite temperature path integral Langevin equation (PILE) method (J. Chem. Phys. 2010, 133, 124104) to the case of open Feynman paths. Such open paths are necessary for a ground state formulation. We illustrate the applicability of the method using model systems and the weakly bound water-parahydrogen dimer. We show that the method can lead to converged zero point energies and structural properties.
Feynman Path Integrals Over Entangled States
Green, A G; Keeling, J; Simon, S H
2016-01-01
The saddle points of a conventional Feynman path integral are not entangled, since they comprise a sequence of classical field configurations. We combine insights from field theory and tensor networks by constructing a Feynman path integral over a sequence of matrix product states. The paths that dominate this path integral include some degree of entanglement. This new feature allows several insights and applications: i. A Ginzburg-Landau description of deconfined phase transitions. ii. The emergence of new classical collective variables in states that are not adiabatically continuous with product states. iii. Features that are captured in product-state field theories by proliferation of instantons are encoded in perturbative fluctuations about entangled saddles. We develop a general formalism for such path integrals and a couple of simple examples to illustrate their utility.
Path integral representations on the complex sphere
Grosche, C. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik
2007-08-15
In this paper we discuss the path integral representations for the coordinate systems on the complex sphere S{sub 3C}. The Schroedinger equation, respectively the path integral, separates in exactly 21 orthogonal coordinate systems. We enumerate these coordinate systems and we are able to present the path integral representations explicitly in the majority of the cases. In each solution the expansion into the wave-functions is stated. Also, the kernel and the corresponding Green function can be stated in closed form in terms of the invariant distance on the sphere, respectively on the hyperboloid. (orig.)
Rigorous time slicing approach to Feynman path integrals
Fujiwara, Daisuke
2017-01-01
This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schrödinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives of order equal to or higher than two are bounded. The semi-classical asymptotic formula up to the second term of the fundamental solution is also proved by a method different from that of Birkhoff. A bound of the remainder term is also proved. The Feynman path integral is a method of quantization using the Lagrangian function, whereas Schrödinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger's method, there is still much to be done concerning rigorous mathematical treatment of Feynman's method. Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces which is obtained by...
Path integration in relativistic quantum mechanics
Redmount, I H; Redmount, Ian H.; Suen, Wai-Mo
1993-01-01
The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories. The exact quantum-mechanical propagator is calculated here for a particle described by the simple relativistic action proportional to its proper time. This propagator is nonvanishing outside the light cone, implying that spacelike trajectories must be included in the path integral. The propagator matches the WKB approximation to the corresponding configuration-space path integral far from the light cone; outside the light cone that approximation consists of the contribution from a single spacelike geodesic. This propagator also has the unusual property that its short-time limit does not coincide with the WKB approximation, making the construction of a concrete skeletonized version of the path integral more complicated than in nonrelativistic theory.
Fajun Yu
2014-01-01
Full Text Available We generate complex integrable couplings from zero curvature equations associated with matrix spectral problems in this paper. A direct application to the WKI spectral problem leads to a novel soliton equation hierarchy of integrable coupling system; then we consider the Hamiltonian structure of the integrable coupling system. We select the U¯, V¯ and generate the nonlinear composite parts, which generate new extended WKI integrable couplings. It is also indicated that the method of block matrix is an efficient and straightforward way to construct the integrable coupling system.
Mori, Toshifumi; Hamers, Robert J; Pedersen, Joel A; Cui, Qiang
2014-07-17
Motivated by specific applications and the recent work of Gao and co-workers on integrated tempering sampling (ITS), we have developed a novel sampling approach referred to as integrated Hamiltonian sampling (IHS). IHS is straightforward to implement and complementary to existing methods for free energy simulation and enhanced configurational sampling. The method carries out sampling using an effective Hamiltonian constructed by integrating the Boltzmann distributions of a series of Hamiltonians. By judiciously selecting the weights of the different Hamiltonians, one achieves rapid transitions among the energy landscapes that underlie different Hamiltonians and therefore an efficient sampling of important regions of the conformational space. Along this line, IHS shares similar motivations as the enveloping distribution sampling (EDS) approach of van Gunsteren and co-workers, although the ways that distributions of different Hamiltonians are integrated are rather different in IHS and EDS. Specifically, we report efficient ways for determining the weights using a combination of histogram flattening and weighted histogram analysis approaches, which make it straightforward to include many end-state and intermediate Hamiltonians in IHS so as to enhance its flexibility. Using several relatively simple condensed phase examples, we illustrate the implementation and application of IHS as well as potential developments for the near future. The relation of IHS to several related sampling methods such as Hamiltonian replica exchange molecular dynamics and λ-dynamics is also briefly discussed.
Dressed coordinates: The path-integral approach
Casana, R.; Flores-Hidalgo, G.; Pimentel, B. M.
2007-02-01
The recently introduced dressed coordinates are studied in the path-integral approach. These coordinates are defined in the context of a harmonic oscillator linearly coupled to massless scalar field and it is shown that in this model the dressed coordinates appear as a coordinate transformation preserving the path-integral functional measure. The analysis also generalizes the sum rules established in a previous work.
Anomalous paths in quantum mechanical path-integrals
Grimsmo, Arne L., E-mail: arne.grimsmo@ntnu.no [Department of Physics, The Norwegian University of Science and Technology, N-7491 Trondheim (Norway); Department of Physics, The University of Auckland, Private Bag 92019, Auckland (New Zealand); Klauder, John R., E-mail: klauder@phys.ufl.edu [Departments of Physics and Mathematics, University of Florida, Gainesville, FL 32611 (United States); Skagerstam, Bo-Sture K., E-mail: bo-sture.skagerstam@ntnu.no [Department of Physics, The Norwegian University of Science and Technology, N-7491 Trondheim (Norway); Kavli Institute for Theoretical Physics, Kohn Hall, University of California at Santa Barbara, CA 93106-4030 (United States); CREOL, The College of Optics and Photonics at the University of Central Florida, 4000 Central Florida Boulevard, Orlando, FL 32816 (United States)
2013-11-25
We investigate modifications of the discrete-time lattice action, for a quantum mechanical particle in one spatial dimension, that vanish in the naïve continuum limit but which, nevertheless, induce non-trivial effects due to quantum fluctuations. These effects are seen to modify the geometry of the paths contributing to the path-integral describing the time evolution of the particle, which we investigate through numerical simulations. In particular, we demonstrate the existence of a modified lattice action resulting in paths with any fractal dimension, d{sub f}, between one and two. We argue that d{sub f}=2 is a critical value, and we exhibit a type of lattice modification where the fluctuations in the position of the particle becomes independent of the time step, in which case the paths are interpreted as superdiffusive Lévy flights. We also consider the jaggedness of the paths, and show that this gives an independent classification of lattice theories.
Matrix factorization method for the Hamiltonian structure of integrable systems
S Ghosh; B Talukdar; S Chakraborti
2003-07-01
We demonstrate that the process of matrix factorization provides a systematic mathematical method to investigate the Hamiltonian structure of non-linear evolution equations characterized by hereditary operators with Nijenhuis property.
张玉峰
2003-01-01
A subalgebra of loop algebra A2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions,a formalism similar to the well-known Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and a generalized standard form of the Schrodinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra A2 into loop algebra A1. Furthermore,a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.
Local-time representation of path integrals.
Jizba, Petr; Zatloukal, Václav
2015-12-01
We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization of the local-time representation that includes arbitrary functionals of the local time is also provided. We argue that the results obtained represent a powerful alternative to the traditional Feynman-Kac formula, particularly in the high- and low-temperature regimes. To illustrate this point, we apply our local-time representation to analyze the asymptotic behavior of the Bloch density matrix at low temperatures. Further salient issues, such as connections with the Sturm-Liouville theory and the Rayleigh-Ritz variational principle, are also discussed.
无
2010-01-01
The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method.First,a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay.Then,stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system,and after that the Lyapunov function of the averaged It? equation is taken as the optimal linear combination of the corresponding independent first integrals in involution.Finally,the stability of the system is determined by using the largest eigenvalue of the linearized system.Two examples are used to illustrate the proposed procedure and the effects of delayed time on the Lyapunov stability are discussed as well.
Multi-hamiltonian formulation for a class of degenerate completely integrable systems
Bueken, P
1994-01-01
: Generalizing a construction of P. Vanhaecke, we introduce a large class of degenerate (i.e., associated to a degenerate Poisson bracket) completely integrable systems on (a dense subset of) the space \\R^{2d+n+1}, called the generalized master systems. It turns out that certain generalized master systems (with different Poisson brackets and different Hamiltonians) determine the same Hamiltonian vector fields (and are therefore different descriptions of the same Hamiltonian system), and that the Poisson brackets of these systems are compatible. Consequently, our class of generalized master systems actually consists of a (smaller) class of completely integrable systems, and our construction yields a multi-Hamiltonian structure for these systems. As an application, we construct a multi-Hamiltonian structure for the so-called master systems introduced by D. Mumford.
张素英; 邓子辰
2004-01-01
For the constrained generalized Hamiltonian system with dissipation, by introducing Lagrange multiplier and using projection technique, the Lie group integration method was presented, which can preserve the inherent structure of dynamic system and the constraint-invariant. Firstly, the constrained generalized Hamiltonian system with dissipative was converted to the non-constraint generalized Hamiltonian system, then Lie group integration algorithm for the non-constraint generalized Hamiltonian system was discussed, finally the projection method for generalized Hamiltonian system with constraint was given. It is found that the constraint invariant is ensured by projection technique, and after introducing Lagrange multiplier the Lie group character of the dynamic system can't be destroyed while projecting to the constraint manifold. The discussion is restricted to the case of holonomic constraint. A presented numerical example shows the effectiveness of the method.
Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations
Weinstein, Marvin; /SLAC
2009-02-12
It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I begin with an incredibly easy way to derive the time evolution of a Gaussian wave-packet for the case free and harmonic motion without any need to know the eigenstates of the Hamiltonian. This discussion is completely analytic and I will later use it to relate the solution for the behavior of the Gaussian packet to the Feynman path-integral and stationary phase approximation. It will be clear that using the information about the evolution of the Gaussian in this way goes far beyond what the stationary phase approximation tells us. Next, I introduce the concept of the bucket brigade approach to dealing with problems that cannot be handled totally analytically. This approach combines the intuition obtained in the initial discussion, as well as the intuition obtained from the path-integral, with simple numerical tools. My goal is to show that, for any specific process, there is a simple Hilbert space interpretation of the stationary phase approximation. I will then argue that, from the point of view of numerical approximations, the trajectory obtained from my generalization of the stationary phase approximation specifies that subspace of the full Hilbert space that is needed to compute the time evolution of the particular state under the full Hamiltonian. The prescription I will give is totally non-perturbative and we will see, by the grace of Maple animations computed for the case of the anharmonic oscillator Hamiltonian, that this approach allows surprisingly accurate computations to be performed with very little work. I think of this approach to the path-integral as defining what I call a guided numerical approximation scheme. After the discussion of the anharmonic oscillator I will
Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R+4
Llibre, Jaume; Xiao, Dongmei
2017-02-01
In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R+4 are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka-Volterra Hamiltonian systems in R+4. As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R+4 defined by the hypersurfaces axy + bzw + cx2 y + dxy2 + ez2 w + fzw2 = h, where a , b , c , d , e , f , w and h are real constants.
Path integral distance for data interpretation
Volchenkov, D
2015-01-01
The process of data interpretation is always based on the implicit introduction of equivalence relations on the set of walks over the database. Every equivalence relation on the set of walks specifies a Markov chain describing the transitions of a discrete time random walk. In order to geometrize and interpret the data, we propose the new distance between data units defined as a "Feynman path integral", in which all possible paths between any two nodes in a graph model of the data are taken into account, although some paths are more preferable than others. Such a path integral distance approach to the analysis of databases has proven its efficiency and success, especially on multivariate strongly correlated data where other methods fail to detect structural components (urban planning, historical language phylogenies, music, street fashion traits analysis, etc. ). We believe that it would become an invaluable tool for the intelligent complexity reduction and big data interpretation.
Noncommutative integrability, paths and quasi-determinants
Di Francesco, Philippe
2010-01-01
In previous work, we showed that the solution of certain systems of discrete integrable equations, notably $Q$ and $T$-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras [BZ], the Kontsevich evolution [DFK09b] and the $T$-systems themselves [DFK09a]. In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positiv...
The Liouville integrable coupling system of the m-AKNS hierarchy and its Hamiltonian structure
Yue Chao; Yang Geng-Wen; Xu Yue-Cai
2007-01-01
In this paper a type of 9-dimensional vector loop algebra (F) is constructed,which is devoted to establish an isospectral problem.It follows that a Liouville integrable coupling system of the m-AKNS hierarchy is obtained by employing the Tu scheme,whose Hamiltonian structure is worked out by making use of constructed quadratic identity.The method given in the paper can be used to obtain many other integrable couplings and their Hamiltonian structures.
ANALYSIS OF LIMIT CYCLES TO A PERTURBED INTEGRABLE NON-HAMILTONIAN SYSTEM
无
2012-01-01
Bifurcation of limit cycles to a perturbed integrable non-Hamiltonian system is investigated using both qualitative analysis and numerical exploration.The investigation is based on detection functions which are particularly effective for the perturbed integrable non-Hamiltonian system.The study reveals that the system has 3 limit cycles.By the method of numerical simulation,the distributed orderliness of the 3 limitcycles is observed,and their nicety places are determined.The study also indicates that each ...
Path Integral Techniques in Conformal Field Theory
Van Tonder, A J
2004-01-01
We present the theory of a two-dimensional conformal scalar field using path integral techniques. We derive the conformal anomaly using an adaptation of the method of Fujikawa, and compare the result with a derivation based on a Pauli-Villars measure, where the anomaly is shifted from the path integral measure to the energy-momentum trace. In the path integral approach the energy-momentum is a true coordinate-invariant tensor quantity, and we explain how it is related to the corresponding non-tensor object arising in the operator approach, obtaining an intuitive explanation within the context of the path integral approach for the anomalous transformation law and anomalous Ward identities of the latter. After carefully calculating nontrivial contact terms arising in certain energy-momentum products, we use these to provide a simple consistency check confirming the change of variables formula for the path integral and to review the relationship between the conformal anomaly and the energy-momentum two-point fun...
Nonlinear Super Integrable Couplings of Super Dirac Hierarchy and Its Super Hamiltonian Structures
尤福财
2012-01-01
We construct nonlinear super integrable couplings of the super integrable Dirac hierarchy based on an enlarged matrix Lie superalgebra. Then its super Hamiltonian structure is furnished by super trace identity. As its reduction, we gain the nonlinear integrable couplings of the classical integrable Dirac hierarchy.
STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS~
DengMaolin; HongMingchao; ZhuWeiqiu
2003-01-01
A strategy is proposed based on the stochastic averaging method for quasi nonintegrable Hamiltonian systems and the stochastic dynamical programming principle. The proposed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation. By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional averaged Ito stochastic differential equation. By using the stochastic dynamical programming principle the dynamical programming equation for minimizing the response of the system is formulated.The optimal control law is derived from the dynamical programming equation and the bounded control constraints. The response of optimally controlled systems is predicted through solving the FPK equation associated with It5 stochastic differential equation. An example is worked out in detail to illustrate the application of the control strategy proposed.
Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems
无
2010-01-01
The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. The response of the controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation and the Riccati equation for the estimated error of system states. As an example to illustrate the procedure and effectiveness of the proposed method, the stochastic optimal control problem of a partially observable two-degree-of-freedom quasi-integrable Hamiltonian system is worked out in detail.
Path integral quantization corresponding to the deformed Heisenberg algebra
Pramanik, Souvik, E-mail: souvick.in@gmail.com [Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108 (India); Moussa, Mohamed, E-mail: mohamed.ibrahim@fsc.bu.edu.eg [Department of Physics, Faculty of Sciences, Benha University, Benha 13518 (Egypt); Faizal, Mir, E-mail: f2mir@uwaterloo.ca [Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1 (Canada); Ali, Ahmed Farag, E-mail: ahmed.ali@fsc.bu.edu.eg [Department of Physics, Faculty of Sciences, Benha University, Benha 13518 (Egypt)
2015-11-15
In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity show very much different result.
Classical and quantum dynamics from classical paths to path integrals
Dittrich, Walter
2016-01-01
Graduate students who want to become familiar with advanced computational strategies in classical and quantum dynamics will find here both the fundamentals of a standard course and a detailed treatment of the time-dependent oscillator, Chern-Simons mechanics, the Maslov anomaly and the Berry phase, to name a few. Well-chosen and detailed examples illustrate the perturbation theory, canonical transformations, the action principle and demonstrate the usage of path integrals. This new edition has been revised and enlarged with chapters on quantum electrodynamics, high energy physics, Green’s functions and strong interaction.
Field theory a path integral approach
Das, Ashok
2006-01-01
This unique book describes quantum field theory completely within the context of path integrals. With its utility in a variety of fields in physics, the subject matter is primarily developed within the context of quantum mechanics before going into specialized areas.Adding new material keenly requested by readers, this second edition is an important expansion of the popular first edition. Two extra chapters cover path integral quantization of gauge theories and anomalies, and a new section extends the supersymmetry chapter, where singular potentials in supersymmetric systems are described.
High-order Path Integral Monte Carlo methods for solving quantum dot problems
Chin, Siu A
2014-01-01
The conventional second-order Path Integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of anti-symmetric free fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work, we show that optimized fourth-order Path Integral Monte Carlo methods, which use no more than 5 free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.
Current Algebra in the Path Integral framework
Cardenas, V H; Saavedra, J
1998-01-01
In this letter we describe an approach to the current algebra based in the Path Integral formalism. We use this method for abelian and non-abelian quantum field theories in 1+1 and 2+1 dimensions and the correct expressions are obtained. Our results show the independence of the regularization of the current algebras.
Path-integral molecular dynamics simulation of diamond
Ramírez, Rafael; Herrero, Carlos P.; Hernández, Eduardo R.
2006-06-01
Diamond is studied by path-integral molecular dynamics simulations of the atomic nuclei in combination with a tight-binding Hamiltonian to describe its electronic structure and total energy. This approach allows us to quantify the influence of quantum zero-point vibrations and finite temperatures on both the electronic and vibrational properties of diamond. The electron-phonon coupling mediated by the zero-point vibration reduces the direct electronic gap of diamond by 10%. The calculated decrease of the direct gap with temperature shows good agreement with the experimental data available up to 700K . Anharmonic vibrational frequencies of the crystal have been obtained from a linear-response approach based on the path integral formalism. In particular, the temperature dependence of the zone-center optical phonon has been derived from the simulations. The anharmonicity of the interatomic potential produces a red shift of this phonon frequency. At temperatures above 500K , this shift is overestimated in comparison to available experimental data. The predicted temperature shift of the elastic constant c44 displays reasonable agreement with the available experimental results.
Bead-Fourier path integral molecular dynamics
Ivanov, Sergei D.; Lyubartsev, Alexander P.; Laaksonen, Aatto
2003-06-01
Molecular dynamics formulation of Bead-Fourier path integral method for simulation of quantum systems at finite temperatures is presented. Within this scheme, both the bead coordinates and Fourier coefficients, defining the path representing the quantum particle, are treated as generalized coordinates with corresponding generalized momenta and masses. Introduction of the Fourier harmonics together with the center-of-mass thermostating scheme is shown to remove the ergodicity problem, known to pose serious difficulties in standard path integral molecular dynamics simulations. The method is tested for quantum harmonic oscillator and hydrogen atom (Coulombic potential). The simulation results are compared with the exact analytical solutions available for both these systems. Convergence of the results with respect to the number of beads and Fourier harmonics is analyzed. It was shown that addition of a few Fourier harmonics already improves the simulation results substantially, even for a relatively small number of beads. The proposed Bead-Fourier path integral molecular dynamics is a reliable and efficient alternative to simulations of quantum systems.
Path integration in tactile perception of shapes.
Moscatelli, Alessandro; Naceri, Abdeldjallil; Ernst, Marc O
2014-11-01
Whenever we move the hand across a surface, tactile signals provide information about the relative velocity between the skin and the surface. If the system were able to integrate the tactile velocity information over time, cutaneous touch may provide an estimate of the relative displacement between the hand and the surface. Here, we asked whether humans are able to form a reliable representation of the motion path from tactile cues only, integrating motion information over time. In order to address this issue, we conducted three experiments using tactile motion and asked participants (1) to estimate the length of a simulated triangle, (2) to reproduce the shape of a simulated triangular path, and (3) to estimate the angle between two-line segments. Participants were able to accurately indicate the length of the path, whereas the perceived direction was affected by a direction bias (inward bias). The response pattern was thus qualitatively similar to the ones reported in classical path integration studies involving locomotion. However, we explain the directional biases as the result of a tactile motion aftereffect.
Path Integral Bosonization of Massive GNO Fermions
Park, Q H
1997-01-01
We show the quantum equivalence between certain symmetric space sine-Gordon models and the massive free fermions. In the massless limit, these fermions reduce to the free fermions introduced by Goddard, Nahm and Olive (GNO) in association with symmetric spaces $K/G$. A path integral formulation is given in terms of the Wess-Zumino-Witten action where the field variable $g$ takes value in the orthogonal, unitary, and symplectic representations of the group $G$ in the basis of the symmetric space. We show that, for example, such a path integral bosonization is possible when the symmetric spaces $K/G$ are $SU(N) the relation between massive GNO fermions and the nonabelian solitons, and explain the restriction imposed on the fermion mass matrix due to the integrability of the bosonic model.
Classical and quantum dynamics from classical paths to path integrals
Dittrich, Walter
2017-01-01
Graduate students who wish to become familiar with advanced computational strategies in classical and quantum dynamics will find in this book both the fundamentals of a standard course and a detailed treatment of the time-dependent oscillator, Chern-Simons mechanics, the Maslov anomaly and the Berry phase, to name just a few topics. Well-chosen and detailed examples illustrate perturbation theory, canonical transformations and the action principle, and demonstrate the usage of path integrals. The fifth edition has been revised and enlarged to include chapters on quantum electrodynamics, in particular, Schwinger’s proper time method and the treatment of classical and quantum mechanics with Lie brackets and pseudocanonical transformations. It is shown that operator quantum electrodynamics can be equivalently described with c-numbers, as demonstrated by calculating the propagation function for an electron in a prescribed classical electromagnetic field.
A family of affine quantum group invariant integrable extensions of the Hubbard Hamiltonian
Avakyan, A. [Erevanskij Fizicheskij Inst., Erevan (Armenia); Hakobyan, T. [Erevanskij Fizicheskij Inst., Erevan (Armenia); Sedrakyan, A. [Erevanskij Fizicheskij Inst., Erevan (Armenia)
1997-04-21
We construct a family of spin chain Hamiltonians, which have the affine quantum group symmetry U{sub q}g. Their eigenvalues coincide with the eigenvalues of the usual spin chain Hamiltonians, but have the degeneracy of levels, corresponding to the affine U{sub q}g. The space of states of these spin chains is formed by the tensor product of the fully reducible representations of the quantum group. The fermionic representations of the constructed spin chain Hamiltonians show that we have obtained new extensions of the Hubbard Hamiltonians. All of them are integrable and have the affine quantum group symmetry. The exact ground state of such type of model is presented, exhibiting superconducting behavior via the {eta}-pairing mechanism. (orig.).
Modification of logarithmic Hamiltonians and application of explicit symplectic-like integrators
Li, Dan; Wu, Xin
2017-08-01
We modify the logarithmic Hamiltonian of Mikkola and Tanikawa by adding a constant (or function) to both the kinetic energy and the force function. Explicit symplectic algorithms are available when the logarithmic Hamiltonian has two separable parts of coordinates and momenta. However, they are not if the logarithmic Hamiltonian is inseparable. Fortunately, they are still efficient by manipulating the logarithmic Hamiltonian as a new separable Hamiltonian in an extended phase space. In fact, they belong to symplectic-like integrators. The choice of mixing maps affects the performance of the considered symplectic-like integrators. It is shown that two maps about sequent permutations of coordinates and momenta are inferior to a map with mid-point permutations in some cases. The choice of the constant (or function) added also exerts some influence on the performance of the algorithms. As a result, with the help of the mid-point permutations and a suitable choice for the constant (or function) included, the logarithmic Hamiltonian methods bring an increase in accuracy compared to the non-logarithmic ones, particularly for highly eccentric orbits.
Xuncheng Huang; Guizhang Tu
2006-01-01
The Hamiltonian equation provides us an alternate description of the basic physical laws of motion, which is used to be described by Newton's law. The research on Hamiltonian integrable systems is one of the most important topics in the theory of solitons. This article proposes a new hierarchy of integrable systems of 1+2 dimensions with its Hamiltonian form by following the residue approach of Fokas and Tu. The new hierarchy of integrable system is of fundamental intere...
Path integral quantization of parametrised field theory
Varadarajan, M
2004-01-01
Free scalar field theory on a flat spacetime can be cast into a generally covariant form known as parametrised field theory in which the action is a functional of the scalar field as well as the embedding variables which describe arbitrary, in general curved, foliations of the flat spacetime. We construct the path integral quantization of parametrised field theory in order to analyse issues at the interface of quantum field theory and general covariance in a path integral context. We show that the measure in the Lorentzian path integral is non-trivial and is the analog of the Fradkin- Vilkovisky measure for quantum gravity. We construct Euclidean functional integrals in the generally covariant setting of parametrised field theory using key ideas of Schleich and show that our constructions imply the existence of non-standard `Wick rotations' of the standard free scalar field 2 point function. We develop a framework to study the problem of time through computations of scalar field 2 point functions. We illustra...
An Introduction to Control of Chaos for Quasi-Integrable Hamiltonian Systems
Silva, Vilarbo da
2013-01-01
Quasi-integrable Hamiltonian systems are of great interest in many research fields of physics and mathematics. In these systems, the phase space has regular and chaotic trajectories. These trajectories depend in part on the magnitude of perturbation that breaks the integrability of the system. The value of the critical perturbation responsible for this transition is a key element in the control of chaos . In this paper, we explore a procedure for the control in quasi-integrable Hamiltonian systems via canonical map. Initially, we present the basic tools for this study: Hamiltonian map, linearization of the map and Chirikov criterion. Subsequently, we investigated the behavior of a wave-particle interaction front perturbation. Finally, we confront with a numerical analytical approach (iteration of the map) results, showing a good agreement.
Quantum gravitation the Feynman path integral approach
Hamber, Herbert W
2009-01-01
The book covers the theory of Quantum Gravitation from the point of view of Feynman path integrals. These provide a manifestly covariant approach in which fundamental quantum aspects of the theory such as radiative corrections and the renormalization group can be systematically and consistently addressed. The path integral method is suitable for both perturbative as well as non-perturbative studies, and is known to already provide a framework of choice for the theoretical investigation of non-abelian gauge theories, the basis for three of the four known fundamental forces in nature. The book thus provides a coherent outline of the present status of the theory gravity based on Feynman’s formulation, with an emphasis on quantitative results. Topics are organized in such a way that the correspondence to similar methods and results in modern gauge theories becomes apparent. Covariant perturbation theory are developed using the full machinery of Feynman rules, gauge fixing, background methods and ghosts. The ren...
Path integral measure for gravitational interactions
Kazuo Fujikawa
1983-10-01
Full Text Available It is pointed out that the path-integral variables as well as the local measure for gravitational interactions are uniquely specified if one imposes the anomaly-free condition on the Becchi-Rouet-Stora supersymmetry associated with general coordinate transformations. This prescription is briefly illustrated for the Einstein gravity and supergravity in four space-time dimensions and the relativistic string theory in two dimensions.
A Path Integral Approach To Noncommutative Superspace
Chepelev, I; Chepelev, Iouri; Ciocarlie, Calin
2003-01-01
A path integral formula for the associative star-product of two superfields is proposed. It is a generalization of the Kontsevich-Cattaneo-Felder's formula for the star-product of functions of bosonic coordinates. The associativity of the star-product imposes certain conditions on the background of our sigma model. For generic background the action is not supersymmetric. The supersymmetry invariance of the action constrains the background and leads to a simple formula for the star-product.
Path integrals for dimerized quantum spin systems
Foussats, Adriana, E-mail: afoussats@gmail.co [Facultad de Ciencias Exactas, Ingenieria y Agrimensura and Instituto de Fisica Rosario (UNR-CONICET), Av. Pellegrini 250, 2000 Rosario (Argentina); Greco, Andres [Facultad de Ciencias Exactas, Ingenieria y Agrimensura and Instituto de Fisica Rosario (UNR-CONICET), Av. Pellegrini 250, 2000 Rosario (Argentina); Muramatsu, Alejandro [Institut fuer Theoretische Physik III, Universitaet Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart (Germany)
2011-01-11
Dimerized quantum spin systems may appear under several circumstances, e.g. by a modulation of the antiferromagnetic exchange coupling in space, or in frustrated quantum antiferromagnets. In general, such systems display a quantum phase transition to a Neel state as a function of a suitable coupling constant. We present here two path-integral formulations appropriate for spin S=1/2 dimerized systems. The first one deals with a description of the dimers degrees of freedom in an SO(4) manifold, while the second one provides a path-integral for the bond-operators introduced by Sachdev and Bhatt. The path-integral quantization is performed using the Faddeev-Jackiw symplectic formalism for constrained systems, such that the measures and constraints that result from the algebra of the operators is provided in both cases. As an example we consider a spin-Peierls chain, and show how to arrive at the corresponding field-theory, starting with both an SO(4) formulation and bond-operators.
An integrable case of the p + ip pairing Hamiltonian interacting with its environment
Lukyanenko, Inna; Isaac, Phillip S.; Links, Jon
2016-02-01
We consider a generalization of the p + ip pairing Hamiltonian, with external interaction terms of a particular form. These terms allow for the exchange of particles between the system and its environment. As a result the {u}(1) symmetry associated with conservation of particle number, present in the p + ip Hamiltonian, is broken. Nonetheless the generalized model is integrable. We establish integrability using the boundary quantum inverse scattering method, with one of the reflection matrices chosen to be non-diagonal. We also derive the corresponding Bethe ansatz equations, the roots of which parametrize the exact solution for the energy spectrum.
An Energy-Work Relationship Integration Scheme for Nonconservative Hamiltonian Systems
Fu Jingli
2008-01-01
Full Text Available This letter focuses on studying a new energy-work relationship numerical integration scheme of nonconservative Hamiltonian systems. The signal-stage, multistage, and parallel composition numerical integration schemes are presented for this system. The high-order energy-work relation scheme of the system is constructed by a parallel connection of n multistage scheme of order 2 which its order of accuracy is 2n. The connection, which is discrete analog of usual case, between the change of energy and work of nonconservative force is obtained for nonconservative Hamiltonian systems.This letter also shows that the more the stages of the schemes are, the less the error rate of the scheme is for nonconservative Hamiltonian systems. Finally, an applied example is discussed to illustrate these results.
Purely geometric path integral for spin foams
Shirazi, Atousa Chaharsough
2013-01-01
Spin-foams are a proposal for defining the dynamics of loop quantum gravity via path integral. In order for a path integral to be at least formally equivalent to the corresponding canonical quantization, at each point in the space of histories it is important that the integrand have not only the correct phase -- a topic of recent focus in spin-foams -- but also the correct modulus, usually referred to as the measure factor. The correct measure factor descends from the Liouville measure on the reduced phase space, and its calculation is a task of canonical analysis. The covariant formulation of gravity from which spin-foams are derived is the Plebanski-Holst formulation, in which the basic variables are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integ...
Quantum Measurement and Extended Feynman Path Integral
文伟; 白彦魁
2012-01-01
Quantum measurement problem has existed many years and inspired a large of literature in both physics and philosophy, but there is still no conclusion and consensus on it. We show it can be subsumed into the quantum theory if we extend the Feynman path integral by considering the relativistic effect of Feynman paths. According to this extended theory, we deduce not only the Klein-Gordon equation, but also the wave-function-collapse equation. It is shown that the stochastic and instantaneous collapse of the quantum measurement is due to the ＂potential noise＂ of the apparatus or environment and ＂inner correlation＂ of wave function respectively. Therefore, the definite-status of the macroscopic matter is due to itself and this does not disobey the quantum mechanics. This work will give a new recognition for the measurement problem.
Two-Component Super AKNS Equations and Their Finite-Dimensional Integrable Super Hamiltonian System
Jing Yu; Jingwei Han
2014-01-01
Starting from a matrix Lie superalgebra, two-component super AKNS system is constructed. By making use of monononlinearization technique of Lax pairs, we find that the obtained two-component super AKNS system is a finite-dimensional integrable super Hamiltonian system. And its Lax representation and $r$ -matrix are also given in this paper.
Two-Component Super AKNS Equations and Their Finite-Dimensional Integrable Super Hamiltonian System
Jing Yu
2014-01-01
Full Text Available Starting from a matrix Lie superalgebra, two-component super AKNS system is constructed. By making use of monononlinearization technique of Lax pairs, we find that the obtained two-component super AKNS system is a finite-dimensional integrable super Hamiltonian system. And its Lax representation and r-matrix are also given in this paper.
Geometric quantization of completely integrable Hamiltonian systems in the action-angle variables
Giachetta, G; Sardanashvily, G
2002-01-01
We provide geometric quantization of a completely integrable Hamiltonian system in the action-angle variables around an invariant torus. These variables enable one to choose the angle polarization which otherwise is not easily determined. It is spanned by almost-Hamiltonian vector fields of angle variables. The associated quantum algebra consists of functions affine in action coordinates. We show that this algebra has a continuum set of nonequivalent representations in the separable pre-Hilbert space of smooth complex functions on the torus. The spectrum of the action operators is obtained.
Path integral for multi-field inflation
Gong, Jinn-Ouk; Shiu, Gary
2016-01-01
We develop the path integral formalism for studying cosmological perturbations in multi-field inflation, which is particularly well suited to study quantum theories with gauge symmetries such as diffeomorphism invariance. We formulate the gauge fixing conditions based on the Poisson brackets of the constraints, from which we derive two convenient gauges that are appropriate for multi-field inflation. We then adopt the in-in formalism to derive the most general expression for the power spectrum of the curvature perturbation including the corrections from the interactions of the curvature mode with other light degrees of freedom. We also discuss the contributions of the interactions to the bispectrum.
Path integral for multi-field inflation
Gong, Jinn-Ouk; Seo, Min-Seok; Shiu, Gary
2016-07-01
We develop the path integral formalism for studying cosmological perturbations in multi-field inflation, which is particularly well suited to study quantum theories with gauge symmetries such as diffeomorphism invariance. We formulate the gauge fixing conditions based on the Poisson brackets of the constraints, from which we derive two convenient gauges that are appropriate for multi-field inflation. We then adopt the in-in formalism to derive the most general expression for the power spectrum of the curvature perturbation including the corrections from the interactions of the curvature mode with other light degrees of freedom. We also discuss the contributions of the interactions to the bispectrum.
THE HAMILTONIAN STRUCTURE OF TWO INTEGRABLE EXPANDING MODELS
Liu Bin; Dong Huanhe; Li Zhu
2007-01-01
In this paper,an extended loop algebra is constructed from which an isospectral problem established.It follows that the integrable couplings of the Tu hierarchy and M-AKNS-KN hierarchy are obtained.and their Hamilton structures are presented by the quadratic-form identity.Moreover,we guarantee that the expanding model we obtained are also Liouville integrable.
Quantum-classical path integral with self-consistent solvent-driven reference propagators.
Banerjee, Tuseeta; Makri, Nancy
2013-10-24
Efficient procedures for evaluating the quantum-classical path integral (QCPI) [J. Chem. Phys. 2013, 137, 22A552] are described. The main idea is to identify a trajectory-specific reference Hamiltonian that captures the dominant effects of the classical "solvent" degrees of freedom on the dynamics of the quantum "system". This time-dependent reference is used to construct a system propagator that is valid for large time increments. Residual "quantum memory" interactions are included via the path integral representation of the density matrix, which converges with large time steps. Two physically motivated reference schemes are considered. The first involves the dynamics of the solvent unperturbed by the system, which forms the basis for the "classical path" approximation. The second is based on solvent trajectories determined self-consistently with the evolution of the system, according to the time-dependent self-consistent field or Ehrenfest model. Application to dissipative two-level systems indicates that both reference schemes allow a substantial increase of the path integral time step, leading to rapid convergence of the path sum. In addition, the time-dependent reference propagators automatically weigh state-to-state coupling against solvent reorganization in the determination of transition probabilities, further enhancing the convergence of the path integral.
Note on integrability of certain homogeneous Hamiltonian systems in 2D constant curvature spaces
Maciejewski, Andrzej J.; Szumiński, Wojciech; Przybylska, Maria
2017-02-01
We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous potential in flat spaces. Thanks to this property Hamilton equations admit, in a general case, a particular solution. Using this solution we derive necessary integrability conditions investigating differential Galois group of variational equations.
On integrability of some bi-Hamiltonian two field systems of partial differential equations
De Sole, Alberto; Kac, Victor G.; Turhan, Refik
2015-05-01
We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H0, H1), where H0 is a strongly skew-adjoint operator. This is applied to the construction of some new two field integrable systems of PDE by taking the pair (H0, H1) in the family of compatible Poisson structures that arose in the study of cohomology of moduli spaces of curves.
A 2+1-Dimensional Non-Isothermal Magnetogasdynamic System. Hamiltonian-Ermakov Integrable Reduction
Hongli An
2012-08-01
Full Text Available A 2+1-dimensional anisentropic magnetogasdynamic system with a polytropic gas law is shown to admit an integrable elliptic vortex reduction when γ=2 to a nonlinear dynamical subsystem with underlying integrable Hamiltonian-Ermakov structure. Exact solutions of the magnetogasdynamic system are thereby obtained which describe a rotating elliptic plasma cylinder. The semi-axes of the elliptical cross-section, remarkably, satisfy a Ermakov-Ray-Reid system.
High order path integrals made easy
Kapil, Venkat; Behler, Jörg; Ceriotti, Michele
2016-12-01
The precise description of quantum nuclear fluctuations in atomistic modelling is possible by employing path integral techniques, which involve a considerable computational overhead due to the need of simulating multiple replicas of the system. Many approaches have been suggested to reduce the required number of replicas. Among these, high-order factorizations of the Boltzmann operator are particularly attractive for high-precision and low-temperature scenarios. Unfortunately, to date, several technical challenges have prevented a widespread use of these approaches to study the nuclear quantum effects in condensed-phase systems. Here we introduce an inexpensive molecular dynamics scheme that overcomes these limitations, thus making it possible to exploit the improved convergence of high-order path integrals without having to sacrifice the stability, convenience, and flexibility of conventional second-order techniques. The capabilities of the method are demonstrated by simulations of liquid water and ice, as described by a neural-network potential fitted to the dispersion-corrected hybrid density functional theory calculations.
The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo
Hoffman, Matthew D
2011-01-01
Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior and sensitivity to correlated parameters that plague many MCMC methods by taking a series of steps informed by first-order gradient information. These features allow it to converge to high-dimensional target distributions much more quickly than simpler methods such as random walk Metropolis or Gibbs sampling. However, HMC's performance is highly sensitive to two user-specified parameters: a step size {\\epsilon} and a desired number of steps L. In particular, if L is too small then the algorithm exhibits undesirable random walk behavior, while if L is too large the algorithm wastes computation. We introduce the No-U-Turn Sampler (NUTS), an extension to HMC that eliminates the need to set a number of steps L. NUTS uses a recursive algorithm to build a set of likely candidate points that spans a wide swath of the target distribution, stopping automatically when it starts to double back and retrace it...
Macroscopic diffusive transport in a microscopically integrable Hamiltonian system.
Prosen, Tomaž; Zunkovič, Bojan
2013-07-26
We demonstrate that a completely integrable classical mechanical model, namely the lattice Landau-Lifshitz classical spin chain, supports diffusive spin transport with a finite diffusion constant in the easy-axis regime, while in the easy-plane regime, it displays ballistic transport in the absence of any known relevant local or quasilocal constant of motion in the symmetry sector of the spin current. This surprising finding should open the way towards analytical computation of diffusion constants for integrable interacting systems and hints on the existence of new quasilocal classical conservation laws beyond the standard soliton theory.
van Zon, Ramses; Hernández de la Peña, Lisandro; Peslherbe, Gilles H; Schofield, Jeremy
2008-10-01
In this paper, the imaginary-time path-integral representation of the canonical partition function of a quantum system and nonequilibrium work fluctuation relations are combined to yield methods for computing free-energy differences in quantum systems using nonequilibrium processes. The path-integral representation is isomorphic to the configurational partition function of a classical field theory, to which a natural but fictitious Hamiltonian dynamics is associated. It is shown that if this system is prepared in an equilibrium state, after which a control parameter in the fictitious Hamiltonian is changed in a finite time, then formally the Jarzynski nonequilibrium work relation and the Crooks fluctuation relation hold, where work is defined as the change in the energy as given by the fictitious Hamiltonian. Since the energy diverges for the classical field theory in canonical equilibrium, two regularization methods are introduced which limit the number of degrees of freedom to be finite. The numerical applicability of the methods is demonstrated for a quartic double-well potential with varying asymmetry. A general parameter-free smoothing procedure for the work distribution functions is useful in this context.
Building a cognitive map by assembling multiple path integration systems.
Wang, Ranxiao Frances
2016-06-01
Path integration and cognitive mapping are two of the most important mechanisms for navigation. Path integration is a primitive navigation system which computes a homing vector based on an animal's self-motion estimation, while cognitive map is an advanced spatial representation containing richer spatial information about the environment that is persistent and can be used to guide flexible navigation to multiple locations. Most theories of navigation conceptualize them as two distinctive, independent mechanisms, although the path integration system may provide useful information for the integration of cognitive maps. This paper demonstrates a fundamentally different scenario, where a cognitive map is constructed in three simple steps by assembling multiple path integrators and extending their basic features. The fact that a collection of path integration systems can be turned into a cognitive map suggests the possibility that cognitive maps may have evolved directly from the path integration system.
Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
Gloria Marí Beffa
2008-03-01
Full Text Available In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998, 161-213; 55 (1999, 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection.
Development Path of Urban-rural Integration
2012-01-01
The urban and rural areas are regarded as two major components of the regional economic system. Only through joint balanced development of the two can we achieve overall economic optimization and social welfare maximization. But the great social division of labor has separated urban areas from rural areas,which casts the shadow of city-oriented theory on cooperative relations between urban and rural areas. Mutual separation between urban and rural settlements and independent development trigger off a range of social problems. We must undertake guidance through rational development path of urban-rural integration,to eliminate the phenomenon of urban-rural dual structure,and promote the sustainable development of population,resources and environment in urban and rural areas as soon as possible.
Path Integral Monte Carlo Approach to the U(1) Lattice Gauge Theory in (2+1) Dimensions
Loan, M; Sloggett, C; Hamer, C; Loan, Mushtaq; Brunner, Michael; Sloggett, Clare; Hamer, Chris
2003-01-01
Path Integral Monte Carlo simulations have been performed for U(1) lattice gauge theory in (2+1) dimensions on anisotropic lattices. We extract the static quark potential, the string tension and the low-lying "glueball" spectrum. The Euclidean string tension and mass gap decrease exponentially at weak coupling in excellent agreement with the predictions of Polyakov and G{\\" o}pfert and Mack, but their magnitudes are five times bigger than predicted. Extrapolations are made to the extreme anisotropic or Hamiltonian limit, and comparisons are made with previous estimates obtained in the Hamiltonian formulation.
Polymer quantum mechanics some examples using path integrals
Parra, Lorena [Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., México and Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen (Netherlands); Vergara, J. David [Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F. (Mexico)
2014-01-14
In this work we analyze several physical systems in the context of polymer quantum mechanics using path integrals. First we introduce the group averaging method to quantize constrained systems with path integrals and later we use this procedure to compute the effective actions for the polymer non-relativistic particle and the polymer harmonic oscillator. We analyze the measure of the path integral and we describe the semiclassical dynamics of the systems.
An integrable generalization of the super AKNS hierarchy and its bi-Hamiltonian formulation
Yu, Jing; Ma, Wen-Xiu; Han, Jingwei; Chen, Shouting
2017-02-01
Based on a Lie super-algebra B(0, 1), an integrable generalization of the super AKNS iso-spectral problem is introduced and its corresponding generalized super AKNS hierarchy is generated. By making use of the super-trace identity (or the super variational identity), the resulting super soliton hierarchy can be put into a super bi-Hamiltonian form. A generalized super AKNS soliton hierarchy with self-consistent sources is also presented.
Classical Affine W-Algebras for gl_N and Associated Integrable Hamiltonian Hierarchies
De Sole, Alberto; Kac, Victor G.; Valeri, Daniele
2016-11-01
We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper to show that all W-algebras W({gl}N, f)} carry such a hierarchy. As an application, we show that all vector constrained KP hierarchies and their matrix generalizations are obtained from these hierarchies by Dirac reduction, which provides the former with a bi-Poisson structure.
Compensation for time-delayed feedback bang-bang control of quasi-integrable Hamiltonian systems
无
2009-01-01
The stochastic averaging method for quasi-integrable Hamiltonian systems with time-delayed feedback bang-bang control is first introduced. Then, two time delay compensation methods, namely the method of changing control force amplitude (CFA) and the method of changing control delay time (CDT), are proposed. The conditions applicable to each compensation method are discussed. Finally, an example is worked out in detail to illustrate the application and effectiveness of the proposed methods and the two compensation methods in combination.
Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems
Maolin Deng; Weiqiu Zhu
2007-01-01
A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems (multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged It6 stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximiza-tion of reliability and mean first-passage time are formulated.The optimal control law is derived from the dynamical pro-gramming equations and the control constraints. The final dynamical programming equations for these control prob-lems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynami-cal programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illus-trate the application and effectiveness of the proposed control strategy.
Lagrangian and Hamiltonian structures in an integrable hierarchy and space-time duality
Avan, Jean; Doikou, Anastasia; Kundu, Anjan
2016-01-01
We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schr\\"odinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zero-curvature representation of the NLEE can be realized by two Hamiltonian formulations stemming from two inequivalent Poisson structures and two distinct Hamiltonians. This is different from the standard bi-Hamiltonian structure. One is well-known and based on the standard Poisson structure for NLS. The other one is new and based on a different Poisson structure at each level of the hierarchy, yielding the corresponding NLEE as a {\\it space} evolution equation. We emphasize the role of the Lagrangian formulation as a unifying framework for deriving both Poisson structures, using ideas from covariant field theory. One of our main results is to show that the two matrices of the Lax pair satisfy the same form of ultralocal Poisson algebr...
Optimal control strategies for stochastically excited quasi partially integrable Hamiltonian systems
Ronghua Huan; Maolin Deng; Weiqiu Zhu
2007-01-01
In this paper two different control strategies designed to alleviate the response of quasi partially integrable Hamiltonian systems subjected to stochastic excitation are proposed. First, by using the stochastic averaging method for quasi partially integrable Hamiltonian systems, an n-DOF controlled quasi partially integrable Hamiltonian system with stochastic excitation is converted into a set of partially averaged Ito stochastic differential equations. Then, the dynamical programming equation associated with the partially averaged Ito equations is formulated by applying the stochastic dynamical programming principle. In the first control strategy, the optimal control law is derived from the dynamical programming equation and the control constraints without solving the dynamical programming equation. In the second control strategy, the optimal control law is obtained by solving the dynamical programming equation. Finally, both the responses of controlled and uncontrolled systems are predicted through solving the Fokker-Plank-Kolmogorov equation associated with fully averaged Ito equations. An example is worked out to illustrate the application and effectiveness of the two proposed control strategies.
Path Integral Solution by Sum Over Perturbation Series
Lin, D H
1999-01-01
A method for calculating the relativistic path integral solution via sum over perturbation series is given. As an application the exact path integral solution of the relativistic Aharonov-Bohm-Coulomb system is obtained by the method. Different from the earlier treatment based on the space-time transformation and infinite multiple-valued trasformation of Kustaanheimo-Stiefel in order to perform path integral, the method developed in this contribution involves only the explicit form of a simple Green's function and an explicit path integral is avoided.
Looping probabilities of elastic chains: a path integral approach.
Cotta-Ramusino, Ludovica; Maddocks, John H
2010-11-01
We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e.g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end
Lagrangian and Hamiltonian structures in an integrable hierarchy and space–time duality
Avan, Jean, E-mail: Jean.Avan@u-cergy.fr [Laboratoire de Physique Théorique et Modélisation (CNRS UMR 8089), Université de Cergy-Pontoise, F-95302 Cergy-Pontoise (France); Caudrelier, Vincent, E-mail: v.caudrelier@city.ac.uk [Department of Mathematics, City University London, Northampton Square, EC1V 0HB London (United Kingdom); Doikou, Anastasia, E-mail: A.Doikou@hw.ac.uk [Department of Mathematics, Heriot-Watt University, EH14 4AS, Edinburgh (United Kingdom); Kundu, Anjan, E-mail: Anjan.Kundu@saha.ac.in [Saha Institute of Nuclear Physics, Theory Division, Kolkata (India)
2016-01-15
We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schrödinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zero-curvature representation of the NLEE can be realized by two Hamiltonian formulations stemming from two distinct choices of the configuration space, yielding two inequivalent Poisson structures on the corresponding phase space and two distinct Hamiltonians. This is fundamentally different from the standard bi-Hamiltonian or generally multitime structure. The first formulation chooses purely space-dependent fields as configuration space; it yields the standard Poisson structure for NLS. The other one is new: it chooses purely time-dependent fields as configuration space and yields a different Poisson structure at each level of the hierarchy. The corresponding NLEE becomes a space evolution equation. We emphasize the role of the Lagrangian formulation as a unifying framework for deriving both Poisson structures, using ideas from covariant field theory. One of our main results is to show that the two matrices of the Lax pair satisfy the same form of ultralocal Poisson algebra (up to a sign) characterized by an r-matrix structure, whereas traditionally only one of them is involved in the classical r-matrix method. We construct explicit dual hierarchies of Hamiltonians, and Lax representations of the triggered dynamics, from the monodromy matrices of either Lax matrix. An appealing procedure to build a multi-dimensional lattice of Lax pair, through successive uses of the dual Poisson structures, is briefly introduced.
Anco, S C
2010-01-01
A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin-vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, ...
Mochon, C
2006-01-01
Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In this limit, the problem of finding the optimal query algorithm can be mapped into the problem of finding shortest paths on a manifold. The study of these shortest paths leads to lower bounds of the original unitary oracle problem. A number of example Hamiltonian oracles are studied in this paper, including oracle interrogation and the problem of computing the XOR of the hidden bits. Both of these problems are related to the study of geodesics on spheres with non-round metrics. For the case of two hidden bits a complete description of the geodesics is given. For n hidden bits a simple lower bound is proven that shows the problems require a query time proportional to n, even in the continuum limit. Finally, the problem of continuous Grover search is reexamined leading to a modest improvement to the protocol of Farhi and Gutmann.
Path Integrals and Lorentz Violation in Polymer Quantized Scalar Fields
Kajuri, Nirmalya
2014-01-01
We obtain a path integral formulation of polymer quantized scalar field theory, starting from the Hilbert Space framework. This brings the polymer quantized scalar field theory under the ambit of Feynman diagrammatic techniques. The path integral formulation also shows that Lorentz invariance is lost for the Klein-Gordon field.
Towards a Realistic Parsing of the Feynman Path Integral
Ken Wharton
2016-01-01
Full Text Available The Feynman path integral does not allow a one real path interpretation, because the quantum amplitudes contribute to probabilities in a non-separable manner. The opposite extreme, all paths happen, is not a useful or informative account. In this paper it is shown that an intermediate parsing of the path integral, into realistic non-interfering possibilities, is always available. Each realistic possibility formally corresponds to numerous particle paths, but is arguably best interpreted as a spacetime-valued field. Notably, one actual field history can always be said to occur, although it will generally not have an extremized action. The most obvious concerns with this approach are addressed, indicating necessary follow-up research. But without obvious showstoppers, it seems plausible that the path integral might be reinterpreted to explain quantum phenomena in terms of Lorentz covariant field histories.Quanta 2016; 5: 1–11.
Two-path plasmonic interferometer with integrated detector
Dyer, Gregory Conrad; Shaner, Eric A.; Aizin, Gregory
2016-03-29
An electrically tunable terahertz two-path plasmonic interferometer with an integrated detection element can down convert a terahertz field to a rectified DC signal. The integrated detector utilizes a resonant plasmonic homodyne mixing mechanism that measures the component of the plasma waves in-phase with an excitation field that functions as the local oscillator in the mixer. The plasmonic interferometer comprises two independently tuned electrical paths. The plasmonic interferometer enables a spectrometer-on-a-chip where the tuning of electrical path length plays an analogous role to that of physical path length in macroscopic Fourier transform interferometers.
Path Integrals and the WKB approximation in Loop Quantum Cosmology
Ashtekar, Abhay; Henderson, Adam
2010-01-01
We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.
Path integrals and the WKB approximation in loop quantum cosmology
Ashtekar, Abhay; Campiglia, Miguel; Henderson, Adam
2010-12-01
We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.
FIRST-PASSAGE TIME OF QUASI-NON-INTEGRABLE-HAMILTONIAN SYSTEM
甘春标; 徐博侯
2000-01-01
Studies on first-passage failure are extended to the multi-degree-offreedom quasi-non～integrable-Hamiltonian systems under parametric excitations of Gaussian white noises in this paper. By the stochastic averaging method of energy envelope, the system's energy can be modeled as a one-dimensional approximate diffusion process by which the classical Pontryagin equation with suitable boundary conditions is applicable to analyzing the statistical moments of the first-passage time of an arbitrary order. An example is studied in detail and some numerical results are given to illustrate the above procedure.
Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on S^2×S^3
Charles P. Boyer
2011-06-01
Full Text Available I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold S^2×S^3. In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, Y^{p,q}, discovered by physicists by showing that Y^{p,q} and Y^{p',q'} are inequivalent as contact structures if and only if p≠p'.
Leclerc, A.; Jolicard, G.; Viennot, D.; Killingbeck, J. P.
2012-01-01
The constrained adiabatic trajectory method (CATM) is reexamined as an integrator for the Schrödinger equation. An initial discussion places the CATM in the context of the different integrators used in the literature for time-independent or explicitly time-dependent Hamiltonians. The emphasis is put on adiabatic processes and within this adiabatic framework the interdependence between the CATM, the wave operator, the Floquet, and the (t, t') theories is presented in detail. Two points are then more particularly analyzed and illustrated by a numerical calculation describing the H_2^+ ion submitted to a laser pulse. The first point is the ability of the CATM to dilate the Hamiltonian spectrum and thus to make the perturbative treatment of the equations defining the wave function possible, possibly by using a Krylov subspace approach as a complement. The second point is the ability of the CATM to handle extremely complex time-dependencies, such as those which appear when interaction representations are used to integrate the system.
Accelerated nuclear quantum effects sampling with open path integrals
Mazzola, Guglielmo
2016-01-01
We numericaly demonstrate that, in double well models, the autocorrelation time of open path integral Monte Carlo simulations can be much smaller compared to standard ones using ring polymers. We also provide an intuitive explanation based on the role of instantons as transition states of the path integral pseudodynamics. Therefore we propose that, in all cases when the ground state approximation to the finite temperature partition function holds, open path integral simulations can be used to accelerate the sampling in realistic simulations aimed to explore nuclear quantum effects.
Hamiltonian Structures and Integrability for a Discrete Coupled KdV-Type Equation Hierarchy
ZHAO Hai-Qiong; ZHU Zuo-Nong; ZHANG Jing-Li
2011-01-01
@@ Coupled Korteweg-de Vries(KdV) systems have many important physical applications.By considering a 4 × 4spectral problem,we derive a discrete coupled KdV-type equation hierarchy.Our hierarchy includes the coupled Volterra system proposed by Lou et al.(e-print arXiv:0711.0420) as the first member which is a discrete version of the coupled KdV equation.We also investigate the integrability in the Liouville sense and the multi-Hamiltonian structures for the obtained hierarchy.%Coupled Korteweg-de Vries (KdV) systems have many important physical applications.By considering a 4 × 4 spectral problem, we derive a discrete coupled KdV-type equation hierarchy.Our hierarchy includes the coupled Volterra system proposed by Lou et al.(e-print arXiv: 0711.0420) as the first member which is a discrete version of the coupled KdV equation.We also investigate the integrability in the Liouville sense and the multi-Hamiltonian structures for the obtained hierarchy.
Leclerc, Arnaud; Viennot, David; Killingbeck, John P; 10.1063/1.3673320
2012-01-01
The Constrained Adiabatic Trajectory Method (CATM) is reexamined as an integrator for the Schr\\"odinger equation. An initial discussion places the CATM in the context of the different integrators used in the literature for time-independent or explicitly time-dependent Hamiltonians. The emphasis is put on adiabatic processes and within this adiabatic framework the interdependence between the CATM, the wave operator, the Floquet and the (t,t') theories is presented in detail. Two points are then more particularly analysed and illustrated by a numerical calculation describing the $H_2^+$ ion submitted to a laser pulse. The first point is the ability of the CATM to dilate the Hamiltonian spectrum and thus to make the perturbative treatment of the equations defining the wave function possible, possibly by using a Krylov subspace approach as a complement. The second point is the ability of the CATM to handle extremely complex time-dependencies, such as those which appear when interaction representations are used to...
Ab-initio path integral techniques for molecules
Shin, D; Shumway, J; Ho, Ming-Chak; Shin, Daejin
2006-01-01
Path integral Monte Carlo with Green's function analysis allows the sampling of quantum mechanical properties of molecules at finite temperature. While a high-precision computation of the energy of the Born-Oppenheimer surface from path integral Monte Carlo is quite costly, we can extract many properties without explicitly calculating the electronic energies. We demonstrate how physically relevant quantities, such as bond-length, vibrational spectra, and polarizabilities of molecules may be sampled directly from the path integral simulation using Matsubura (temperature) Green's functions (imaginary-time correlation functions). These calculations on the hydrogen molecule are a proof-of-concept, designed to motivate new work on fixed-node path-integral calculations for molecules.
Master equations and the theory of stochastic path integrals
Weber, Markus F
2016-01-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approxima...
Remembered landmarks enhance the precision of path integration
Shannon O´Leary
2005-01-01
Full Text Available When navigating by path integration, knowledge of ones position becomes increasingly uncertain as one walks from a known location. This uncertainty decreases if one perceives a known landmark location nearby. We hypothesized that remembering landmarks might serve a similar purpose for path integration as directly perceiving them. If this is true, walking near a remembered landmark location should enhance response consistency in path integration tasks. To test this, we asked participants to view a target and then attempt to walk to it without vision. Some participants saw the target plus a landmark during the preview. Compared with no-landmark trials, response consistency nearly doubled when participants passed near the remembered landmark location. Similar results were obtained when participants could audibly perceive the landmark while walking. A control experiment ruled out perceptual context effects during the preview. We conclude that remembered landmarks can enhance path integration even though they are not directly perceived.
Variational path integral molecular dynamics study of a water molecule
Miura, Shinichi
2013-08-01
In the present study, a variational path integral molecular dynamics method developed by the author [Chem. Phys. Lett. 482, 165 (2009)] is applied to a water molecule on the adiabatic potential energy surface. The method numerically generates an exact wavefunction using a trial wavefunction of the target system. It has been shown that even if a poor trial wavefunction is employed, the exact quantum distribution is numerically extracted, demonstrating the robustness of the variational path integral method.
Characterizing regulatory path motifs in integrated networks using perturbational data
Joshi, Anagha Madhusudan; Van Parys, Thomas; de Peer, Yves Van; Michoel, Tom
2010-01-01
We introduce Pathicular http://bioinformatics.psb.ugent.be/software/details/Pathicular, a Cytoscape plugin for studying the cellular response to perturbations of transcription factors by integrating perturbational expression data with transcriptional, protein-protein and phosphorylation networks. Pathicular searches for 'regulatory path motifs', short paths in the integrated physical networks which occur significantly more often than expected between transcription factors and their targets in...
Emergent symmetry in a thermal pure state path integral
Sasa, Shin-ichi; Yokokura, Yuki
2016-01-01
We study a thermally isolated quantum many-body system with an external control represented by a time-dependent parameter. By formulating a thermal pure state path integral, we derive an effective action for trajectories in a thermodynamic state space, where the entropy appears with its conjugate variable. In particular, when operations are quasi-static, the symmetry for the uniform translation of the conjugate variable emerges in the path integral. This leads to the entropy as a Noether invariant.
Sensory feedback in a bump attractor model of path integration.
Poll, Daniel B; Nguyen, Khanh; Kilpatrick, Zachary P
2016-04-01
Mammalian spatial navigation systems utilize several different sensory information channels. This information is converted into a neural code that represents the animal's current position in space by engaging place cell, grid cell, and head direction cell networks. In particular, sensory landmark (allothetic) cues can be utilized in concert with an animal's knowledge of its own velocity (idiothetic) cues to generate a more accurate representation of position than path integration provides on its own (Battaglia et al. The Journal of Neuroscience 24(19):4541-4550 (2004)). We develop a computational model that merges path integration with feedback from external sensory cues that provide a reliable representation of spatial position along an annular track. Starting with a continuous bump attractor model, we explore the impact of synaptic spatial asymmetry and heterogeneity, which disrupt the position code of the path integration process. We use asymptotic analysis to reduce the bump attractor model to a single scalar equation whose potential represents the impact of asymmetry and heterogeneity. Such imperfections cause errors to build up when the network performs path integration, but these errors can be corrected by an external control signal representing the effects of sensory cues. We demonstrate that there is an optimal strength and decay rate of the control signal when cues appear either periodically or randomly. A similar analysis is performed when errors in path integration arise from dynamic noise fluctuations. Again, there is an optimal strength and decay of discrete control that minimizes the path integration error.
Vehicle path tracking by integrated chassis control
Saman Salehpour; Yaghoub Pourasad; Seyyed Hadi Taheri
2015-01-01
The control problem of trajectory based path following for passenger vehicles is studied. Comprehensive nonlinear vehicle model is utilized for simulation vehicle response during various maneuvers in MATLAB/Simulink. In order to follow desired path, a driver model is developed to enhance closed loop driver/vehicle model. Then, linear quadratic regulator (LQR) controller is developed which regulates direct yaw moment and corrective steering angle on wheels. Particle swam optimization (PSO) method is utilized to optimize the LQR controller for various dynamic conditions. Simulation results indicate that, over various maneuvers, side slip angle and lateral acceleration can be reduced by 10%and 15%, respectively, which sustain the vehicle stable. Also, anti-lock brake system is designed for longitudinal dynamics of vehicle to achieve desired slip during braking and accelerating. Proposed comprehensive controller demonstrates that vehicle steerability can increase by about 15% during severe braking by preventing wheel from locking and reducing stopping distance.
Qiuping A. Wang
2014-02-01
Full Text Available A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields an extended formalism of mechanics for random dynamics. Within this theory, Liouville’s theorem of conservation of phase density distribution must be modified to allow time evolution of phase density and consequently the Boltzmann H theorem. We argue that the gap between the regular Newtonian dynamics and the random dynamics was not considered in the criticisms of the H theorem.
The perturbative approach to path integrals: A succinct mathematical treatment
Nguyen, Timothy
2016-09-01
We study finite-dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick's theorem allows one to evaluate integrals perturbatively, i.e., as a series expansion in a formal parameter irrespective of convergence properties. We establish invariance properties of such a Wick expansion under coordinate changes and the action of a Lie group of symmetries, and we use this to study essential features of path integral manipulations, including coordinate changes, Ward identities, Schwinger-Dyson equations, Faddeev-Popov gauge-fixing, and eliminating fields by their equation of motion. We also discuss the asymptotic nature of the Wick expansion and the implications this has for defining path integrals perturbatively and nonperturbatively.
Medial temporal lobe roles in human path integration.
Naohide Yamamoto
Full Text Available Path integration is a process in which observers derive their location by integrating self-motion signals along their locomotion trajectory. Although the medial temporal lobe (MTL is thought to take part in path integration, the scope of its role for path integration remains unclear. To address this issue, we administered a variety of tasks involving path integration and other related processes to a group of neurosurgical patients whose MTL was unilaterally resected as therapy for epilepsy. These patients were unimpaired relative to neurologically intact controls in many tasks that required integration of various kinds of sensory self-motion information. However, the same patients (especially those who had lesions in the right hemisphere walked farther than the controls when attempting to walk without vision to a previewed target. Importantly, this task was unique in our test battery in that it allowed participants to form a mental representation of the target location and anticipate their upcoming walking trajectory before they began moving. Thus, these results put forth a new idea that the role of MTL structures for human path integration may stem from their participation in predicting the consequences of one's locomotor actions. The strengths of this new theoretical viewpoint are discussed.
Lecture notes in topics in path integrals and string representations
Botelho, Luiz C L
2017-01-01
Functional Integrals is a well-established method in mathematical physics, especially those mathematical methods used in modern non-perturbative quantum field theory and string theory. This book presents a unique, original and modern treatment of strings representations on Bosonic Quantum Chromodynamics and Bosonization theory on 2d Gauge Field Models, besides of rigorous mathematical studies on the analytical regularization scheme on Euclidean quantum field path integrals and stochastic quantum field theory. It follows an analytic approach based on Loop space techniques, functional determinant exact evaluations and exactly solubility of four dimensional QCD loop wave equations through Elfin Botelho fermionic extrinsic self avoiding string path integrals.
INTEGRATED LAYOUT DESIGN OF CELLS AND FLOW PATHS
Li Zhihua; Zhong Yifang; Zhou Ji
2003-01-01
The integrated layout problem in manufacturing systems is investigated. An integrated model for concurrent layout design of cells and flow paths is formulated. A hybrid approach combined an enhanced branch-and-bound algorithm with a simulated annealing scheme is proposed to solve this problem. The integrated layout method is applied to re-layout the gear pump shop of a medium-size manufacturer of hydraulic pieces. Results show that the proposed layout method can concurrently provide good solutions of the cell layouts and the flow path layouts.
Mielke, Steven L; Truhlar, Donald G
2016-01-21
Using Feynman path integrals, a molecular partition function can be written as a double integral with the inner integral involving all closed paths centered at a given molecular configuration, and the outer integral involving all possible molecular configurations. In previous work employing Monte Carlo methods to evaluate such partition functions, we presented schemes for importance sampling and stratification in the molecular configurations that constitute the path centroids, but we relied on free-particle paths for sampling the path integrals. At low temperatures, the path sampling is expensive because the paths can travel far from the centroid configuration. We now present a scheme for importance sampling of whole Feynman paths based on harmonic information from an instantaneous normal mode calculation at the centroid configuration, which we refer to as harmonically guided whole-path importance sampling (WPIS). We obtain paths conforming to our chosen importance function by rejection sampling from a distribution of free-particle paths. Sample calculations on CH4 demonstrate that at a temperature of 200 K, about 99.9% of the free-particle paths can be rejected without integration, and at 300 K, about 98% can be rejected. We also show that it is typically possible to reduce the overhead associated with the WPIS scheme by sampling the paths using a significantly lower-order path discretization than that which is needed to converge the partition function.
Path integration and the neural basis of the 'cognitive map'.
McNaughton, B.L.; Battaglia, F.P.; Jensen, O.; Moser, E.I.; Moser, M.B
2006-01-01
The hippocampal formation can encode relative spatial location, without reference to external cues, by the integration of linear and angular self-motion (path integration). Theoretical studies, in conjunction with recent empirical discoveries, suggest that the medial entorhinal cortex (MEC) might pe
On the Path Integral Loop Representation of (2+1) Lattice Non-Abelian Theory
Aroca, J M; Gambini, R
1998-01-01
A gauge invariant Hamiltonian representation for SU(2) in terms of a spin network basis is introduced. The vectors of the spin network basis are independent and the electric part of the Hamiltonian is diagonal in this representation. The corresponding path integral for SU(2) lattice gauge theory is expressed as a sum over colored surfaces, i.e. only involving the $j_p$ attached to the lattice plaquettes. This surfaces may be interpreted as the world sheets of the spin networks In 2+1 dimensions, this can be accomplished by working in a lattice dual to a tetrahedral lattice constructed on a face centered cubic Bravais lattice. On such a lattice, the integral of gauge variables over boundaries or singular lines - which now always bound three coloured surfaces - only contributes when four singular lines intersect at one vertex and can be explicitly computed producing a 6-j or Racah symbol. We performed a strong coupling expansion for the free energy. The convergence of the series expansions is quite different fr...
The quantum bouncer by the path integral method
Goodings, D. A.; Szeredi, T.
1991-10-01
The path integral formulation of quantum mechanics in the semiclassical or WKB approximation provides a physically intuitive way of relating a classical system to its quantum analog. A fruitful way of studying quantum chaos is based upon applying the Gutzwiller periodic orbit sum rule, a result derived by the path integral method in the WKB approximation. This provides some motivation for learning about path integral techniques. In this paper a pedagogical example of the path integral formalism is presented in the hope of conveying the basic physical and mathematical concepts. The ``quantum bouncer'' is studied—the quantum version of a particle moving in one dimension above a perfectly reflecting surface while subject to a constant force directed toward the surface. The classical counterpart of this system is a ball bouncing on a floor in a constant gravitational field, collisions with the floor being assumed to be elastic. Path integration is used to derive the energy eigenvalues and eigenfunctions of the quantum bouncer in the WKB or semiclassical approximation. The results are shown to be the same as those obtained by solving the Schrödinger equation in the same approximation.
PathSys: integrating molecular interaction graphs for systems biology
Raval Alpan
2006-02-01
Full Text Available Abstract Background The goal of information integration in systems biology is to combine information from a number of databases and data sets, which are obtained from both high and low throughput experiments, under one data management scheme such that the cumulative information provides greater biological insight than is possible with individual information sources considered separately. Results Here we present PathSys, a graph-based system for creating a combined database of networks of interaction for generating integrated view of biological mechanisms. We used PathSys to integrate over 14 curated and publicly contributed data sources for the budding yeast (S. cerevisiae and Gene Ontology. A number of exploratory questions were formulated as a combination of relational and graph-based queries to the integrated database. Thus, PathSys is a general-purpose, scalable, graph-data warehouse of biological information, complete with a graph manipulation and a query language, a storage mechanism and a generic data-importing mechanism through schema-mapping. Conclusion Results from several test studies demonstrate the effectiveness of the approach in retrieving biologically interesting relations between genes and proteins, the networks connecting them, and of the utility of PathSys as a scalable graph-based warehouse for interaction-network integration and a hypothesis generator system. The PathSys's client software, named BiologicalNetworks, developed for navigation and analyses of molecular networks, is available as a Java Web Start application at http://brak.sdsc.edu/pub/BiologicalNetworks.
Momentum and Hamiltonian in Complex Action Theory
Nagao, Keiichi; Nielsen, Holger Bech
In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view based on the complex coordinate formalism of our foregoing paper. After reviewing the formalism briefly, we describe in FPI with a Lagrangian the time development of a ξ-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator. Solving this eigenvalue problem, we derive the momentum and Hamiltonian. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum relation again via the saddle point for p. This study confirms that the momentum and Hamiltonian in the CAT have the same forms as those in the real action theory. We also show the third derivation of the momentum relation via the saddle point for q.
Polymer density functional approach to efficient evaluation of path integrals
Brukhno, Andrey; Vorontsov-Velyaminov, Pavel N.; Bohr, Henrik
2005-01-01
A polymer density functional theory (P-DFT) has been extended to the case of quantum statistics within the framework of Feynman path integrals. We start with the exact P-DFT formalism for an ideal open chain and adapt its efficient numerical solution to the case of a ring. We show that, similarly......, the path integral problem can, in principle, be solved exactly by making use of the two-particle pair correlation function (2p-PCF) for the ends of an open polymer, half of the original. This way the exact data for one-dimensional quantum harmonic oscillator are reproduced in a wide range of temperatures......-consistent iteration so as to correctly account for the interparticle interactions. The algorithm is speeded up by taking convolutions with the aid of fast Fourier transforms. We apply this approximate path integral DFT (PI-DFT) method to systems within spherical symmetry: 3D harmonic oscillator, atoms of hydrogen...
Low-temperature anharmonicity of barium titanate: A path-integral molecular-dynamics study
Geneste, Grégory; Dammak, Hichem; Hayoun, Marc; Thiercelin, Mickael
2013-01-01
We investigate the influence of quantum effects on the dielectric and piezoelectric properties of barium titanate in its (low-temperature) rhombohedral phase, and show the strongly anharmonic character of this system even at low temperature. For this purpose, we perform path-integral molecular-dynamics simulations under fixed pressure and fixed temperature, using an efficient Langevin thermostat-barostat, and an effective Hamiltonian derived from first-principles calculations. The quantum fluctuations are shown to significantly enhance the static dielectric susceptibility (≈ by a factor of 2) and the piezoelectric constants, reflecting the strong anharmonicity of this ferroelectric system even at very low temperature. The slow temperature-evolution of the dielectric properties observed below ≈100 K is attributed (i) to zero-point energy contributions and (ii) to harmonic behavior if the quantum effects are turned off.
Path integral approach for quantum motion on spaces of non-constant curvature according to Koenigs
Grosche, C. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik
2006-08-15
In this contribution I discuss a path integral approach for the quantum motion on two-dimensional spaces according to Koenigs, for short ''Koenigs-Spaces''. Their construction is simple: One takes a Hamiltonian from two-dimensional flat space and divides it by a two-dimensional superintegrable potential. These superintegrable potentials are the isotropic singular oscillator, the Holt-potential, and the Coulomb potential. In all cases a non-trivial space of non-constant curvature is generated. We can study free motion and the motion with an additional superintegrable potential. For possible bound-state solutions we find in all three cases an equation of eighth order in the energy E. The special cases of the Darboux spaces are easily recovered by choosing the parameters accordingly. (orig.)
A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results
Baaquie, Belal E.
1997-12-01
The Black-Scholes formula for pricing options on stocks and other securities has been generalized by Merton and Garman to the case when stock volatility is stochastic. The derivation of the price of a security derivative with stochastic volatility is reviewed starting from the first principles of finance. The equation of Merton and Garman is then recast using the path integration technique of theoretical physics. The price of the stock option is shown to be the analogue of the Schrödinger wavefunction of quantum mechanics and the exact Hamiltonian and Lagrangian of the system is obtained. The results of Hull and White are generalized to the case when stock price and volatility have non-zero correlation. Some exact results for pricing stock options for the general correlated case are derived.
Minary, Peter; Martyna, Glenn J.; Tuckerman, Mark E.
2003-02-01
In this paper (Paper I) and a companion paper (Paper II), novel new algorithms and applications of the isokinetic ensemble as generated by Gauss' principle of least constraint, pioneered for use with molecular dynamics 20 years ago, are presented for biophysical, path integral, and Car-Parrinello based ab initio molecular dynamics. In Paper I, a new "extended system" version of the isokinetic equations of motion that overcomes the ergodicity problems inherent in the standard approach, is developed using a new theory of non-Hamiltonian phase space analysis [M. E. Tuckerman et al., Europhys. Lett. 45, 149 (1999); J. Chem. Phys. 115, 1678 (2001)]. Reversible multiple time step integrations schemes for the isokinetic methods, first presented by Zhang [J. Chem. Phys. 106, 6102 (1997)] are reviewed. Next, holonomic constraints are incorporated into the isokinetic methodology for use in fast efficient biomolecular simulation studies. Model and realistic examples are presented in order to evaluate, critically, the performance of the new isokinetic molecular dynamic schemes. Comparisons are made to the, now standard, canonical dynamics method, Nosé-Hoover chain dynamics [G. J. Martyna et al., J. Chem. Phys. 97, 2635 (1992)]. The new isokinetic techniques are found to yield more efficient sampling than the Nosé-Hoover chain method in both path integral molecular dynamics and biophysical molecular dynamics calculations. In Paper II, the use of isokinetic methods in Car-Parrinello based ab initio molecular dynamics calculations is presented.
Master equations and the theory of stochastic path integrals.
Weber, Markus F; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon
Master equations and the theory of stochastic path integrals
Weber, Markus F.; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from
Grosche, C. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik
2007-08-15
In this contribution a path integral approach for the quantum motion on three-dimensional spaces according to Koenigs, for short''Koenigs-Spaces'', is discussed. Their construction is simple: One takes a Hamiltonian from three-dimensional flat space and divides it by a three-dimensional superintegrable potential. Such superintegrable potentials will be the isotropic singular oscillator, the Holt-potential, the Coulomb potential, or two centrifugal potentials, respectively. In all cases a non-trivial space of non-constant curvature is generated. In order to obtain a proper quantum theory a curvature term has to be incorporated into the quantum Hamiltonian. For possible bound-state solutions we find equations up to twelfth order in the energy E. (orig.)
Age differences in virtual environment and real world path integration
Diane E Adamo
2012-09-01
Full Text Available Accurate path integration requires the integration of visual, proprioceptive, and vestibular self-motion cues and age effects associated with alterations in processing information from these systems may contribute to declines in path integration abilities. The present study investigated age-related differences in path integration in conditions that varied as a function of available sources of sensory information. Twenty-two healthy, young (23.8 ± 3.0 yrs. and 16 older (70.1 ± 6.4 yrs. adults participated in distance reproduction and triangle completion tasks performed in a virtual environment and two real world conditions: guided walking and wheelchair propulsion. For walking and wheelchair propulsion conditions, participants wore a blindfold and wore noise-blocking headphones and were guided through the workspace by the experimenter. For the virtual environment (VE condition, participants viewed self-motion information on a computer monitor and used a joystick to navigate through the environment. For triangle completion tasks, older compared to younger individuals showed greater errors in rotation estimations performed in the wheelchair condition; and for rotation and distance estimations in the VE condition. Distance reproduction tasks, in contrast, did not show any age effects. These findings demonstrate that age differences in path integration vary as a function of the available sources of information and by the complexity of outbound pathway.
Path Integrals and the Statistical Thermodynamics of Black Holes.
Martinez, Erik Andres
The path integral is an important element in modern approaches to the quantization of the gravitational field. Path integral representations of partition functions for static and stationary black hole systems as well as path integrals for minisuperspace models of cosmology are presented. The functional integral is defined throughout as a sum over Lorentzian histories. A consistent formulation of Feynman's prescription to construct partition functions in terms of path integrals for general gravitational systems is presented and contrasted with other "Euclideanization" prescriptions. It is shown that the central object in the description of black hole systems is the gravitational action. In particular, the additivity of the entropies of matter and black holes in thermal equilibrium is a consequence of the additivity of their corresponding actions, and thermodynamic potentials like the energy or the pressure are not in general addivite when gravity plays an important role. Partition functions as stationary phase approximations of functional integrals for all the thermodynamic ensembles are then constructed by including gravitation as a part of the thermodynamical system. We show that a complex geometry is required to derive the thermodynamic properties of stationary geometries from the sum over histories. The corresponding real "thermodynamical" action is calculated explicitly and the thermodynamical data that imply thermal equilibrium in the presence of a rotating black hole in interaction with matter fields are presented and related to geometrical data. Some of the consequences for Kerr-Newman black hole systems are also discussed. For minisuperspace cosmologies the Lorentzian path integral is a Green function for the Wheeler-DeWitt operator, and its real part is a solution to the Wheeler -DeWitt equation. It is computed explicitly for the de Sitter minisuperspace model. The resulting Green function is then related to both the Hartle-Hawking and tunneling wave
On the Hamiltonian integrability of the bi-Yang-Baxter sigma-model
Delduc, Francois; Magro, Marc; Vicedo, Benoit
2015-01-01
The bi-Yang-Baxter sigma-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimcik who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset sigma-model on G x G / G_diag, we show that it also admits a Lax matrix whose Poisson bracket is of the standard r/s-form characterised by a twist function which we determine. A number of results immediately follow from this, including the identification of certain complex Poisson commuting Kac-Moody currents as well as an explicit description of the q-deformed symmetries of the model. Moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations o...
On the Hamiltonian integrability of the bi-Yang-Baxter σ-model
Delduc, F.; Lacroix, S.; Magro, M. [Laboratoire de Physique, ENS de Lyon et CNRS UMR 5672, Université de Lyon,46, allée d’Italie, 69364 LYON Cedex 07 (France); Vicedo, B. [School of Physics, Astronomy and Mathematics, University of Hertfordshire,College Lane, Hatfield AL10 9AB (United Kingdom)
2016-03-15
The bi-Yang-Baxter σ-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimčík who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset σ-model on G×G/G{sub diag}, we show that it also admits a Lax matrix whose Poisson bracket is of the standard r/s-form characterised by a twist function which we determine. A number of results immediately follow from this, including the identification of certain complex Poisson commuting Kac-Moody currents as well as an explicit description of the q-deformed symmetries of the model. Moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations of σ-models. Finally, we show that although the Poisson bracket of the Lax matrix still takes the r/s-form after fixing the G{sub diag} gauge symmetry, it is no longer characterised by a twist function.
Path Integral Understanding in the Context of the Electromagnetic Theory
Gonzalez, Maria D.
2006-12-01
Introductory electromagnetic courses at the University of Juarez are in general identified by the use of a traditional instruction. The path integral is a fundamental mathematical knowledge to understand the properties of conservative fields such that the electric field. Many students in these courses do not develop the necessary scientific skills and mathematical formalism to understand the fact that the potential difference does not depend on the path followed from one point to another one inside an electric field. It is fundamental to probe the student understanding difficulties to apply the concept of path integral in an electromagnetic context. The use of the software CABRI could become an important didactic choice during the development of the potential difference concept. It was necessary the recollection of data related to the student procedural difficulties in the use of the designed CABRI activities. Sponsor: member Sergio Flores
The path integral formulation of climate dynamics.
Antonio Navarra
Full Text Available The chaotic nature of the atmospheric dynamics has stimulated the applications of methods and ideas derived from statistical dynamics. For instance, ensemble systems are used to make weather predictions recently extensive, which are designed to sample the phase space around the initial condition. Such an approach has been shown to improve substantially the usefulness of the forecasts since it allows forecasters to issue probabilistic forecasts. These works have modified the dominant paradigm of the interpretation of the evolution of atmospheric flows (and oceanic motions to some extent attributing more importance to the probability distribution of the variables of interest rather than to a single representation. The ensemble experiments can be considered as crude attempts to estimate the evolution of the probability distribution of the climate variables, which turn out to be the only physical quantity relevant to practice. However, little work has been done on a direct modeling of the probability evolution itself. In this paper it is shown that it is possible to write the evolution of the probability distribution as a functional integral of the same kind introduced by Feynman in quantum mechanics, using some of the methods and results developed in statistical physics. The approach allows obtaining a formal solution to the Fokker-Planck equation corresponding to the Langevin-like equation of motion with noise. The method is very general and provides a framework generalizable to red noise, as well as to delaying differential equations, and even field equations, i.e., partial differential equations with noise, for example, general circulation models with noise. These concepts will be applied to an example taken from a simple ENSO model.
The Path Integral Formulation of Climate Dynamics
Navarra, Antonio; Tribbia, Joe; Conti, Giovanni
2013-01-01
The chaotic nature of the atmospheric dynamics has stimulated the applications of methods and ideas derived from statistical dynamics. For instance, ensemble systems are used to make weather predictions recently extensive, which are designed to sample the phase space around the initial condition. Such an approach has been shown to improve substantially the usefulness of the forecasts since it allows forecasters to issue probabilistic forecasts. These works have modified the dominant paradigm of the interpretation of the evolution of atmospheric flows (and oceanic motions to some extent) attributing more importance to the probability distribution of the variables of interest rather than to a single representation. The ensemble experiments can be considered as crude attempts to estimate the evolution of the probability distribution of the climate variables, which turn out to be the only physical quantity relevant to practice. However, little work has been done on a direct modeling of the probability evolution itself. In this paper it is shown that it is possible to write the evolution of the probability distribution as a functional integral of the same kind introduced by Feynman in quantum mechanics, using some of the methods and results developed in statistical physics. The approach allows obtaining a formal solution to the Fokker-Planck equation corresponding to the Langevin-like equation of motion with noise. The method is very general and provides a framework generalizable to red noise, as well as to delaying differential equations, and even field equations, i.e., partial differential equations with noise, for example, general circulation models with noise. These concepts will be applied to an example taken from a simple ENSO model. PMID:23840577
On Duru-Kleinert Path Integral In Quantum Cosmology
Jafarizadeh, M A; Rastegar, A R
1998-01-01
We show that the Duru-Kleinert fixed energy amplitude leads to the path integral for the propagation amplitude in the closed FRW quantum cosmology with scale factor as one degree of freedom. Then, using the Duru-Kleinert equivalence of corresponding actions, we calculate the tunneling rate, with exact prefactor, through the dilute-instanton approximation to first order in
Path Integration Applied to Structural Systems with Uncertain Properties
Nielsen, Søren R.K.; Köylüoglu, H. Ugur
Path integration (cell-to-cell mapping) method is applied to evaluate the joint probability density function (jpdf) of the response of the structural systems, with uncertain properties, subject to white noise excitation. A general methodology to deal with uncertainties is outlined and applied...
Quantum tunneling splittings from path-integral molecular dynamics
Mátyus, Edit; Wales, David J.; Althorpe, Stuart C.
2016-03-01
We illustrate how path-integral molecular dynamics can be used to calculate ground-state tunnelling splittings in molecules or clusters. The method obtains the splittings from ratios of density matrix elements between the degenerate wells connected by the tunnelling. We propose a simple thermodynamic integration scheme for evaluating these elements. Numerical tests on fully dimensional malonaldehyde yield tunnelling splittings in good overall agreement with the results of diffusion Monte Carlo calculations.
Dvornikov, Maxim; Gitman, D. M.
2012-11-01
We study massive 1/2-spin particles in various external backgrounds keeping in mind applications to neutrino physics. We are mainly interested in massive Majorana (Weyl) fields. However, massive neutral Dirac particles are also considered. We formulate classical Lagrangian theory of the massive Weyl field in terms of Grassmann-odd two-component spinors. Then we construct the Hamiltonian formulation of such a theory, which turns out to be a theory with second-class constraints. Using this formulation we canonically quantize the massive free Weyl field. We derive propagators of the Weyl field and relate them to the propagator of a massive Dirac particle. We also study the massive Weyl particles propagating in the background mater. We find the path integral representation for the propagator of such a field, as well as the corresponding pseudoclassical particle action. The massless limit of the Weyl field interacting with the matter is considered and compared with results of other works. Finally, the path integral representation for the propagator of the neutral massive Dirac particle with an anomalous magnetic moment moving in the background matter and external electromagnetic field, as well as the corresponding pseudoclassical particle action are constructed.
Tackling Higher Derivative Ghosts with the Euclidean Path Integral
Fontanini, Michele
2011-01-01
An alternative to the effective field theory approach to treat ghosts in higher derivative theories is to attempt to integrate them out via the Euclidean path integral formalism. It has been suggested that this method could provide a consistent framework within which we might tolerate the ghost degrees of freedom that plague, among other theories, the higher derivative gravity models that have been proposed to explain cosmic acceleration. We consider the extension of this idea to treating a class of terms with order six derivatives, and find that for a general term the Euclidean path integral approach works in the most trivial background, Minkowski. Moreover we see that even in de Sitter background, despite some difficulties, it is possible to define a probability distribution for tensorial perturbations of the metric.
Beam spread functions calculated using Feynman path integrals
Kilgo, Paul; Tessendorf, Jerry
2017-07-01
A method of solving the radiative transfer equation using Feynman path integrals (FPIs) is discussed. The FPI approach is a mathematical framework for computing multiple scattering in participating media. Its numerical behavior is not well known, and techniques are being developed to solve the FPI approach numerically. A missing numerical technique is detailed and used to calculate beam spread functions (BSFs), a commonly studied experimental property of many types of media. The calculations are compared against measured BSFs of sea ice. Analysis shows differently-shaped BSFs, and suggests the width parameter of the calculated BSF's Gaussian fit approaches a value in the limit of the number of path segments. A projection is attempted, but suggests a larger number of path segments would not increase the width of the calculated BSF. The trial suggests the approach is numerically stable, but requires further testing to ensure scientific accuracy.
Vilasi, Gaetano
2001-01-01
This is both a textbook and a monograph. It is partially based on a two-semester course, held by the author for third-year students in physics and mathematics at the University of Salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems. As a textbook, it provides a systematic and self-consistent formulation of Hamiltonian dynamics both in a rigorous coordinate language and in the modern language of differential geometry. It also presents powerful mathematical methods of theoretical physics, especially in gauge theories and general relativity. As a m
BOOK REVIEW: Path Integrals in Field Theory: An Introduction
Ryder, Lewis
2004-06-01
In the 1960s Feynman was known to particle physicists as one of the people who solved the major problems of quantum electrodynamics, his contribution famously introducing what are now called Feynman diagrams. To other physicists he gained a reputation as the author of the Feynman Lectures on Physics; in addition some people were aware of his work on the path integral formulation of quantum theory, and a very few knew about his work on gravitation and Yang--Mills theories, which made use of path integral methods. Forty years later the scene is rather different. Many of the problems of high energy physics are solved; and the standard model incorporates Feynman's path integral method as a way of proving the renormalisability of the gauge (Yang--Mills) theories involved. Gravitation is proving a much harder nut to crack, but here also questions of renormalisability are couched in path-integral language. What is more, theoretical studies of condensed matter physics now also appeal to this technique for quantisation, so the path integral method is becoming part of the standard apparatus of theoretical physics. Chapters on it appear in a number of recent books, and a few books have appeared devoted to this topic alone; the book under review is a very recent one. Path integral techniques have the advantage of enormous conceptual appeal and the great disadvantage of mathematical complexity, this being partly the result of messy integrals but more fundamentally due to the notions of functional differentiation and integration which are involved in the method. All in all this subject is not such an easy ride. Mosel's book, described as an introduction, is aimed at graduate students and research workers in particle physics. It assumes a background knowledge of quantum mechanics, both non-relativistic and relativistic. After three chapters on the path integral formulation of non-relativistic quantum mechanics there are eight chapters on scalar and spinor field theory, followed
Baxter, Mathew; Choudhury, S. Roy; Van Gorder, Robert A.
2015-06-01
In the present paper, we present an integrable hierarchy for the Zakharov-Ito system. We first construct the Lenard recursion sequence and zero curvature representation for the Zakharov-Ito system, following Cao's method as significantly generalized by other authors. We then construct the bi-Hamiltonian structures employing variational trace identities but woven together with the Lenard recursion sequences. From this, we are in a position to construct an integrable hierarchy of equations from the Zakharov-Ito system, and we obtain the recursion operator and Poisson brackets for constructing this hierarchy. Finally, we demonstrate that the obtained hierarchy is indeed Liouville integrable.
Mathematical theory of Feynman path integrals an introduction
Albeverio, Sergio A; Mazzucchi, Sonia
2008-01-01
Feynman path integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non-relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ideas based on Feynman path integrals have also played an important role in areas of mathematics like low-dimensional topology and differential geometry, algebraic geometry, infinite-dimensional analysis and geometry, and number theory. The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. To take care of the many developments since then, an entire new chapter on the current forefront of research has been added. Except for this new chapter and the correction of a few misprints, the basic material and presentation of the first edition has been maintained. At the end of each chapter the reader will also find notes with further bibliographical information.
State Space Path Integrals for Electronically Nonadiabatic Reaction Rates
Duke, Jessica Ryan
2016-01-01
We present a state-space-based path integral method to calculate the rate of electron transfer (ET) in multi-state, multi-electron condensed-phase processes. We employ an exact path integral in discrete electronic states and continuous Cartesian nuclear variables to obtain a transition state theory (TST) estimate to the rate. A dynamic recrossing correction to the TST rate is then obtained from real-time dynamics simulations using mean field ring polymer molecular dynamics. We employ two different reaction coordinates in our simulations and show that, despite the use of mean field dynamics, the use of an accurate dividing surface to compute TST rates allows us to achieve remarkable agreement with Fermi's golden rule rates for nonadiabatic ET in the normal regime of Marcus theory. Further, we show that using a reaction coordinate based on electronic state populations allows us to capture the turnover in rates for ET in the Marcus inverted regime.
Path integral quantization of the relativistic Hopfield model
Belgiorno, F; Piazza, F Dalla; Doronzo, M
2016-01-01
The path integral quantization method is applied to a relativistically covariant version of the Hopfield model, which represents a very interesting mesoscopic framework for the description of the interaction between quantum light and dielectric quantum matter, with particular reference to the context of analogue gravity. In order to take into account the constraints occurring in the model, we adopt the Faddeev-Jackiw approach to constrained quantization in the path integral formalism. In particular we demonstrate that the propagator obtained with the Faddeev-Jackiw approach is equivalent to the one which, in the framework of Dirac canonical quantization for constrained systems, can be directly computed as the vacuum expectation value of the time ordered product of the fields. Our analysis also provides an explicit example of quantization of the electromagnetic field in a covariant gauge and coupled with the polarization field, which is a novel contribution to the literature on the Faddeev-Jackiw procedure.
A discrete history of the Lorentzian path integral
Loll, R
2003-01-01
In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a well-defined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to_convergent_ sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d=2 and d=3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry.
Remarks on the Origin of Path Integration: Einstein and Feynman
Sauer, Tilman
2008-01-01
I offer some historical comments about the origins of Feynman's path integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's thesis supervisor John Wheeler, it is argued that, contrary to what one might expect, the significance of the interaction between Einstein and Feynman pertained to a critique of classical field theory, rather than to a direct critique of quantum mechanics itself...
A path to integration in an academic health science center.
Panko, W B; Wilson, W
1992-01-01
This article describes a networking and integration strategy in use at the University of Michigan Medical Center. This strategy builds upon the existing technology base and is designed to provide a roadmap that will direct short-term development along a productive, long-term path. It offers a way to permit the short-term development of incremental solutions to current problems while at the same time maximizing the likelihood that these incremental efforts can be recycled into a more comprehensive approach.
冯奇; 周雪忠; 黄厚宽; 张小平
2011-01-01
Trial-based value iteration is a class of efficient algorithms to solve partially observable Markov decision process (POMDP), among which FSVI is one of the fastest algorithms. But the overhead of computing MDP value function by FSVI is not negligible for large-scale POMDP problems. In this paper, we propose a new value iteration method based on the shortest Hamiltonian path (shortest Hamiltonian path-based value iteration, SHP-VI). This method explores an optimal belief trajectory using the shortest Hamiltonian path resulting from ant colony optimization, and updates value function over the encountered belief states in reversed order. Compared with FSVI, the experimental results show that SHP-VI accelerates the computation of belief trajectory greatly in trial-based algorithms.%基于试探(trial-based)的值迭代算法是求解部分可观察Markov决策过程(partially observable Markov decision process,POMDP)模型的一类有效算法,其中FSVI算法是目前最快的算法之一.然而对于较大规模的POMDP问题,FSVI计算MDP值函数的时间是不容忽视的.提出一种基于最短哈密顿通路(shortest Hamiltonian path)的值迭代算法(shortest Hamiltonian path-based value iteration,SHP-VI).该方法用求解最短哈密顿通路问题的蚁群算法计算一条最优信念状态轨迹,然后在这些信念状态上反向更新值函数.通过与FSVI算法的实验比较,结果表明SHP-VI算法很大程度地提高了基于试探的算法计算信念状态轨迹的效率.
Anco, Stephen C.; Myrzakulov, R.
2010-10-01
A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrödinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrödinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.
Efficient stochastic thermostatting of path integral molecular dynamics
Ceriotti, Michele; Parrinello, Michele; Markland, Thomas E.; Manolopoulos, David E.
2010-09-01
The path integral molecular dynamics (PIMD) method provides a convenient way to compute the quantum mechanical structural and thermodynamic properties of condensed phase systems at the expense of introducing an additional set of high frequency normal modes on top of the physical vibrations of the system. Efficiently sampling such a wide range of frequencies provides a considerable thermostatting challenge. Here we introduce a simple stochastic path integral Langevin equation (PILE) thermostat which exploits an analytic knowledge of the free path integral normal mode frequencies. We also apply a recently developed colored noise thermostat based on a generalized Langevin equation (GLE), which automatically achieves a similar, frequency-optimized sampling. The sampling efficiencies of these thermostats are compared with that of the more conventional Nosé-Hoover chain (NHC) thermostat for a number of physically relevant properties of the liquid water and hydrogen-in-palladium systems. In nearly every case, the new PILE thermostat is found to perform just as well as the NHC thermostat while allowing for a computationally more efficient implementation. The GLE thermostat also proves to be very robust delivering a near-optimum sampling efficiency in all of the cases considered. We suspect that these simple stochastic thermostats will therefore find useful application in many future PIMD simulations.
Path integral approach to the quantum fidelity amplitude.
Vaníček, Jiří; Cohen, Doron
2016-06-13
The Loschmidt echo is a measure of quantum irreversibility and is determined by the fidelity amplitude of an imperfect time-reversal protocol. Fidelity amplitude plays an important role both in the foundations of quantum mechanics and in its applications, such as time-resolved electronic spectroscopy. We derive an exact path integral formula for the fidelity amplitude and use it to obtain a series of increasingly accurate semiclassical approximations by truncating an exact expansion of the path integral exponent. While the zeroth-order expansion results in a remarkably simple, yet non-trivial approximation for the fidelity amplitude, the first-order expansion yields an alternative derivation of the so-called 'dephasing representation,' circumventing the use of a semiclassical propagator as in the original derivation. We also obtain an approximate expression for fidelity based on the second-order expansion, which resolves several shortcomings of the dephasing representation. The rigorous derivation from the path integral permits the identification of sufficient conditions under which various approximations obtained become exact. © 2016 The Authors.
Baskan, O; Speetjens, M F M; Metcalfe, G; Clercx, H J H
2015-10-01
Countless theoretical/numerical studies on transport and mixing in two-dimensional (2D) unsteady flows lean on the assumption that Hamiltonian mechanisms govern the Lagrangian dynamics of passive tracers. However, experimental studies specifically investigating said mechanisms are rare. Moreover, they typically concern local behavior in specific states (usually far away from the integrable state) and generally expose this indirectly by dye visualization. Laboratory experiments explicitly addressing the global Hamiltonian progression of the Lagrangian flow topology entirely from integrable to chaotic state, i.e., the fundamental route to efficient transport by chaotic advection, appear non-existent. This motivates our study on experimental visualization of this progression by direct measurement of Poincaré sections of passive tracer particles in a representative 2D time-periodic flow. This admits (i) accurate replication of the experimental initial conditions, facilitating true one-to-one comparison of simulated and measured behavior, and (ii) direct experimental investigation of the ensuing Lagrangian dynamics. The analysis reveals a close agreement between computations and observations and thus experimentally validates the full global Hamiltonian progression at a great level of detail.
Baskan, O.; Clercx, H. J. H [Fluid Dynamics Laboratory, Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven (Netherlands); Speetjens, M. F. M. [Energy Technology Laboratory, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven (Netherlands); Metcalfe, G. [Commonwealth Scientific and Industrial Research Organisation, Melbourne, Victoria 3190 (Australia); Swinburne University of Technology, Department of Mechanical Engineering, Hawthorn VIC 3122 (Australia)
2015-10-15
Countless theoretical/numerical studies on transport and mixing in two-dimensional (2D) unsteady flows lean on the assumption that Hamiltonian mechanisms govern the Lagrangian dynamics of passive tracers. However, experimental studies specifically investigating said mechanisms are rare. Moreover, they typically concern local behavior in specific states (usually far away from the integrable state) and generally expose this indirectly by dye visualization. Laboratory experiments explicitly addressing the global Hamiltonian progression of the Lagrangian flow topology entirely from integrable to chaotic state, i.e., the fundamental route to efficient transport by chaotic advection, appear non-existent. This motivates our study on experimental visualization of this progression by direct measurement of Poincaré sections of passive tracer particles in a representative 2D time-periodic flow. This admits (i) accurate replication of the experimental initial conditions, facilitating true one-to-one comparison of simulated and measured behavior, and (ii) direct experimental investigation of the ensuing Lagrangian dynamics. The analysis reveals a close agreement between computations and observations and thus experimentally validates the full global Hamiltonian progression at a great level of detail.
BACKWARD ERROR ANALYSIS OF SYMPLECTIC INTEGRATORS FOR LINEAR SEPARABLE HAMILTONIAN SYSTEMS
Peter G(o)rtz
2002-01-01
Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge Kutta, and Runge-Kutta-Nystrom methods applied to the simple Hamiltonian system p = -vq, q = κp are studied. Some new results in connection with P-stability are pre sented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is the exact solution of a perturbed Hamiltonian system at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.
Butko, Yana A., E-mail: yanabutko@yandex.ru, E-mail: kinderknecht@math.uni-sb.de [Bauman Moscow State Technical University, 2nd Baumanskaya street, 5, Moscow 105005, Russia and University of Saarland, Postfach 151150, D-66041 Saarbrücken (Germany); Grothaus, Martin, E-mail: grothaus@mathematik.uni-kl.de [University of Kaiserslautern, 67653 Kaiserslautern (Germany); Smolyanov, Oleg G., E-mail: Smolyanov@yandex.ru [Lomonosov Moscow State University, Vorob’evy gory 1, Moscow 119992 (Russian Federation)
2016-02-15
Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.
Koda, Shin-Ichi; Takatsuka, Kazuo
2011-03-01
The coherent path integral is generalized such that the identity operator represented in a complete (actually overcomplete) set of the coherent states with the “time-variable” exponents are inserted between two consecutive short-time propagators. Since such a complete set of any given exponent can constitute the identity operator, the exponent may be varied from time to time without loss of generality as long as it is set common to all the Gaussians. However, a finite truncation of the coherent state expansion should result in different values of the propagator depending on the choice of the exponents. Furthermore, approximation methodology to treat with the exact propagator can also depend on this choice, and thereby many different semiclassical propagators may emerge from these combinations. Indeed, we show that the well-known semiclassical propagators such as those of Van Vleck, Herman-Kluk, Heller’s thawed Gaussian, and many others can be derived in a systematic manner, which enables one to comprehend these semiclassical propagators from a unified point of view. We are particularly interested in our generalized form of the Herman-Kluk propagator, since the relative accuracy of this propagator has been well established by Kay, and since, nevertheless, its derivation was not necessarily clear. Thus our generalized Herman-Kluk propagator replaces the classical Hamiltonian with a Gaussian averaged quantum Hamiltonian, generating non-Newtonian trajectories. We perform a numerical test to assess the quality of such a family of generalized Herman-Kluk propagators and find that the original Herman-Kluk gives an accurate result. The reason why this has come about is also discussed.
Direct path integral estimators for isotope fractionation ratios
Cheng, Bingqing
2014-01-01
Fractionation of isotopes among distinct molecules or phases is a quantum effect which is often exploited to obtain insights on reaction mechanisms, biochemical, geochemical and atmospheric phenomena. Accurate evaluation of isotope ratios in atomistic simulations is challenging, because one needs to perform a thermodynamic integration with respect to the isotope mass, along with time-consuming path integral calculations. By re-formulating the problem as a particle exchange in the ring polymer partition function, we derive new estimators giving direct access to the differential partitioning of isotopes, which can simplify the calculations by avoiding thermodynamic integration. We demonstrate the efficiency of these estimators by applying them to investigate the isotope fractionation ratios in the gas-phase Zundel cation, and in a few simple hydrocarbons.
Ryan, M.
1972-01-01
The study of cosmological models by means of equations of motion in Hamiltonian form is considered. Hamiltonian methods applied to gravity seem to go back to Rosenfeld (1930), who constructed a quantum-mechanical Hamiltonian for linearized general relativity theory. The first to notice that cosmologies provided a simple model in which to demonstrate features of Hamiltonian formulation was DeWitt (1967). Applications of the ADM formalism to homogeneous cosmologies are discussed together with applications of the Hamiltonian formulation, giving attention also to Bianchi-type universes. Problems involving the concept of superspace and techniques of quantization are investigated.
Path integrals and symmetry breaking for optimal control theory
Kappen, H J
2005-01-01
This paper considers linear-quadratic control of a non-linear dynamical system subject to arbitrary cost. I show that for this class of stochastic control problems the non-linear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation. The transformation is similar to the transformation used to relate the classical Hamilton-Jacobi equation to the Schr\\"odinger equation. As a result of the linearity, the usual backward computation can be replaced by a forward diffusion process, that can be computed by stochastic integration or by the evaluation of a path integral. It is shown, how in the deterministic limit the PMP formalism is recovered. The significance of the path integral approach is that it forms the basis for a number of efficient computational methods, such as MC sampling, the Laplace approximation and the variational approximation. We show the effectiveness of the first two methods in number of examples. Examples are given that show the qualitative difference between stochastic and d...
Potential theory, path integrals and the Laplacian of the indicator
Lange, Rutger-Jan
2012-11-01
This paper links the field of potential theory — i.e. the Dirichlet and Neumann problems for the heat and Laplace equation — to that of the Feynman path integral, by postulating the following seemingly ill-defined potential: V(x):=∓ {{σ^2}}/2nabla_x^2{1_{{xin D}}} where the volatility is the reciprocal of the mass (i.e. m = 1/ σ 2) and ħ = 1. The Laplacian of the indicator can be interpreted using the theory of distributions: it is the d-dimensional analogue of the Dirac δ'-function, which can formally be defined as partial_x^2{1_{x>0 }} . We show, first, that the path integral's perturbation series (or Born series) matches the classical single and double boundary layer series of potential theory, thereby connecting two hitherto unrelated fields. Second, we show that the perturbation series is valid for all domains D that allow Green's theorem (i.e. with a finite number of corners, edges and cusps), thereby expanding the classical applicability of boundary layers. Third, we show that the minus (plus) in the potential holds for the Dirichlet (Neumann) boundary condition; showing for the first time a particularly close connection between these two classical problems. Fourth, we demonstrate that the perturbation series of the path integral converges as follows:Table Float="No" ID="Taba"> mode of convergence absorbed propagator reflected propagator convex domain alternating monotone concave domain monotone alternating Table> We also discuss the third boundary problem (which poses Robin boundary conditions) and discuss an extension to moving domains.
Mastromatteo, Michael; Jackson, Bret
2013-11-21
Electronic structure methods based on density functional theory are used to construct a reaction path Hamiltonian for CH4 dissociation on the Ni(100) and Ni(111) surfaces. Both quantum and quasi-classical trajectory approaches are used to compute dissociative sticking probabilities, including all molecular degrees of freedom and the effects of lattice motion. Both approaches show a large enhancement in sticking when the incident molecule is vibrationally excited, and both can reproduce the mode specificity observed in experiments. However, the quasi-classical calculations significantly overestimate the ground state dissociative sticking at all energies, and the magnitude of the enhancement in sticking with vibrational excitation is much smaller than that computed using the quantum approach or observed in the experiments. The origin of this behavior is an unphysical flow of zero point energy from the nine normal vibrational modes into the reaction coordinate, giving large values for reaction at energies below the activation energy. Perturbative assumptions made in the quantum studies are shown to be accurate at all energies studied.
Path-integral formula for local thermal equilibrium
Hongo, Masaru
2016-01-01
We develop a complete path-integral formulation of relativistic quantum fields in local thermal equilibrium, which brings about the emergence of thermally induced curved spacetime. The resulting action is shown to have full diffeomorphism invariance and gauge invariance in thermal spacetime with imaginary-time independent backgrounds. This leads to the notable symmetry properties of emergent thermal spacetime: Kaluza-Klein gauge symmetry, spatial diffeomorphism symmetry, and gauge symmetry. A thermodynamic potential in local thermal equilibrium, or the so-called Masseiu-Planck functional, is identified as a generating functional for conserved currents such as the energy-momentum tensor and the electric current.
Remarks on the Origin of Path Integration: Einstein and Feynman
Sauer, Tilman
2008-01-01
I offer some historical comments about the origins of Feynman's path integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's thesis supervisor John Wheeler, it is argued that, contrary to what one might expect, the significance of the interaction between Einstein and Feynman pertained to a critique of classical field theory, rather than to a direct critique of quantum mechanics itself. Nevertheless, the critical perspective on classical field theory became a motivation and point of departure for Feynman's space-time approach to non-relativistic quantum mechanics.
Remarks on the Origin of Path Integration:. Einstein and Feynman
Sauer, T.
2008-11-01
I offer some historical comments about the origins of Feynman's path-integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's thesis supervisor John Wheeler, it is argued that, contrary to what one might expect, the significance of the interaction between Einstein and Feynman pertained to a critique of classical field theory, rather than to a direct critique of quantum mechanics itself. Nevertheless, the critical perspective on classical field theory became a motivation and point of departure for Feynman's space-time approach to non-relativistic quantum mechanics.
Self-gravitating stellar collapse: explicit geodesics and path integration
Jayashree Balakrishna
2016-11-01
Full Text Available We extend the work of Oppenheimer-Synder to model the gravitational collapse of a star to a black hole by including quantum mechanical effects. We first derive closed-form solutions for classical paths followed by a particle on the surface of the collapsing star in Schwarzschild and Kruskal coordinates for space-like, time-like and light-like geodesics. We next present an application of these paths to model the collapse of ultra-light dark matter particles, which necessitates incorporating quantum effects. To do so we treat a particle on the surface of the star as a wavepacket and integrate over all possible paths taken by the particle. The waveform is computed in Schwarzschild coordinates and found to exhibit an ingoing and an outgoing component, where the former contains the probability of collapse, while the latter contains the probability that the star will disperse. These calculations pave the way for investigating the possibility of quantum collapse that does not lead to black hole formation as well as for exploring the nature of the wavefunction inside r = 2M.
A Hamiltonian treatment of stimulated Brillouin scattering in nanoscale integrated waveguides
Sipe, J E
2015-01-01
We present a multimode Hamiltonian formulation for the problem of opto-acoustic interactions in optical waveguides. We establish a Hamiltonian representation of the acoustic field and then introduce a full system with a simple opto-acoustic coupling that includes both photoelastic/electrostrictive and radiation pressure/moving boundary effects. The Heisenberg equations of motion are used to obtain coupled mode equations for quantized envelope operators for the optical and acoustic fields. We show that the coupling coefficients obtained coincide with those established earlier, but our formalism provides a much simpler demonstration of the connection between radiation pressure and moving boundary effects than in previous work [C. Wolff et al, Physical Review A 92, 013836 (2015)].
2010-10-21
transfor- mation, the hybrid potential VAB remains well-defined. Therefore, the associated reversible work can be computed. This reversible work formally...corresponds to changing the potential of the system VAB from the state that represents the original molecule A to the state that corresponds to molecule...constructed. Typically, the hybrid Hamiltonian of the system (HAB) that includes the potential VAB is linearly interpolated between the end points, molecules
CHEN Tong; WU Ning; YU Yue
2011-01-01
We have developed a path integral formalism of the quantum mechanics in the rotating frame of reference, and proposed a path integral description of spin degrees of freedom, which is connected to the Schwinger bosons realization of the angular momenta. We
N-slit interference: Path integrals, Bohmian trajectories
Sbitnev, Valeriy I
2010-01-01
Path integrals give a possibility to compute in details routes of particles from particle sources through slit gratings and further to detectors. The path integral for a particle passing through the Gaussian slit results in the Gaussian wavepacket. The wavepackets prepared on N slits and superposed together give rise to interference pattern in the near-field zone. It transforms to diffraction in the far-field zone represented by divergent principal rays, at that all rays are partitioned from each other by (N-2) subsidiary rays. The Bohmian trajectories in the near-field zone of N-slit gratings show wavy behavior. And they become straight in the far-field zone. The trajectories show zigzag behavior on the interference Talbot carpet (ratio of particle wavelength to a distance between slits are much smaller than 1 and N >> 1). Namely, the trajectories prefer to pass through caustics and avoid lacunae, i.e., places with small probability densities. Monochromatic thermal neutrons (wavelength=0.5 nm) simulate radia...
A note on the path integral representation for Majorana fermions
Greco, Andrés
2016-04-01
Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev-Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.
Path-integral molecular dynamics simulation of 3C-SiC
Ramírez, Rafael; Herrero, Carlos P.; Hernández, Eduardo R.; Cardona, Manuel
2008-01-01
Molecular dynamics simulations of 3C-SiC have been performed as a function of pressure and temperature. These simulations treat both electrons and atomic nuclei by quantum mechanical methods. While the electronic structure of the solid is described by an efficient tight-binding Hamiltonian, the nuclei dynamics is treated by the path-integral formulation of statistical mechanics. To assess the relevance of nuclear quantum effects, the results of quantum simulations are compared to others where either the Si nuclei, the C nuclei, or both atomic nuclei are treated as classical particles. We find that the experimental thermal expansion of 3C-SiC is realistically reproduced by our simulations. The calculated bulk modulus of 3C-SiC and its pressure derivative at room temperature show also good agreement with the available experimental data. The effect of the electron-phonon interaction on the direct electronic gap of 3C-SiC has been calculated as a function of temperature and related to results obtained for bulk diamond and Si. Comparison to available experimental data shows satisfactory agreement, although we observe that the employed tight-binding model tends to overestimate the magnitude of the electron-phonon interaction. The effect of treating the atomic nuclei as classical particles on the direct gap of 3C-SiC has been assessed. We find that nonlinear quantum effects related to the atomic masses are particularly relevant at temperatures below 250K .
Dvornikov, Maxim
2012-01-01
We study massive 1/2-spin particles in various external backgrounds keeping in mind applications to neutrino physics. We are mainly interested in massive Majorana (Weyl) fields. However, massive neutral Dirac particles have been also considered. We have formulated classical Lagrangian theory of the massive Weyl field in terms of Grassmann-odd two component spinors. Then we construct the Hamiltonian formulation of such a theory, which turns out to be a theory with second-class constraints. Using this formulation we canonically quantize the massive free Weyl field. We derive propagators of the Weyl field and relate them to the propagator of a massive Dirac particle. We also study the massive Weyl particles propagating in the background mater. We find the path integral representation for the propagator of such a field as well as the corresponding pseudoclassical particle action. The massless limit of the Weyl field interacting with the matter is considered and compared with results of other works. Finally, the p...
Atmospheric Refraction Path Integrals in Ground-Based Interferometry
Mathar, R J
2004-01-01
The basic effect of the earth's atmospheric refraction on telescope operation is the reduction of the true zenith angle to the apparent zenith angle, associated with prismatic aberrations due to the dispersion in air. If one attempts coherent superposition of star images in ground-based interferometry, one is in addition interested in the optical path length associated with the refracted rays. In a model of a flat earth, the optical path difference between these is not concerned as the translational symmetry of the setup means no net effect remains. Here, I evaluate these interferometric integrals in the more realistic arrangement of two telescopes located on the surface of a common earth sphere and point to a star through an atmosphere which also possesses spherical symmetry. Some focus is put on working out series expansions in terms of the small ratio of the baseline over the earth radius, which allows to bypass some numerics which otherwise is challenged by strong cancellation effects in building the opti...
Complex Nonlinearity Chaos, Phase Transitions, Topology Change and Path Integrals
Ivancevic, Vladimir G
2008-01-01
Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals is a book about prediction & control of general nonlinear and chaotic dynamics of high-dimensional complex systems of various physical and non-physical nature and their underpinning geometro-topological change. The book starts with a textbook-like expose on nonlinear dynamics, attractors and chaos, both temporal and spatio-temporal, including modern techniques of chaos–control. Chapter 2 turns to the edge of chaos, in the form of phase transitions (equilibrium and non-equilibrium, oscillatory, fractal and noise-induced), as well as the related field of synergetics. While the natural stage for linear dynamics comprises of flat, Euclidean geometry (with the corresponding calculation tools from linear algebra and analysis), the natural stage for nonlinear dynamics is curved, Riemannian geometry (with the corresponding tools from nonlinear, tensor algebra and analysis). The extreme nonlinearity – chaos – corresponds to th...
Spin And Curvature In The Worldline Path Integral
Dilkes, F A
1999-01-01
Several aspects of worldline path-integrals are discussed in the context of quantum field theory. It is shown how “near-diagonal” elements of the Seeley-Gilkey coefficients can be computed both in the presence of an arbitrary Riemann metric, a gauge- potential and a scalar potential. These are connected with derivative expansions and ultraviolet properties of field theories. Recently resolved subtleties connected with curvature and curvilinear coordinate systems are taken into account and non-covariant terms in the worldline action are shown to be a necessary ingredient for a correct expansion. This is contrasted with the success of older formal methods. Rudimentary symbolic algebra is shown to be a practical tool for tracking the combinatorics of higher-order calculations. A significant generalization of the Parker-Toms conjecture and the form of the single-particle effective action in curved space results. Some aspects of spin are also considered and it is shown how the spinning particle...
Thermal momentum distribution from path integrals with shifted boundary conditions
Giusti, Leonardo
2011-01-01
For a thermal field theory formulated in the grand canonical ensemble, the distribution of the total momentum is an observable characterizing the thermal state. We show that its cumulants are related to thermodynamic potentials. In a relativistic system for instance, the thermal variance of the total momentum is a direct measure of the enthalpy. We relate the generating function of the cumulants to the ratio of (a) a partition function expressed as a Matsubara path integral with shifted boundary conditions in the compact direction, and (b) the ordinary partition function. In this form the generating function is well suited for Monte-Carlo evaluation, and the cumulants can be extracted straightforwardly. We test the method in the SU(3) Yang-Mills theory and obtain the entropy density at three different temperatures.
High-resolution path-integral development of financial options
Ingber, L
2000-01-01
The Black-Scholes theory of option pricing has been considered for many years as an important but very approximate zeroth-order description of actual market behavior. We generalize the functional form of the diffusion of these systems and also consider multi-factor models including stochastic volatility. Daily Eurodollar futures prices and implied volatilities are fit to determine exponents of functional behavior of diffusions using methods of global optimization, Adaptive Simulated Annealing (ASA), to generate tight fits across moving time windows of Eurodollar contracts. These short-time fitted distributions are then developed into long-time distributions using a robust non-Monte Carlo path-integral algorithm, PATHINT, to generate prices and derivatives commonly used by option traders.
Quantum Field Theory: From Operators to Path Integrals
Huang, Kerson
1998-07-01
A unique approach to quantum field theory, with emphasis on the principles of renormalization Quantum field theory is frequently approached from the perspective of particle physics. This book adopts a more general point of view and includes applications of condensed matter physics. Written by a highly respected writer and researcher, it first develops traditional concepts, including Feynman graphs, before moving on to key topics such as functional integrals, statistical mechanics, and Wilson's renormalization group. The connection between the latter and conventional perturbative renormalization is explained. Quantum Field Theory is an exceptional textbook for graduate students familiar with advanced quantum mechanics as well as physicists with an interest in theoretical physics. It features: * Coverage of quantum electrodynamics with practical calculations and a discussion of perturbative renormalization * A discussion of the Feynman path integrals and a host of current subjects, including the physical approach to renormalization, spontaneous symmetry breaking and superfluidity, and topological excitations * Nineteen self-contained chapters with exercises, supplemented with graphs and charts
Theory of extreme correlations using canonical Fermions and path integrals
Shastry, B. Sriram, E-mail: sriram@physics.ucsc.edu
2014-04-15
The t–J model is studied using a novel and rigorous mapping of the Gutzwiller projected electrons, in terms of canonical electrons. The mapping has considerable similarity to the Dyson–Maleev transformation relating spin operators to canonical Bosons. This representation gives rise to a non Hermitian quantum theory, characterized by minimal redundancies. A path integral representation of the canonical theory is given. Using it, the salient results of the extremely correlated Fermi liquid (ECFL) theory, including the previously found Schwinger equations of motion, are easily rederived. Further, a transparent physical interpretation of the previously introduced auxiliary Greens function and the ‘caparison factor’, is obtained. The low energy electron spectral function in this theory, with a strong intrinsic asymmetry, is summarized in terms of a few expansion coefficients. These include an important emergent energy scale Δ{sub 0} that shrinks to zero on approaching the insulating state, thereby making it difficult to access the underlying very low energy Fermi liquid behavior. The scaled low frequency ECFL spectral function, related simply to the Fano line shape, has a peculiar energy dependence unlike that of a Lorentzian. The resulting energy dispersion obtained by maximization is a hybrid of a massive and a massless Dirac spectrum E{sub Q}{sup ∗}∼γQ−√(Γ{sub 0}{sup 2}+Q{sup 2}), where the vanishing of Q, a momentum type variable, locates the kink minimum. Therefore the quasiparticle velocity interpolates between (γ∓1) over a width Γ{sub 0} on the two sides of Q=0, implying a kink there that strongly resembles a prominent low energy feature seen in angle resolved photoemission spectra (ARPES) of cuprate materials. We also propose novel ways of analyzing the ARPES data to isolate the predicted asymmetry between particle and hole excitations. -- Highlights: •Spectral function of the Extremely Correlated Fermi Liquid theory at low energy.
Fluctuating paths and fields Festschrift Kleinert (Hagen)
Bachmann, M; Schmidt, H J; Janke, W
2001-01-01
This volume covers the following fields: path integrals, quantum field theory, variational perturbation theory, phase transitions and critical phenomena, topological defects, strings and membranes, gravitation and cosmology. Contents: Path Integrals and Quantum Mechanics: Semiclassical Quantum Mechanics: A Path-Integral Approach (B R Holstein); Conjecture on the Reality of Spectra of Non-Hermitian Hamiltonians (C M Bender et al.); Time-Transformation Approach to q -Deformed Objects (A Inomata); Characterizing Volume Forms (P Cartier et al.); Vassiliev Invariants and Functional Integration (L H
Path integrals, matter waves, and the double slit
Jones, Eric R.; Bach, Roger A.; Batelaan, Herman
2015-11-01
Basic explanations of the double slit diffraction phenomenon include a description of waves that emanate from two slits and interfere. The locations of the interference minima and maxima are determined by the phase difference of the waves. An optical wave, which has a wavelength λ and propagates a distance L, accumulates a phase of 2π L/λ . A matter wave, also having wavelength λ that propagates the same distance L, accumulates a phase of π L/λ , which is a factor of two different from the optical case. Nevertheless, in most situations, the phase difference, {{Δ }}\\varphi , for interfering matter waves that propagate distances that differ by {{Δ }}L, is approximately 2π {{Δ }}L/λ , which is the same value computed in the optical case. The difference between the matter and optical case hinders conceptual explanations of diffraction from two slits based on the matter-optics analogy. In the following article we provide a path integral description for matter waves with a focus on conceptual explanation. A thought experiment is provided to illustrate the validity range of the approximation {{Δ }}\\varphi ≈ 2π {{Δ }}L/λ .
Nearest neighbor interaction in the Path Integral Renormalization Group method
de Silva, Wasanthi; Clay, R. Torsten
2014-03-01
The Path Integral Renormalization Group (PIRG) method is an efficient numerical algorithm for studying ground state properties of strongly correlated electron systems. The many-body ground state wave function is approximated by an optimized linear combination of Slater determinants which satisfies the variational principle. A major advantage of PIRG is that is does not suffer the Fermion sign problem of quantum Monte Carlo. Results are exact in the noninteracting limit and can be enhanced using space and spin symmetries. Many observables can be calculated using Wick's theorem. PIRG has been used predominantly for the Hubbard model with a single on-site Coulomb interaction U. We describe an extension of PIRG to the extended Hubbard model (EHM) including U and a nearest-neighbor interaction V. The EHM is particularly important in models of charge-transfer solids (organic superconductors) and at 1/4-filling drives a charge-ordered state. The presence of lattice frustration also makes studying these systems difficult. We test the method with comparisons to small clusters and long one dimensional chains, and show preliminary results for a coupled-chain model for the (TMTTF)2X materials. This work was supported by DOE grant DE-FG02-06ER46315.
Theory of Atom Optics: Feynman's Path Integral Approach
DENG Lü-bi
2006-01-01
The present theory of atom optics is established mainly on the Schr(o)dinger equations or the matrix mechanics equation.The authors present a new theoretical formulation of atom optics: Feynman's path integral theory.Its advantage is that one can describe the diffraction and interference of atoms passing through slits (or grating),apertures,and standing wave laser field in Earth's gravitational field by using a type of wave function and calculation is simple.For this reason,we derive the wave functions of particles in the following configurations: single slit (and slit with the van der Waals interaction),double slit,N slit,rectangular aperture,circular aperture,the Mach-Zehndertype interferometer,the interferometer with the Raman beams,the Sagnac effect,the Aharonov-Casher effect,the Kapitza-Dirac diffraction effect,and the Aharonov-Bohm effect.The authors give a wave function of the state of particles on the screen in abovementioned configurations.Our formulas show good agreement with present experimental measurements.
Quantum effects in graphene monolayers: Path-integral simulations
Herrero, Carlos P.; Ramírez, Rafael
2016-12-01
Path-integral molecular dynamics (PIMD) simulations have been carried out to study the influence of quantum dynamics of carbon atoms on the properties of a single graphene layer. Finite-temperature properties were analyzed in the range from 12 to 2000 K, by using the LCBOPII effective potential. To assess the magnitude of quantum effects in structural and thermodynamic properties of graphene, classical molecular dynamics simulations have been also performed. Particular emphasis has been laid on the atomic vibrations along the out-of-plane direction. Even though quantum effects are present in these vibrational modes, we show that at any finite temperature classical-like motion dominates over quantum delocalization, provided that the system size is large enough. Vibrational modes display an appreciable anharmonicity, as derived from a comparison between kinetic and potential energies of the carbon atoms. Nuclear quantum effects are found to be appreciable in the interatomic distance and layer area at finite temperatures. The thermal expansion coefficient resulting from PIMD simulations vanishes in the zero-temperature limit, in agreement with the third law of thermodynamics.
Momentum and Hamiltonian in Complex Action Theory
Nagao, Keiichi
2011-01-01
In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view. In arXiv:1104.3381[quant-ph], introducing a philosophy to keep the analyticity in parameter variables of FPI and defining a modified set of complex conjugate, hermitian conjugates and bras, we have extended $| q >$ and $| p >$ to complex $q$ and $p$ so that we can deal with a complex coordinate $q$ and a complex momentum $p$. After reviewing them briefly, we describe in terms of the newly introduced devices the time development of a $\\xi$-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator $\\hat{p}$, in FPI with a starting Lagrangian. Solving the eigenvalue problem, we derive the momentum and Hamiltonian. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum again via the saddle point for $p$. This study confirms that the momentum and Hamiltonian in the CAT have t...
Path integral approach to eikonal and next-to-eikonal exponentiation
Laenen, E.; Stavenga, G.; White, C.D.
2009-01-01
We approach the issue of exponentiation of soft gauge boson corrections to scattering amplitudes from a path integral point of view. We show that if one represents the amplitude as a first quantized path integral in a mixed coordinate-momentum space representation, a charged particle interacting wit
Path Integral and Solutions of the Constraint Equations The Case of Reducible Gauge Theories
Ferraro, R; Puchin, M
1994-01-01
It is shown that the BRST path integral for reducible gauge theories, with appropriate boundary conditions on the ghosts, is a solution of the constraint equations. This is done by relating the BRST path integral to the kernel of the evolution operator projected on the physical subspace.
On the coordinate (in)dependence of the formal path integral
Johnson-Freyd, Theo
. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the “formal” or “Feynman-diagrammatic” path integral construction. We then outline a proof of the following claim: the formal......When path integrals are discussed in quantum field theory, it is almost always assumed that the fields take values in a vector bundle. When the fields are instead valued in a possibly-curved fiber bundle, the independence of the formal path integral on the coordinates becomes much less obvious...... path integral does not depend on the choice of coordinates, but only on a choice of fiberwise volume form. Our outline is an honest proof when the formal path integral is defined without ultraviolet divergences....
Self-Dual Yang-Mills and the Hamiltonian Structures of Integrable Systems
Schiff, J
1992-01-01
In recent years it has been shown that many, and possibly all, integrable systems can be obtained by dimensional reduction of self-dual Yang-Mills. I show how the integrable systems obtained this way naturally inherit bihamiltonian structure. I also present a simple, gauge-invariant formulation of the self-dual Yang-Mills hierarchy proposed by several authors, and I discuss the notion of gauge equivalence of integrable systems that arises from the gauge invariance of the self-duality equations (and their hierarchy); this notion of gauge equivalence may well be large enough to unify the many diverse existing notions.
Orsucci, Davide [Scuola Normale Superiore, I-56126 Pisa (Italy); Burgarth, Daniel [Department of Mathematics, Aberystwyth University, Aberystwyth SY23 3BZ (United Kingdom); Facchi, Paolo; Pascazio, Saverio [Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari (Italy); INFN, Sezione di Bari, I-70126 Bari (Italy); Nakazato, Hiromichi; Yuasa, Kazuya [Department of Physics, Waseda University, Tokyo 169-8555 (Japan); Giovannetti, Vittorio [NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa (Italy)
2015-12-15
The problem of Hamiltonian purification introduced by Burgarth et al. [Nat. Commun. 5, 5173 (2014)] is formalized and discussed. Specifically, given a set of non-commuting Hamiltonians (h{sub 1}, …, h{sub m}) operating on a d-dimensional quantum system ℋ{sub d}, the problem consists in identifying a set of commuting Hamiltonians (H{sub 1}, …, H{sub m}) operating on a larger d{sub E}-dimensional system ℋ{sub d{sub E}} which embeds ℋ{sub d} as a proper subspace, such that h{sub j} = PH{sub j}P with P being the projection which allows one to recover ℋ{sub d} from ℋ{sub d{sub E}}. The notions of spanning-set purification and generator purification of an algebra are also introduced and optimal solutions for u(d) are provided.
Meeds, E.; Leenders, R.; Welling, M.; Meila, M.; Heskes, T.
2015-01-01
Approximate Bayesian computation (ABC) is a powerful and elegant framework for performing inference in simulation-based models. However, due to the difficulty in scaling likelihood estimates, ABC remains useful for relatively lowdimensional problems. We introduce Hamiltonian ABC (HABC), a set of lik
Gülden Gün
2013-01-01
Full Text Available We analyze Noether and -symmetries of the path equation describing the minimum drag work. First, the partial Lagrangian for the governing equation is constructed, and then the determining equations are obtained based on the partial Lagrangian approach. For specific altitude functions, Noether symmetry classification is carried out and the first integrals, conservation laws and group invariant solutions are obtained and classified. Then, secondly, by using the mathematical relationship with Lie point symmetries we investigate -symmetry properties and the corresponding reduction forms, integrating factors, and first integrals for specific altitude functions of the governing equation. Furthermore, we apply the Jacobi last multiplier method as a different approach to determine the new forms of -symmetries. Finally, we compare the results obtained from different classifications.
Power Series Expansion of Propagator for Path Integral and Its Applications
无
2007-01-01
In this paper we obtain a propagator of path integral for a harmonic oscillator and a driven harmonic oscillator by using the power series expansion. It is shown that our result for the harmonic oscillator is more exact than the previous one obtained with other approximation methods. By using the same method, we obtain a propagator of path integral for the driven harmonic oscillator, which does not have any exact expansion. The more exact propagators may improve the path integral results for these systems.
A systematic study of finite BRST-BFV transformations in generalized Hamiltonian formalism
Batalin, Igor A.; Lavrov, Peter M.; Tyutin, Igor V.
2014-09-01
We study systematically finite BRST-BFV transformations in the generalized Hamiltonian formalism. We present explicitly their Jacobians and the form of a solution to the compensation equation determining the functional field dependence of finite Fermionic parameters, necessary to generate an arbitrary finite change of gauge-fixing functions in the path integral.
A systematic study of finite BRST-BFV transformations in generalized Hamiltonian formalism
Batalin, Igor A; Tyutin, Igor V
2014-01-01
We study systematically finite BRST-BFV transformations in the generalized Hamiltonian formalism. We present explicitly their Jacobians and the form of a solution to the compensation equation determining the functional field dependence of finite Fermionic parameters, necessary to generate arbitrary finite change of gauge-fixing functions in the path integral.
Batalin, Igor A; Tyutin, Igor V
2014-01-01
We study systematically finite BRST-BFV transformations in $Sp(2)$-extended generalized Hamiltonian formalism. We present explicitly their Jacobians and the form of a solution to the compensation equation determining the functional field dependence of finite Fermionic parameters, necessary to generate arbitrary finite change of gauge-fixing functions in the path integral.
Batalin, Igor A.; Lavrov, Peter M.; Tyutin, Igor V.
2014-09-01
We study systematically finite BRST-BFV transformations in Sp(2)-extended generalized Hamiltonian formalism. We present explicitly their Jacobians and the form of a solution to the compensation equation determining the functional field dependence of finite Fermionic parameters, necessary to generate arbitrary finite change of gauge-fixing functions in the path integral.
Piloting and Path Integration within and across Boundaries
Mou, Weimin; Wang, Lin
2015-01-01
Three experiments investigated whether navigation is less efficient across boundaries than within boundaries. In an immersive virtual environment, participants learned objects' locations in a large room or a small room. Participants then pointed to the objects' original locations after physically walking a circuitous path without vision.…
Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac
Fine, Dana
2016-01-01
Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N=1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.
A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat
Liu, Jian; Li, Dezhang; Liu, Xinzijian
2016-07-01
We introduce a novel simple algorithm for thermostatting path integral molecular dynamics (PIMD) with the Langevin equation. The staging transformation of path integral beads is employed for demonstration. The optimum friction coefficients for the staging modes in the free particle limit are used for all systems. In comparison to the path integral Langevin equation thermostat, the new algorithm exploits a different order of splitting for the phase space propagator associated to the Langevin equation. While the error analysis is made for both algorithms, they are also employed in the PIMD simulations of three realistic systems (the H2O molecule, liquid para-hydrogen, and liquid water) for comparison. It is shown that the new thermostat increases the time interval of PIMD by a factor of 4-6 or more for achieving the same accuracy. In addition, the supplementary material shows the error analysis made for the algorithms when the normal-mode transformation of path integral beads is used.
A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat.
Liu, Jian; Li, Dezhang; Liu, Xinzijian
2016-07-14
We introduce a novel simple algorithm for thermostatting path integral molecular dynamics (PIMD) with the Langevin equation. The staging transformation of path integral beads is employed for demonstration. The optimum friction coefficients for the staging modes in the free particle limit are used for all systems. In comparison to the path integral Langevin equation thermostat, the new algorithm exploits a different order of splitting for the phase space propagator associated to the Langevin equation. While the error analysis is made for both algorithms, they are also employed in the PIMD simulations of three realistic systems (the H2O molecule, liquid para-hydrogen, and liquid water) for comparison. It is shown that the new thermostat increases the time interval of PIMD by a factor of 4-6 or more for achieving the same accuracy. In addition, the supplementary material shows the error analysis made for the algorithms when the normal-mode transformation of path integral beads is used.
Feng, Ju; Ying, Zu-Guang; Zhu, Wei-Qiu
2012-01-01
A minimax stochastic optimal semi-active control strategy for stochastically excited quasi-integrable Hamiltonian systems with parametric uncertainty by using magneto-rheological (MR) dampers is proposed. Firstly, the control problem is formulated as an n-degree-of-freedom (DOF) controlled......, uncertain quasi-integrable Hamiltonian system and the control forces produced by MR dampers are split into the passive part and the semi-active part. Then the passive part is incorporated into the uncontrolled system. After that, the stochastic optimal semi-active control problem is solved by applying...... the minimax stochastic optimal control strategy based on the stochastic averaging method and stochastic differential game. The worst-case disturbances and the optimal controls are obtained by the minimax dynamical programming equation with the constraints of disturbance bounds and MR damper dynamics. Finally...
A variational path integral molecular dynamics study of a solid helium-4
Miura, Shinichi
2011-01-01
In the present study, a variational path integral molecular dynamics method developed by the author [Chem. Phys. Lett. 482 (2009) 165] is applied to a solid helium-4 in the ground state. The method is a molecular dynamics algorithm for a variational path integral method which can be used to generate the exact ground state numerically. The solid state is shown to successfully be realized by the method, although a poor trial wavefunction that cannot describe the solid state is used.
Which way and how far? Tracking of translation and rotation information for human path integration.
Chrastil, Elizabeth R; Sherrill, Katherine R; Hasselmo, Michael E; Stern, Chantal E
2016-10-01
Path integration, the constant updating of the navigator's knowledge of position and orientation during movement, requires both visuospatial knowledge and memory. This study aimed to develop a systems-level understanding of human path integration by examining the basic building blocks of path integration in humans. To achieve this goal, we used functional imaging to examine the neural mechanisms that support the tracking and memory of translational and rotational components of human path integration. Critically, and in contrast to previous studies, we examined movement in translation and rotation tasks with no defined end-point or goal. Navigators accumulated translational and rotational information during virtual self-motion. Activity in hippocampus, retrosplenial cortex (RSC), and parahippocampal cortex (PHC) increased during both translation and rotation encoding, suggesting that these regions track self-motion information during path integration. These results address current questions regarding distance coding in the human brain. By implementing a modified delayed match to sample paradigm, we also examined the encoding and maintenance of path integration signals in working memory. Hippocampus, PHC, and RSC were recruited during successful encoding and maintenance of path integration information, with RSC selective for tasks that required processing heading rotation changes. These data indicate distinct working memory mechanisms for translation and rotation, which are essential for updating neural representations of current location. The results provide evidence that hippocampus, PHC, and RSC flexibly track task-relevant translation and rotation signals for path integration and could form the hub of a more distributed network supporting spatial navigation. Hum Brain Mapp 37:3636-3655, 2016. © 2016 Wiley Periodicals, Inc.
Path integral methods for the dynamics of stochastic and disordered systems
Hertz, John A.; Roudi, Yasser; Sollich, Peter
2017-01-01
We review some of the techniques used to study the dynamics of disordered systems subject to both quenched and fast (thermal) noise. Starting from the Martin–Siggia–Rose/Janssen–De Dominicis–Peliti path integral formalism for a single variable stochastic dynamics, we provide a pedagogical survey...... in studying the disorder-averaged dynamics. Finally, we discuss the path integral approach for the case of hard Ising spins and review some recent developments in the dynamics of such kinetic Ising models....
Scalable fiber integrated source for higher-dimensional path-entangled photonic quNits
Schaeff, Christoph; Lapkiewicz, Radek; Fickler, Robert; Ramelow, Sven; Zeilinger, Anton
2012-01-01
Integrated photonic circuits offer the possibility for complex quantum optical experiments in higher-dimensional photonic systems. However, the advantages of integration and scalability can only be fully utilized with the availability of a source for higher-dimensional entangled photons. Here, a novel fiber integrated source for path-entangled photons in the telecom band at 1.55\\mum using only standard fiber technology is presented. Due to the special design the source shows good scalability towards higher-dimensional entangled photonic states (quNits), while path entanglement offers direct compatibility with on-chip path encoding. We present an experimental realization of a path-entangled two-qubit source. A very high quality of entanglement is verified by various measurements, i.a. a tomographic state reconstruction is performed leading to a background corrected fidelity of (99.45+-0.06)%. Moreover, we describe an easy method for extending our source to arbitrarily high dimensions.
Trouvé, Hélène; Couturier, Yves; Etheridge, Francis; Saint-Jean, Olivier; Somme, Dominique
2010-01-01
Background The literature on integration indicates the need for an enhanced theorization of institutional integration. This article proposes path dependence as an analytical framework to study the systems in which integration takes place. Purpose PRISMA proposes a model for integrating health and social care services for older adults. This model was initially tested in Quebec. The PRISMA France study gave us an opportunity to analyze institutional integration in France. Methods A qualitative approach was used. Analyses were based on semi-structured interviews with actors of all levels of decision-making, observations of advisory board meetings, and administrative documents. Results Our analyses revealed the complexity and fragmentation of institutional integration. The path dependency theory, which analyzes the change capacity of institutions by taking into account their historic structures, allows analysis of this situation. The path dependency to the Bismarckian system and the incomplete reforms of gerontological policies generate the coexistence and juxtaposition of institutional systems. In such a context, no institution has sufficient ability to determine gerontology policy and build institutional integration by itself. Conclusion Using path dependence as an analytical framework helps to understand the reasons why institutional integration is critical to organizational and clinical integration, and the complex construction of institutional integration in France. PMID:20689740
Hélène Trouvé
2010-06-01
Full Text Available Background: The literature on integration indicates the need for an enhanced theorization of institutional integration. This article proposes path dependence as an analytical framework to study the systems in which integration takes place.Purpose: PRISMA proposes a model for integrating health and social care services for older adults. This model was initially tested in Quebec. The PRISMA France study gave us an opportunity to analyze institutional integration in France.Methods: A qualitative approach was used. Analyses were based on semi-structured interviews with actors of all levels of decision-making, observations of advisory board meetings, and administrative documents.Results: Our analyses revealed the complexity and fragmentation of institutional integration. The path dependency theory, which analyzes the change capacity of institutions by taking into account their historic structures, allows analysis of this situation. The path dependency to the Bismarckian system and the incomplete reforms of gerontological policies generate the coexistence and juxtaposition of institutional systems. In such a context, no institution has sufficient ability to determine gerontology policy and build institutional integration by itself.Conclusion: Using path dependence as an analytical framework helps to understand the reasons why institutional integration is critical to organizational and clinical integration, and the complex construction of institutional integration in France.
Hélène Trouvé
2010-06-01
Full Text Available Background: The literature on integration indicates the need for an enhanced theorization of institutional integration. This article proposes path dependence as an analytical framework to study the systems in which integration takes place. Purpose: PRISMA proposes a model for integrating health and social care services for older adults. This model was initially tested in Quebec. The PRISMA France study gave us an opportunity to analyze institutional integration in France. Methods: A qualitative approach was used. Analyses were based on semi-structured interviews with actors of all levels of decision-making, observations of advisory board meetings, and administrative documents. Results: Our analyses revealed the complexity and fragmentation of institutional integration. The path dependency theory, which analyzes the change capacity of institutions by taking into account their historic structures, allows analysis of this situation. The path dependency to the Bismarckian system and the incomplete reforms of gerontological policies generate the coexistence and juxtaposition of institutional systems. In such a context, no institution has sufficient ability to determine gerontology policy and build institutional integration by itself. Conclusion: Using path dependence as an analytical framework helps to understand the reasons why institutional integration is critical to organizational and clinical integration, and the complex construction of institutional integration in France.
The development of path integration: combining estimations of distance and heading.
Smith, Alastair D; McKeith, Laura; Howard, Christina J
2013-12-01
Efficient daily navigation is underpinned by path integration, the mechanism by which we use self-movement information to update our position in space. This process is well understood in adulthood, but there has been relatively little study of path integration in childhood, leading to an underrepresentation in accounts of navigational development. Previous research has shown that calculation of distance and heading both tend to be less accurate in children as they are in adults, although there have been no studies of the combined calculation of distance and heading that typifies naturalistic path integration. In the present study, 5-year-olds and 7-year-olds took part in a triangle-completion task, where they were required to return to the start point of a multi-element path using only idiothetic information. Performance was compared to a sample of adult participants, who were found to be more accurate than children on measures of landing error, heading error, and distance error. Seven-year-olds were significantly more accurate than 5-year-olds on measures of landing error and heading error, although the difference between groups was much smaller for distance error. All measures were reliably correlated with age, demonstrating a clear development of path integration abilities within the age range tested. Taken together, these data make a strong case for the inclusion of path integration within developmental models of spatial navigational processing.
Quantum-classical path integral. I. Classical memory and weak quantum nonlocality.
Lambert, Roberto; Makri, Nancy
2012-12-14
We consider rigorous path integral descriptions of the dynamics of a quantum system coupled to a polyatomic environment, assuming that the latter is well approximated by classical trajectories. Earlier work has derived semiclassical or purely classical expressions for the influence functional from the environment, which should be sufficiently accurate for many situations, but the evaluation of quantum-(semi)classical path integral (QCPI) expressions has not been practical for large-scale simulation because the interaction with the environment introduces couplings nonlocal in time. In this work, we analyze the nature of the effects on a system from its environment in light of the observation [N. Makri, J. Chem. Phys. 109, 2994 (1998)] that true nonlocality in the path integral is a strictly quantum mechanical phenomenon. If the environment is classical, the path integral becomes local and can be evaluated in a stepwise fashion along classical trajectories of the free solvent. This simple "classical path" limit of QCPI captures fully the decoherence of the system via a classical mechanism. Small corrections to the classical path QCPI approximation may be obtained via an inexpensive random hop QCPI model, which accounts for some "back reaction" effects. Exploiting the finite length of nonlocality, we argue that further inclusion of quantum decoherence is possible via an iterative evaluation of the path integral. Finally, we show that the sum of the quantum amplitude factors with respect to the system paths leads to a smooth integrand as a function of trajectory initial conditions, allowing the use of Monte Carlo methods for the multidimensional phase space integral.
National Aeronautics and Space Administration — Develop, integrate and demonstrate a 2-micron pulsed Integrated Path Differential Absorption Lidar (IPDA) instrument CO2 Column Measurement from Airborne platform...
Blondel, Arnaud
2004-05-01
Thermodynamic integration is a widely used method to calculate and analyze the effect of a chemical modification on the free energy of a chemical or biochemical process, for example, the impact of an amino acid substitution on protein association. Numerical fluctuations can introduce large uncertainties, limiting the domain of application of the method. The parametric energy function describing the chemical modification in the thermodynamic integration, the "Alchemical path," determines the amplitudes of the fluctuations. In the present work, I propose a measure of the fluctuations in the thermodynamic integration and an approach to search for a parametric energy path minimizing that measure. The optimal path derived with this approach is very close to the theoretical minimum of the measure, but produces nonergodic sampling. Nevertheless, this path is used to guide the design of a practical and efficient path producing correct sampling. The convergence with this practical path is evaluated on test cases, and compares favorably with that of other methods such as power or polynomial path, soft-core van der Waals, and some other approaches presented in the literature.
Bennett, Ilana J; Stark, Craig E L
2016-03-01
Pattern separation describes the orthogonalization of similar inputs into unique, non-overlapping representations. This computational process is thought to serve memory by reducing interference and to be mediated by the dentate gyrus of the hippocampus. Using ultra-high in-plane resolution diffusion tensor imaging (hrDTI) in older adults, we previously demonstrated that integrity of the perforant path, which provides input to the dentate gyrus from entorhinal cortex, was associated with mnemonic discrimination, a behavioral outcome designed to load on pattern separation. The current hrDTI study assessed the specificity of this perforant path integrity-mnemonic discrimination relationship relative to other cognitive constructs (identified using a factor analysis) and white matter tracts (hippocampal cingulum, fornix, corpus callosum) in 112 healthy adults (20-87 years). Results revealed age-related declines in integrity of the perforant path and other medial temporal lobe (MTL) tracts (hippocampal cingulum, fornix). Controlling for global effects of brain aging, perforant path integrity related only to the factor that captured mnemonic discrimination performance. Comparable integrity-mnemonic discrimination relationships were also observed for the hippocampal cingulum and fornix. Thus, whereas perforant path integrity specifically relates to mnemonic discrimination, mnemonic discrimination may be mediated by a broader MTL network.
Efficient Calculation of Energy Expectation Values in the Path Integral Formalism
Grujic, J
2006-01-01
The path integral formalism, originally introduced by Richard Feynman, represents a powerful general framework for dealing with quantum and statistical theories, as well as an extremely useful tool in many other areas of science. Their numerical integration, however, is notoriously demanding of computer time and it is one of the most challenging computational problems.
Automatic Tool Path Generation for Robot Integrated Surface Sculpturing System
Zhu, Jiang; Suzuki, Ryo; Tanaka, Tomohisa; Saito, Yoshio
In this paper, a surface sculpturing system based on 8-axis robot is proposed, the CAD/CAM software and tool path generation algorithm for this sculpturing system are presented. The 8-axis robot is composed of a 6-axis manipulator and a 2-axis worktable, it carves block of polystyrene foams by heated cutting tools. Multi-DOF (Degree of Freedom) robot benefits from the faster fashion than traditional RP (Rapid Prototyping) methods and more flexibility than CNC machining. With its flexibility driven from an 8-axis configuration, as well as efficient custom-developed software for rough cutting and finish cutting, this surface sculpturing system can carve sculptured surface accurately and efficiently.
Leonel, Edson D; De Oliveira, Juliano A; Saif, Farhan, E-mail: edleonel@rc.unesp.br [Departamento de EstatIstica, Matematica Aplicada e Computacao, UNESP-Univ. Estadual Paulista, Av. 24A, 1515, Bela Vista, 13506-900 Rio Claro, Sao Paulo (Brazil)
2011-07-29
Critical exponents that describe a transition from integrability to non-integrability in a two-dimensional, nonlinear and area-preserving map are obtained via localization of the first invariant spanning curve (invariant tori) in the phase space. In a general class of systems, the position of the first invariant tori is estimated by reducing the mapping of the system to the standard mapping where a transition takes place from local to global chaos. The phase space of the mapping shows a large chaotic sea surrounding periodic islands and limited by a set of invariant tori whose position of the first of them depends on the control parameters. The formalism leads us to obtain analytically critical exponents that describe the behaviour of the average variable (action) along the chaotic sea. The result is compared to several models in the literature confirming the approach is of large interest. The formalism used is general and the procedure can be extended to many other different systems. (fast track communication)
A path-integral Langevin equation treatment of low-temperature doped helium clusters.
Ing, Christopher; Hinsen, Konrad; Yang, Jing; Zeng, Toby; Li, Hui; Roy, Pierre-Nicholas
2012-06-14
We present an implementation of path integral molecular dynamics for sampling low temperature properties of doped helium clusters using Langevin dynamics. The robustness of the path integral Langevin equation and white-noise Langevin equation [M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 133, 124104 (2010)] sampling methods are considered for those weakly bound systems with comparison to path integral Monte Carlo (PIMC) in terms of efficiency and accuracy. Using these techniques, convergence studies are performed to confirm the systematic error reduction introduced by increasing the number of discretization steps of the path integral. We comment on the structural and energetic evolution of He(N)-CO(2) clusters from N = 1 to 20. To quantify the importance of both rotations and exchange in our simulations, we present a chemical potential and calculated band origin shifts as a function of cluster size utilizing PIMC sampling that includes these effects. This work also serves to showcase the implementation of path integral simulation techniques within the molecular modelling toolkit [K. Hinsen, J. Comp. Chem. 21, 79 (2000)], an open-source molecular simulation package.
Utama, Briandhika; Purqon, Acep
2016-08-01
Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.
Hamiltonian analysis of interacting fluids
Banerjee, Rabin; Mitra, Arpan Krishna [S. N. Bose National Centre for Basic Sciences, Kolkata (India); Ghosh, Subir [Indian Statistical Institute, Kolkata (India)
2015-05-15
Ideal fluid dynamics is studied as a relativistic field theory with particular stress on its hamiltonian structure. The Schwinger condition, whose integrated version yields the stress tensor conservation, is explicitly verified both in equal-time and light-cone coordinate systems. We also consider the hamiltonian formulation of fluids interacting with an external gauge field. The complementary roles of the canonical (Noether) stress tensor and the symmetric one obtained by metric variation are discussed. (orig.)
Path integral in area tensor Regge calculus and complex connections
Khatsymovsky, V M
2006-01-01
Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection variables. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but more general quantities and, therefore, these do not appear as arguments of a function to be averaged, but are the integration (dummy) variables. We argue that upon integrating out the latter the resulting measure can be well-defined on physical hypersurface (for the area tensors corresponding to certain edge vectors, i.e. to certain metric) as positive and having exponential cutoff at large areas on condition that we confine ourselves to configurations which do not pass through degenerate metrics.
A path integral approach to asset-liability management
Decamps, M.; de Schepper, A.; Goovaerts, M.J.
2006-01-01
Functional integrals constitute a powerful tool in the investigation of financial models. In the recent econophysics literature, this technique was successfully used for the pricing of a number of derivative securities. In the present contribution, we introduce this approach to the field of asset-li
i-PI: A Python interface for ab initio path integral molecular dynamics simulations
Ceriotti, Michele; Manolopoulos, David E
2014-01-01
Recent developments in path integral methodology have significantly reduced the computational expense of including quantum mechanical effects in the nuclear motion in ab initio molecular dynamics simulations. However, the implementation of these developments requires a considerable programming effort, which has hindered their adoption. Here we describe i-PI, an interface written in Python that has been designed to minimise the effort required to bring state-of-the-art path integral techniques to an electronic structure program. While it is best suited to first principles calculations and path integral molecular dynamics, i-PI can also be used to perform classical molecular dynamics simulations, and can just as easily be interfaced with an empirical forcefield code. To give just one example of the many potential applications of the interface, we use it in conjunction with the CP2K electronic structure package to showcase the importance of nuclear quantum effects in high pressure water.
A Neural Path Integration Mechanism for Adaptive Vector Navigation in Autonomous Agents
Goldschmidt, Dennis; Dasgupta, Sakyasingha; Wörgötter, Florentin
2015-01-01
Animals show remarkable capabilities in navigating their habitat in a fully autonomous and energy-efficient way. In many species, these capabilities rely on a process called path integration, which enables them to estimate their current location and to find their way back home after long-distance......Animals show remarkable capabilities in navigating their habitat in a fully autonomous and energy-efficient way. In many species, these capabilities rely on a process called path integration, which enables them to estimate their current location and to find their way back home after long...... of autonomous agent navigation, but it also reproduces various aspects of animal navigation. Finally, we discuss how the proposed path integration mechanism may be used as a scaffold for spatial learning in terms of vector navigation....
Path integral approach to eikonal and next-to-eikonal exponentiation
Laenen, E; White, C D
2009-01-01
We approach the issue of exponentiation of soft gauge boson corrections to scattering amplitudes from a path integral point of view. We show that if one represents the amplitude as a first quantized path integral in a mixed coordinate-momentum space representation, a charged particle interacting with a soft gauge field is represented as a Wilson line for a semi-infinite line segment, together with calculable fluctuations. Combining such line segments, we show that exponentiation in an abelian field theory follows immediately from standard path-integral combinatorics. In the non-abelian case, we consider color singlet hard interactions with two outgoing external lines, and obtain a new viewpoint for exponentiation in terms of ``webs'', with a closed form solution for their corresponding color factors. We investigate and clarify the structure of next-to-eikonal corrections.
On the path integral representation of the Wigner function and the Barker-Murray ansatz
Sels, Dries; Brosens, Fons; Magnus, Wim
2012-01-01
The propagator of the Wigner function is constructed from the Wigner-Liouville equation as a phase space path integral over a new effective Lagrangian. In contrast to a paper by Barker and Murray (1983) [1], we show that the path integral can in general not be written as a linear superposition of classical phase space trajectories over a family of non-local forces. Instead, we adopt a saddle point expansion to show that the semiclassical Wigner function is a linear superposition of classical solutions for a different set of non-local time dependent forces. As shown by a simple example the specific form of the path integral makes the formulation ideal for Monte Carlo simulation.
Path integration in the field of a topological defect: the case of dispiration
Inomata, Akira; Raynolds, James
2011-01-01
The motion of a particle in the field of dispiration (due to a wedge disclination and a screw dislocation) is studied by path integration. By gauging $SO(2) \\otimes T(1)$, first, we derive the metric, curvature, and torsion of the medium of dispiration. Then we carry out explicitly path integration for the propagator of a particle moving in the non-Euclidean medium under the influence of a scalar potential and a vector potential. We obtain also the winding number representation of the propagator by taking the non-trivial topological structure of the medium into account. We extract the energy spectrum and the eigenfunctions from the propagator. Finally we make some remarks for special cases. Particularly, paying attention to the difference between the result of the path integration and the solution of Schr\\"odinger's equation in the case of disclination, we suggest that Schr\\"odinger equation may have to be modified by a curvature term.
On the Structure of QFT in the Particle Picture of the Path Integral Formulation
Jackson, D M; Morales, A
2008-01-01
In quantum field theory the path integral is usually formulated in the wave picture, i.e., as a sum over field evolutions. This path integral is difficult to define rigorously because of analytic problems whose resolution may ultimately require knowledge of non-perturbative or even Planck scale physics. Alternatively, QFT can be formulated directly in the particle picture, namely as a sum over all multi-particle paths, i.e., over Feynman graphs. This path integral is well-defined, as a map between rings of formal power series. This suggests a program for determining which structures of QFT are provable for this path integral and thus are combinatorial in nature, and which structures are actually sensitive to analytic issues. For a start, we show that the fact that the Legendre transform of the sum of connected graphs yields the effective action is indeed combinatorial in nature and is thus independent of analytic assumptions. Our proof also leads to new methods for the efficient decomposition of Feynman graph...
HAMILTONIAN DECOMPOSITION OF COMPLETE BIPARTITE r-HYPERGRAPHS
吉日木图; 王建方
2001-01-01
In [1] the concepts of paths and cycles of a hypergraph were introduced. In this paper, we give the concepts for bipartite hypergraph and Hamiltonian paths and cycles of a hypergraph,and prove that the complete bipartite 3-hypergraph with q vertices in each part is Hamiltonian decomposable where q is a prime.
Robust path integration in the entorhinal grid cell system with hippocampal feed-back.
Samu, Dávid; Eros, Péter; Ujfalussy, Balázs; Kiss, Tamás
2009-07-01
Animals are able to update their knowledge about their current position solely by integrating the speed and the direction of their movement, which is known as path integration. Recent discoveries suggest that grid cells in the medial entorhinal cortex might perform some of the essential underlying computations of path integration. However, a major concern over path integration is that as the measurement of speed and direction is inaccurate, the representation of the position will become increasingly unreliable. In this paper, we study how allothetic inputs can be used to continually correct the accumulating error in the path integrator system. We set up the model of a mobile agent equipped with the entorhinal representation of idiothetic (grid cell) and allothetic (visual cells) information and simulated its place learning in a virtual environment. Due to competitive learning, a robust hippocampal place code emerges rapidly in the model. At the same time, the hippocampo-entorhinal feed-back connections are modified via Hebbian learning in order to allow hippocampal place cells to influence the attractor dynamics in the entorhinal cortex. We show that the continuous feed-back from the integrated hippocampal place representation is able to stabilize the grid cell code.
Path integral regularization of QED by means of Stueckelberg fields
Jacquot, J L
2005-01-01
With the help of a Stueckelberg field we construct a regularized U(1) gauge invariant action through the introduction of cutoff functions. This action has the property that it converges formally to the unregularized action of QED when the ultraviolet cutoff goes to infinity. Integrating out exactly the Stueckelberg field we obtain a simple effective regularized action, which is fully gauge invariant and gives rise to the same prediction as QED at the tree level and to the one loop order.
Pérez, Alejandro; Tuckerman, Mark E.
2011-08-01
Higher order factorization schemes are developed for path integral molecular dynamics in order to improve the convergence of estimators for physical observables as a function of the Trotter number. The methods are based on the Takahashi-Imada and Susuki decompositions of the Boltzmann operator. The methods introduced improve the averages of the estimators by using the classical forces needed to carry out the dynamics to construct a posteriori weighting factors for standard path integral molecular dynamics. The new approaches are straightforward to implement in existing path integral codes and carry no significant overhead. The Suzuki higher order factorization was also used to improve the end-to-end distance estimator in open path integral molecular dynamics. The new schemes are tested in various model systems, including an ab initio path integral molecular dynamics calculation on the hydrogen molecule and a quantum water model. The proposed algorithms have potential utility for reducing the cost of path integral molecular dynamics calculations of bulk systems.
Constant External Fields in Gauge Theory and the Spin 0, 1/2, 1 Path Integrals
Reuter, M; Schubert, C; Reuter, Martin; Schmidt, Michael G.; Schubert, Christian
1996-01-01
We investigate the usefulness of the ``string-inspired technique'' for gauge theory calculations in a constant external field background. Our approach is based on Strassler's worldline path integral approach to the Bern-Kosower formalism, and on the construction of worldline (super--) Green's functions incorporating external fields as well as internal propagators. The worldline path integral representation of the gluon loop is reexamined in detail. We calculate the two-loop effective actions induced for a constant external field by a scalar and spinor loop, and the corresponding one-loop effective action in the gluon loop case.
Ab initio path integral ring polymer molecular dynamics: Vibrational spectra of molecules
Shiga, Motoyuki; Nakayama, Akira
2008-01-01
The path integral ring polymer molecular dynamics method is combined with 'on-the-fly' ab initio electronic structure calculations and applied to vibrational spectra of small molecules, LiH and H 2O, at the room temperature. The results are compared with those of the numerically exact solution and ab initio path integral centroid molecular dynamics calculation. The peak positions in the calculated spectra are found to be reasonable, showing the red-shift due to potential anharmonicity. This unification enables the investigation of real-time quantum dynamics of chemically complex molecular systems on the ab initio Born-Oppenheimer potential energy surface.
Variational Path-Integral Study on Bound Polarons in Parabolic Quantum Dots and Wires
CHEN Qing-Hu; WANG Zhuang-Bing; WU Fu-Li; LUO Meng-Bo; RUAN Yong-Hong; JIAO Zheng-Kuan
2001-01-01
The expression of the ground-state energy of an electron coupled simultaneously with a Coulomb potential and a longitudinal-optical phonon field in parabolic quantum dots and wires is derived within the framework of Feynman variational path-integral theory. We obtain a general result with arbitrary electron-phonon coupling constant,Coulomb binding parameters, and confining potential strength, which could be used for further numerical calculation of polaron properties. Moreover, it is shown that all the previous path-integral formulae for free polarons,bound polarons, and polarons confined in parabolic quantum dots and wires can be recovered in the present formalism.
Quantum field theory from operators to path integrals
Huang, Kerson
1998-01-01
A unique approach to quantum field theory, with emphasis on the principles of renormalization Quantum field theory is frequently approached from the perspective of particle physics. This book adopts a more general point of view and includes applications of condensed matter physics. Written by a highly respected writer and researcher, it first develops traditional concepts, including Feynman graphs, before moving on to key topics such as functional integrals, statistical mechanics, and Wilson's renormalization group. The connection between the latter and conventional perturbative renormalization is explained
A path integral approach to asset-liability management
Decamps, Marc; De Schepper, Ann; Goovaerts, Marc
2006-05-01
Functional integrals constitute a powerful tool in the investigation of financial models. In the recent econophysics literature, this technique was successfully used for the pricing of a number of derivative securities. In the present contribution, we introduce this approach to the field of asset-liability management. We work with a representation of cash flows by means of a two-dimensional delta-function perturbation, in the case of a Brownian model and a geometric Brownian model. We derive closed-form solutions for a finite horizon ALM policy. The results are numerically and graphically illustrated.
Enzymatic Kinetic Isotope Effects from Path-Integral Free Energy Perturbation Theory.
Gao, J
2016-01-01
Path-integral free energy perturbation (PI-FEP) theory is presented to directly determine the ratio of quantum mechanical partition functions of different isotopologs in a single simulation. Furthermore, a double averaging strategy is used to carry out the practical simulation, separating the quantum mechanical path integral exactly into two separate calculations, one corresponding to a classical molecular dynamics simulation of the centroid coordinates, and another involving free-particle path-integral sampling over the classical, centroid positions. An integrated centroid path-integral free energy perturbation and umbrella sampling (PI-FEP/UM, or simply, PI-FEP) method along with bisection sampling was summarized, which provides an accurate and fast convergent method for computing kinetic isotope effects for chemical reactions in solution and in enzymes. The PI-FEP method is illustrated by a number of applications, to highlight the computational precision and accuracy, the rule of geometrical mean in kinetic isotope effects, enhanced nuclear quantum effects in enzyme catalysis, and protein dynamics on temperature dependence of kinetic isotope effects.
Singular path-independent energy integrals for elastic bodies with thin elastic inclusions
Shcherbakov, V. V.
2016-06-01
An equilibrium problem for a two-dimensional homogeneous linear elastic body containing a thin elastic inclusion and an interfacial crack is considered. The thin inclusion is modeled within the framework of Euler-Bernoulli beam theory. An explicit formula for the first derivative of the energy functional with respect to the crack perturbation along the interface is presented. It is shown that the formulas for the derivative associated with translation and self-similar expansion of the crack are represented as path-independent integrals along smooth contour surrounding one or both crack tips. These path-independent integrals consist of regular and singular terms and are analogs of the well-known Eshelby-Cherepanov-Rice J-integral and Knowles-Sternberg M-integral.
Dorlas, T. C.; Thomas, E. G. F.
2008-01-01
We construct a genuine Radon measure with values in B(l(2)(Z(d))) on the set of paths in Z(d) representing Feynman's integral for the discrete Laplacian on l(2)(Z(d)), and we prove the Feynman integral formula for the solutions of the Schrodinger equation with Hamiltonian H=-1/2 Delta+ V, where Delt
Accelerated path integral methods for atomistic simulations at ultra-low temperatures.
Uhl, Felix; Marx, Dominik; Ceriotti, Michele
2016-08-07
Path integral methods provide a rigorous and systematically convergent framework to include the quantum mechanical nature of atomic nuclei in the evaluation of the equilibrium properties of molecules, liquids, or solids at finite temperature. Such nuclear quantum effects are often significant for light nuclei already at room temperature, but become crucial at cryogenic temperatures such as those provided by superfluid helium as a solvent. Unfortunately, the cost of converged path integral simulations increases significantly upon lowering the temperature so that the computational burden of simulating matter at the typical superfluid helium temperatures becomes prohibitive. Here we investigate how accelerated path integral techniques based on colored noise generalized Langevin equations, in particular the so-called path integral generalized Langevin equation thermostat (PIGLET) variant, perform in this extreme quantum regime using as an example the quasi-rigid methane molecule and its highly fluxional protonated cousin, CH5 (+). We show that the PIGLET technique gives a speedup of two orders of magnitude in the evaluation of structural observables and quantum kinetic energy at ultralow temperatures. Moreover, we computed the spatial spread of the quantum nuclei in CH4 to illustrate the limits of using such colored noise thermostats close to the many body quantum ground state.
On the coordinate (in)dependence of the formal path integral
Johnson-Freyd, Theo
. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the “formal” or “Feynman-diagrammatic” path integral construction. We then outline a proof of the following claim: the formal...
Integrated Flight Path Planning System and Flight Control System for Unmanned Helicopters
Yu-Hsiang Lin
2011-07-01
Full Text Available This paper focuses on the design of an integrated navigation and guidance system for unmanned helicopters. The integrated navigation system comprises two systems: the Flight Path Planning System (FPPS and the Flight Control System (FCS. The FPPS finds the shortest flight path by the A-Star (A* algorithm in an adaptive manner for different flight conditions, and the FPPS can add a forbidden zone to stop the unmanned helicopter from crossing over into dangerous areas. In this paper, the FPPS computation time is reduced by the multi-resolution scheme, and the flight path quality is improved by the path smoothing methods. Meanwhile, the FCS includes the fuzzy inference systems (FISs based on the fuzzy logic. By using expert knowledge and experience to train the FIS, the controller can operate the unmanned helicopter without dynamic models. The integrated system of the FPPS and the FCS is aimed at providing navigation and guidance to the mission destination and it is implemented by coupling the flight simulation software, X-Plane, and the computing software, MATLAB. Simulations are performed and shown in real time three-dimensional animations. Finally, the integrated system is demonstrated to work successfully in controlling the unmanned helicopter to operate in various terrains of a digital elevation model (DEM.
Path Integral Treatment of Proton Transport Processes in BaZrO3
Zhang, Qianfan; Wahnstrom, Goran; Björketun, Mårten
2008-01-01
Nuclear quantum effects on proton transfer and reorientation in BaZrO3 is investigated theoretically using the ab initio path-integral molecular-dynamics simulation technique. The result demonstrates that adding quantum fluctuations has a large effect on, in particular, the transfer barrier...
Quantum mechanical path integrals in curved spaces and the type-A trace anomaly
Bastianelli, Fiorenzo; Corradini, Olindo; Vassura, Edoardo
2017-04-01
Path integrals for particles in curved spaces can be used to compute trace anomalies in quantum field theories, and more generally to study properties of quantum fields coupled to gravity in first quantization. While their construction in arbitrary coordinates is well understood, and known to require the use of a regularization scheme, in this article we take up an old proposal of constructing the path integral by using Riemann normal coordinates. The method assumes that curvature effects are taken care of by a scalar effective potential, so that the particle lagrangian is reduced to that of a linear sigma model interacting with the effective potential. After fixing the correct effective potential, we test the construction on spaces of maximal symmetry and use it to compute heat kernel coefficients and type-A trace anomalies for a scalar field in arbitrary dimensions up to d = 12. The results agree with expected ones, which are reproduced with great efficiency and extended to higher orders. We prove explicitly the validity of the simplified path integral on maximally symmetric spaces. This simplified path integral might be of further use in worldline applications, though its application on spaces of arbitrary geometry remains unclear.
Comment on "Dual path integral representation for finite temperature quantum field theory"
Kazinski, P O
2008-01-01
I show that the novel dual path integral representation for finite temperature quantum field theory proposed in [Phys. Rev. D 77, 105030 (2008), arXiv:0803.1667 ] is a well-known representation of quantum mechanics in terms of symbols of operators.
Factors Affecting Technology Integration in K-12 Classrooms: A Path Model
Inan, Fethi A.; Lowther, Deborah L.
2010-01-01
The purpose of this study was to examine the direct and indirect effects of teachers' individual characteristics and perceptions of environmental factors that influence their technology integration in the classroom. A research-based path model was developed to explain causal relationships between these factors and was tested based on data gathered…
Some comments on rigorous quantum field path integrals in the analytical regularization scheme
Botelho, Luiz C.L. [Universidade Federal Fluminense (UFF), Niteroi, RJ (Brazil). Dept. de Matematica Aplicada]. E-mail: botelho.luiz@superig.com.br
2008-07-01
Through the systematic use of the Minlos theorem on the support of cylindrical measures on R{sup {infinity}}, we produce several mathematically rigorous path integrals in interacting euclidean quantum fields with Gaussian free measures defined by generalized powers of the Laplacian operator. (author)
2012-06-06
...Notice is hereby given that the U.S. International Trade Commission has received a complaint entitled Certain Integrated Circuit Packages Provided With Multiple Heat-Conducting Paths and Products Containing Same, DN 2899; the Commission is soliciting comments on any public interest issues raised by the complaint or complainant's filing under section 210.8(b) of the Commission's Rules of......
Kleinert, H.; Zatloukal, V.
2015-01-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Accelerated path integral methods for atomistic simulations at ultra-low temperatures
Uhl, Felix; Marx, Dominik; Ceriotti, Michele
2016-08-01
Path integral methods provide a rigorous and systematically convergent framework to include the quantum mechanical nature of atomic nuclei in the evaluation of the equilibrium properties of molecules, liquids, or solids at finite temperature. Such nuclear quantum effects are often significant for light nuclei already at room temperature, but become crucial at cryogenic temperatures such as those provided by superfluid helium as a solvent. Unfortunately, the cost of converged path integral simulations increases significantly upon lowering the temperature so that the computational burden of simulating matter at the typical superfluid helium temperatures becomes prohibitive. Here we investigate how accelerated path integral techniques based on colored noise generalized Langevin equations, in particular the so-called path integral generalized Langevin equation thermostat (PIGLET) variant, perform in this extreme quantum regime using as an example the quasi-rigid methane molecule and its highly fluxional protonated cousin, CH5+. We show that the PIGLET technique gives a speedup of two orders of magnitude in the evaluation of structural observables and quantum kinetic energy at ultralow temperatures. Moreover, we computed the spatial spread of the quantum nuclei in CH4 to illustrate the limits of using such colored noise thermostats close to the many body quantum ground state.
Teaching Basic Quantum Mechanics in Secondary School Using Concepts of Feynman Path Integrals Method
Fanaro, Maria de los Angeles; Otero, Maria Rita; Arlego, Marcelo
2012-01-01
This paper discusses the teaching of basic quantum mechanics in high school. Rather than following the usual formalism, our approach is based on Feynman's path integral method. Our presentation makes use of simulation software and avoids sophisticated mathematical formalism. (Contains 3 figures.)
Teaching Basic Quantum Mechanics in Secondary School Using Concepts of Feynman Path Integrals Method
Fanaro, Maria de los Angeles; Otero, Maria Rita; Arlego, Marcelo
2012-01-01
This paper discusses the teaching of basic quantum mechanics in high school. Rather than following the usual formalism, our approach is based on Feynman's path integral method. Our presentation makes use of simulation software and avoids sophisticated mathematical formalism. (Contains 3 figures.)
Fourier Path Integral Monte Carlo Method for the Calculation of the Microcanonical Density of States
Freeman, D L; Freeman, David L.
1994-01-01
Using a Hubbard-Stratonovich transformation coupled with Fourier path integral methods, expressions are derived for the numerical evaluation of the microcanonical density of states for quantum particles obeying Boltzmann statistics. A numerical algorithmis suggested to evaluate the quantum density of states and illustrated on a one-dimensional model system.
Response of Non-Linear Systems to Renewal Impulses by Path Integration
Nielsen, Søren R.K.; Iwankiewicz, R.
The cell-to-cell mapping (path integration) technique has been devised for MDOF non-linear and non-hysteretic systems subjected to random trains of impulses driven by an ordinary renewal point process with gamma-distributed integer parameter interarrival times (an Erlang process). Since the renewal...... additional discrete-valued state variables for which the stochastic equations are also formulated....
Brugnano, Luigi; Trigiante, Donato
2009-01-01
One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. For example, it is well known that standard (even symplectic) methods can only exactly preserve quadratic Hamiltonians. In this paper, a new family of methods, called Hamiltonian Boundary Value Methods (HBVMs), is introduced and analyzed. HBVMs are able to exactly preserve, in the discrete solution, Hamiltonian functions of polynomial type of arbitrarily high degree. These methods turn out to be symmetric, perfectly $A$-stable, and can have arbitrarily high order. A few numerical tests confirm the theoretical results.
Notes on area operator, geometric 2-rough paths and Young integral when p^-1+q^-1=1
Yang, Danyu
2012-01-01
1.When equipped with 2-rough norm and restricted to continuous paths with bounded variation, the area operator is a closable unbounded operator. 2.The area defined through Riemann-Stieltjes integral is the only possible candidate to enhance a path with vanishing 2-variation into a geometric 2-rough path. 3.Young integral is extended to p^-1+q^-1=1 by assigning a finer scale continuity.
Mihai V. Putz
2009-11-01
Full Text Available The density matrix theory, the ancestor of density functional theory, provides the immediate framework for Path Integral (PI development, allowing the canonical density be extended for the many-electronic systems through the density functional closure relationship. Yet, the use of path integral formalism for electronic density prescription presents several advantages: assures the inner quantum mechanical description of the system by parameterized paths; averages the quantum fluctuations; behaves as the propagator for time-space evolution of quantum information; resembles Schrödinger equation; allows quantum statistical description of the system through partition function computing. In this framework, four levels of path integral formalism were presented: the Feynman quantum mechanical, the semiclassical, the Feynman-Kleinert effective classical, and the Fokker-Planck non-equilibrium ones. In each case the density matrix or/and the canonical density were rigorously defined and presented. The practical specializations for quantum free and harmonic motions, for statistical high and low temperature limits, the smearing justification for the Bohr’s quantum stability postulate with the paradigmatic Hydrogen atomic excursion, along the quantum chemical calculation of semiclassical electronegativity and hardness, of chemical action and Mulliken electronegativity, as well as by the Markovian generalizations of Becke-Edgecombe electronic focalization functions – all advocate for the reliability of assuming PI formalism of quantum mechanics as a versatile one, suited for analytically and/or computationally modeling of a variety of fundamental physical and chemical reactivity concepts characterizing the (density driving many-electronic systems.
Exactly solvable path integral for open cavities in terms of quasinormal modes
Maasen van den Brink, A
2000-01-01
We evaluate the finite-temperature Euclidean phase-space path integral for the generating functional of a scalar field inside a leaky cavity. Provided the source is confined to the cavity, one can first of all integrate out the fields on the outside to obtain an effective action for the cavity alone. Subsequently, one uses an expansion of the cavity field in terms of its quasinormal modes (QNMs)-the exact, exponentially damped eigenstates of the classical evolution operator, which previously have been shown to be complete for a large class of models. Dissipation causes the effective cavity action to be nondiagonal in the QNM basis. The inversion of this action matrix inherent in the Gaussian path integral to obtain the generating functional is therefore nontrivial, but can be accomplished by invoking a novel QNM sum rule. The results are consistent with those obtained previously using canonical quantization.
Zhang, Sijie; Schönfeld, Fabian; Wiskott, Laurenz; Manahan-Vaughan, Denise
2014-01-01
Effective spatial navigation is enabled by reliable reference cues that derive from sensory information from the external environment, as well as from internal sources such as the vestibular system. The integration of information from these sources enables dead reckoning in the form of path integration. Navigation in the dark is associated with the accumulation of errors in terms of perception of allocentric position and this may relate to error accumulation in path integration. We assessed this by recording from place cells in the dark under circumstances where spatial sensory cues were suppressed. Spatial information content, spatial coherence, place field size, and peak and infield firing rates decreased whereas sparsity increased following exploration in the dark compared to the light. Nonetheless it was observed that place field stability in darkness was sustained by border information in a subset of place cells. To examine the impact of encountering the environment’s border on navigation, we analyzed the trajectory and spiking data gathered during navigation in the dark. Our data suggest that although error accumulation in path integration drives place field drift in darkness, under circumstances where border contact is possible, this information is integrated to enable retention of spatial representations. PMID:25009477
Bressloff, Paul C
2015-01-01
We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous
Functional integration of vertical flight path and speed control using energy principles
Lambregts, A. A.
1984-01-01
A generalized automatic flight control system was developed which integrates all longitudinal flight path and speed control functions previously provided by a pitch autopilot and autothrottle. In this design, a net thrust command is computed based on total energy demand arising from both flight path and speed targets. The elevator command is computed based on the energy distribution error between flight path and speed. The engine control is configured to produce the commanded net thrust. The design incorporates control strategies and hierarchy to deal systematically and effectively with all aircraft operational requirements, control nonlinearities, and performance limits. Consistent decoupled maneuver control is achieved for all modes and flight conditions without outer loop gain schedules, control law submodes, or control function duplication.
Stochastic averaging of quasi-Hamiltonian systems
朱位秋
1996-01-01
A stochastic averaging method is proposed for quasi-Hamiltonian systems (Hamiltonian systems with light dampings subject to weakly stochastic excitations). Various versions of the method, depending on whether the associated Hamiltonian systems are integrable or nonintegrable, resonant or nonresonant, are discussed. It is pointed out that the standard stochastic averaging method and the stochastic averaging method of energy envelope are special cases of the stochastic averaging method of quasi-Hamiltonian systems and that the results obtained by this method for several examples prove its effectiveness.
Normal Form for Families of Hamiltonian Systems
Zhi Guo WANG
2007-01-01
We consider perturbations of integrable Hamiltonian systems in the neighborhood of normally parabolic invariant tori. Using the techniques of KAM-theory we prove that there exists a canonical transformation that puts the Hamiltonian in normal form up to a remainder of weighted order 2d+1. And some dynamical consequences are obtained.
The Role of Spatial Memory and Frames of Reference in the Precision of Angular Path Integration
Arthur, Joeanna C.; Philbeck, John W.; Kleene, Nicholas J.; Chichka, David
2012-01-01
Angular path integration refers to the ability to maintain an estimate of self-location after a rotational displacement by integrating internally-generated (idiothetic) self-motion signals over time. Previous work has found that non-sensory inputs, namely spatial memory, can play a powerful role in angular path integration (Arthur et al., 2007, 2009). Here we investigated the conditions under which spatial memory facilitates angular path integration. We hypothesized that the benefit of spatial memory is particularly likely in spatial updating tasks in which one’s self-location estimate is referenced to external space. To test this idea, we administered passive, nonvisual body rotations (ranging 40° – 140°) about the yaw axis and asked participants to use verbal reports or open-loop manual pointing to indicate the magnitude of the rotation. Prior to some trials, previews of the surrounding environment were given. We found that when participants adopted an egocentric frame of reference, the previously-observed benefit of previews on within-subject response precision was not manifested, regardless of whether remembered spatial frameworks were derived from vision or spatial language. We conclude that the powerful effect of spatial memory is dependent on one’s frame of reference during self-motion updating. PMID:22885073
The role of spatial memory and frames of reference in the precision of angular path integration.
Arthur, Joeanna C; Philbeck, John W; Kleene, Nicholas J; Chichka, David
2012-09-01
Angular path integration refers to the ability to maintain an estimate of self-location after a rotational displacement by integrating internally-generated (idiothetic) self-motion signals over time. Previous work has found that non-sensory inputs, namely spatial memory, can play a powerful role in angular path integration (Arthur et al., 2007, 2009). Here we investigated the conditions under which spatial memory facilitates angular path integration. We hypothesized that the benefit of spatial memory is particularly likely in spatial updating tasks in which one's self-location estimate is referenced to external space. To test this idea, we administered passive, non-visual body rotations (ranging 40°-140°) about the yaw axis and asked participants to use verbal reports or open-loop manual pointing to indicate the magnitude of the rotation. Prior to some trials, previews of the surrounding environment were given. We found that when participants adopted an egocentric frame of reference, the previously-observed benefit of previews on within-subject response precision was not manifested, regardless of whether remembered spatial frameworks were derived from vision or spatial language. We conclude that the powerful effect of spatial memory is dependent on one's frame of reference during self-motion updating.
Reuschel, Johanna; Rösler, Frank; Henriques, Denise Y P; Fiehler, Katja
2011-04-01
Many studies provide evidence that information from different modalities is integrated following the maximum likelihood estimation model (MLE). For instance, we recently found that visual and proprioceptive path trajectories are optimally combined (Reuschel et al. in Exp Brain Res 201:853-862, 2010). However, other studies have failed to reveal optimal integration of such dynamic information. In the present study, we aim to generalize our previous findings to different parts of the workspace (central, ipsilateral, or contralateral) and to different types of judgments (relative vs. absolute). Participants made relative judgments by judging whether an angular path was acute or obtuse, or they made absolute judgments by judging whether a one-segmented straight path was directed to left or right. Trajectories were presented in the visual, proprioceptive, or combined visual-proprioceptive modality. We measured the bias and the variance of these estimates and predicted both parameters using the MLE. In accordance with the MLE model, participants linearly combined and weighted the unimodal angular path information by their reliabilities irrespective of the side of workspace. However, the precision of bimodal estimates was not greater than that for unimodal estimates, which is inconsistent with the MLE. For the absolute judgment task, participants' estimates were highly accurate and did not differ across modalities. Thus, we were unable to test whether the bimodal percept resulted as a weighted average of the visual and proprioceptive input. Additionally, participants were not more precise in the bimodal compared with the unimodal conditions, which is inconsistent with the MLE. Current findings suggest that optimal integration of visual and proprioceptive information of path trajectory only applies in some conditions.
Path-integral solution of the one-dimensional Dirac quantum cellular automaton
D' Ariano, Giacomo Mauro [QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia, 27100 (Italy); INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia, 27100 (Italy); Mosco, Nicola [QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia, 27100 (Italy); Perinotti, Paolo [QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia, 27100 (Italy); INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia, 27100 (Italy); Tosini, Alessandro [QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia, 27100 (Italy)
2014-09-05
Quantum cellular automata, which describe the discrete and exactly causal unitary evolution of a lattice of quantum systems, have been recently considered as a fundamental approach to quantum field theory and a linear automaton for the Dirac equation in one dimension has been derived. In the linear case a quantum cellular automaton is isomorphic to a quantum walk and its evolution is conveniently formulated in terms of transition matrices. The semigroup structure of the matrices leads to a new kind of discrete path-integral, different from the well known Feynman checkerboard one, that is solved analytically in terms of Jacobi polynomials of the arbitrary mass parameter. - Highlights: • Discrete path integral formulation of linear QCAs in terms of transition matrices. • Derivation of the analytical solution for the one dimensional Dirac QCA. • Solution given in terms of Jacobi polynomials versus the arbitrary mass parameter. • The discrete paths and the transition matrices of the Dirac QCA are binary encoded. • Paths are grouped in equivalence classes according to their overall transition matrix.
The path-independent M Integral around Röthlisberger channels
Meyer, C. R.; Rice, J. R.
2015-12-01
Röthlisberger channels are essential components of subglacial hydrologic systems. Deviations from the Nye creep closure of the ice around a Röthlisberger channel have been long recognized and enhancement factors or a more complex rheology for ice have been suggested as ameliorations to account for channels closing faster than predicted. Here we use the MM integral, a path-independent integral of the equations of continuum mechanics, with a Glen power-law rheology to unify descriptions of creep closure under a variety of stress states surrounding the Röthlisberger channel. The advantage of this approach is that the MM integral around the Röthlisberger channel is equivalent to the integral around the far field. In this way, the creep closure on the channel wall can be determined as a function of the far-field loading, e.g. antiplane shear as well as overburden pressure. We start by analyzing the case of axisymmetric creep closure and we see that the Nye solution is implied by the path-independence of MM integral. We then examine the effects of antiplane shear in several geometries and derive scalings for the creep closure rate based on the MM integral. The results are compared to observations for tunnel closure measurements in a variety of stress states and it is shown that the additional stress components can account for the deviations from the Nye solution. Furthermore, creep closure can be succinctly written in terms of the path-independent MM integral and the variation with applied shear can be found via scalings, which is useful for subglacial hydrology models.
From path integrals to tensor networks for the AdS /CFT correspondence
Miyaji, Masamichi; Takayanagi, Tadashi; Watanabe, Kento
2017-03-01
In this paper, we discuss tensor network descriptions of AdS /CFT from two different viewpoints. First, we start with a Euclidean path-integral computation of ground state wave functions with a UV cutoff. We consider its efficient optimization by making its UV cutoff position dependent and define a quantum state at each length scale. We conjecture that this path integral corresponds to a time slice of anti-de Sitter (AdS) spacetime. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of AdS3 in terms of the dual two-dimensional conformal field theory (CFT). Both approaches support a correspondence between the hyperbolic time slice H2 in AdS3 and a version of continuous multiscale entanglement renormalization ansatz. We also give a heuristic argument about why we can expect a sub-AdS scale bulk locality for holographic CFTs.
Data Assimilation using a GPU Accelerated Path Integral Monte Carlo Approach
Quinn, John C
2011-01-01
The answers to data assimilation questions can be expressed as path integrals over all possible state and parameter histories. We show how these path integrals can be evaluated numerically using a Markov Chain Monte Carlo method designed to run in parallel on a Graphics Processing Unit (GPU). We demonstrate the application of the method to an example with a transmembrane voltage time series of a simulated neuron as an input, and using a Hodgkin-Huxley neuron model. By taking advantage of GPU computing, we gain a parallel speedup factor of up to about 200 times faster than an equivalent serial computation on a CPU, with performance increasing as the length of the observation time used for data assimilation increases.
Ab Initio Path Integral Molecular Dynamics Simulation of Hydrogen in Silicon
Probert, M. I. J.; Glover, M. J.
2006-05-01
We report results of a first-principles theoretical study of an isolated neutral hydrogen atom in crystalline silicon. Spin-polarised density functional theory is used to treat the electrons, and the path-integral molecular dynamics method is used to describe the quantum properties of the nucleus at finite temperature. This is necessary as the hydrogen atom has sufficiently low mass that it exhibits significant nuclear quantum delocalisation and zero-point motion even at room temperature. Unlike post-hoc treatments, such as calculating a static potential energy surface, the path-integral treatment enables such effects to be included "on-the-fly". This is found to be significant, as a coupling is found between the structure of the host silicon lattice and the quantum delocalisation of the hydrogen defect.
A unified scheme for ab initio molecular orbital theory and path integral molecular dynamics
Shiga, Motoyuki; Tachikawa, Masanori; Miura, Shinichi
2001-11-01
We present a general approach for accurate calculation of chemical substances which treats both nuclei and electrons quantum mechanically, adopting ab initio molecular orbital theory for the electronic structure and path integral molecular dynamics for the nuclei. The present approach enables the evaluation of physical quantities dependent on the nuclear configuration as well as the electronic structure, within the framework of Born-Oppenheimer adiabatic approximation. As an application, we give the path integral formulation of electric response properties—dipole moment and polarizability, which characterize the changes both in electronic structure and nuclear configuration at a given temperature when uniform electrostatic field is present. We also demonstrate the calculation of a water molecule using the present approach and the result of temperature and isotope effects is discussed.
Simulations of one- and two-electron systems by Bead-Fourier path integral molecular dynamics
Ivanov, Sergei D.; Lyubartsev, Alexander P.
2005-07-01
The Bead-Fourier path integral molecular dynamics technique introduced earlier [S. D. Ivanov, A. P. Lyubartsev, and A. Laaksonen, Phys. Rev. E 67 066710 (2003)] is applied for simulation of electrons in the simplest molecules: molecular hydrogen, helium atom, and their ions. Special attention is paid to the correct description of electrons in the core region of a nucleus. In an attempt to smooth the Coulomb potential at small distances, a recipe is suggested. The simulation results are in excellent agreement with the analytical solution for the "harmonic helium atom", as well as with the vibrational potential of the H2 molecule and He ionization energies. It is demonstrated, that the Bead-Fourier path integral molecular dynamics technique is able to provide the accuracy required for the description of electron structure and chemical bonds in cases when electron exchange effects need not be taken into account.
Proton momentum distributions in water: A path integral molecular dynamics study
Srinivasan, Varadharajan; Morrone, Joseph A.; Sebastiani, Daniel; Car, Roberto
2007-03-01
Recent neutron Compton scattering experiments have detected the proton momentum distributions of water. This density in momentum space is a quantum mechanical property of the proton, due to the confining anharmonic potential from covalent and hydrogen bonds. The theoretical calculation of this property can be carried out via ``open'' path integral expressions. In this work, we present an extension of the staging path integral molecular dynamics method, which is then employed to calculate the proton momentum distributions of water in the solid, liquid, and supercritical phases. We utilize the SPC/F2 empirical force field to model the system's interactions. The calculated momentum distributions depict both agreement and discrepancies with experiment. The differences may be explained by the deviation of the force field from the true interactions. These distributions provide an abundance of information about the environment and interactions surrounding the proton.
Proton momentum distribution in water: an open path integral molecular dynamics study
Morrone, Joseph A.; Srinivasan, Varadharajan; Sebastiani, Daniel; Car, Roberto
2007-06-01
Recent neutron Compton scattering experiments have detected the proton momentum distribution in water. The theoretical calculation of this property can be carried out via "open" path integral expressions. In this work, present an extension of the staging path integral molecular dynamics method, which is then employed to calculate the proton momentum distributions of water in the solid, liquid, and supercritical phases. We utilize a flexible, single point charge empirical force field to model the system's interactions. The calculated momentum distributions depict both agreement and discrepancies with experiment. The differences may be explained by the deviation of the force field from the true interactions. These distributions provide an abundance of information about the environment and interactions surrounding the proton.
Path integral approach to the pricing of timer options with the Duru-Kleinert time transformation.
Liang, L Z J; Lemmens, D; Tempere, J
2011-05-01
In this paper, a time substitution as used by Duru and Kleinert in their treatment of the hydrogen atom with path integrals is performed to price timer options under stochastic volatility models. We present general pricing formulas for both the perpetual timer call options and the finite time-horizon timer call options. These general results allow us to find closed-form pricing formulas for both the perpetual and the finite time-horizon timer options under the 3/2 stochastic volatility model as well as under the Heston stochastic volatility model. For the treatment of timer options under the 3/2 model we will rely on the path integral for the Morse potential, with the Heston model we will rely on the Kratzer potential. © 2011 American Physical Society
From Path Integrals to Tensor Networks for AdS/CFT
Miyaji, Masamichi; Watanabe, Kento
2016-01-01
In this paper, we discuss tensor network descriptions of AdS/CFT from two different viewpoints. First, we start with an Euclidean path-integral computation of ground state wave functions with a UV cut off. We consider its efficient optimization by making its UV cut off position dependent and define a quantum state at each length scale. We conjecture that this path-integral corresponds to a time slice of AdS. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of AdS3 in terms of the dual 2d CFT. Both approaches support a correspondence between the hyperbolic time slice H2 in AdS3 and a version of continuous MERA (cMERA). We also give a heuristic argument why we can expect a sub-AdS scale bulk locality for holographic CFTs.
Path-integral and Ornstein-Zernike study of quantum fluid structures on the crystallization line
Sesé, Luis M.
2016-03-01
Liquid neon, liquid para-hydrogen, and the quantum hard-sphere fluid are studied with path integral Monte Carlo simulations and the Ornstein-Zernike pair equation on their respective crystallization lines. The results cover the whole sets of structures in the r-space and the k-space and, for completeness, the internal energies, pressures and isothermal compressibilities. Comparison with experiment is made wherever possible, and the possibilities of establishing k-space criteria for quantum crystallization based on the path-integral centroids are discussed. In this regard, the results show that the centroid structure factor contains two significant parameters related to its main peak features (amplitude and shape) that can be useful to characterize freezing.
Semi-classical locality for the non-relativistic path integral in configuration space
Gomes, Henrique
2015-01-01
In an accompanying paper, we have put forward an interpretation of quantum mechanics grounded on a non-relativistic Lagrangian 3+1 formalism of a closed Universe, existing on timeless configuration space. However, not much was said there about the role of locality, which was not assumed. In this paper, I describe how subsystems existing in (spatial) regions with fixed boundary conditions can be represented as submanifolds of the complete configuration space. I show that if the action functional can be put in the form of Riemannian distance element, then dynamical independence of the subsystem implies that the respective submanifolds are totally geodesic. When two regions are mutually independent the semi-classical path integral kernel factorizes, showing cluster decomposition. To exemplify these constructions I then construct a specific gravitational system with two propagating physical degrees of freedom and no refoliation-invariance. Finally, considering the path integral in this 3+1 context, I implement an...
Path-integral action of a particle in the noncommutative phase-space
Gangopadhyay, Sunandan
2016-01-01
In this paper we construct a path integral formulation of quantum mechanics on noncommutative phase-space. We first map the system to an equivalent system on the noncommutative plane. Then by applying the formalism of representing a quantum system in the space of Hilbert-Schmidt operators acting on noncommutative configuration space, the path integral action of a particle is derived. It is observed that the action has a similar form to that of a particle in a magnetic field in the noncommutative plane. From this action the energy spectrum is obtained for the free particle and the harmonic oscillator potential. We also show that the nonlocal nature (in time) of the action yields a second class constrained system from which the noncommutative Heisenberg algebra can be recovered.
Moran, B.; Kulkarni, S.S.; Reeves, H.W.
2007-01-01
A path-independent (conservation) integral is developed for the characterization of solute concentration and flux in a biofilm in the vicinity of a detachment or other flux limiting boundary condition. Steady state conditions of solute diffusion are considered and biofilm kinetics are described by an uptake term which can be expressed in terms of a potential (Michaelis-Menten kinetics). An asymptotic solution for solute concentration at the tip of the detachment is obtained and shown to be analogous to that of antiplane crack problems in linear elasticity. It is shown that the amplitude of the asymptotic solution can be calculated by evaluating a path-independent integral. The special case of a semi-infinite detachment in an infinite strip is considered and the amplitude of the asymptotic field is related to the boundary conditions and problem parameters in closed form for zeroth and first order kinetics and numerically for Michaelis-Menten kinetics. ?? Springer Science+Business Media, Inc. 2007.
Evolution of Arbitrary States under Fock-Darwin Hamiltonian and a Time-Dependent Electric Field
徐晓飞; 杨涛; 翟智远; 潘孝胤
2012-01-01
The method of path integral is employed to calculate the time evolution of the eigenstates of a charged particle under the Fock-Darwin （FD） Hamiltonian subjected to a time-dependent electric field in the plane of the system. An exact analytical expression is established for the evolution of the eigenstates. This result then provides a general solution to the time-dependent Schrodinger equation.
Quantum Adiabatic Evolution Algorithms with Different Paths
Farhi, E; Gutmann, S; Farhi, Edward; Goldstone, Jeffrey; Gutmann, Sam
2002-01-01
In quantum adiabatic evolution algorithms, the quantum computer follows the ground state of a slowly varying Hamiltonian. The ground state of the initial Hamiltonian is easy to construct; the ground state of the final Hamiltonian encodes the solution of the computational problem. These algorithms have generally been studied in the case where the "straight line" path from initial to final Hamiltonian is taken. But there is no reason not to try paths involving terms that are not linear combinations of the initial and final Hamiltonians. We give several proposals for randomly generating new paths. Using one of these proposals, we convert an algorithmic failure into a success.
Calculation rule for Aoyama-Tamra's prescription for path integral with quantum tunneling
Suzuki, H
1995-01-01
We derive a simple calculation rule for Aoyama--Tamra's prescription for path integral with degenerated potential minima. Non-perturbative corrections due to the restricted functional space (fundamental region) can systematically be computed with this rule. It becomes manifest that the prescription might give Borel summable series for finite temperature (or volume) system with quantum tunneling, while the advantage is lost at zero temperature (or infinite volume) limit.
Loredana Teresa Pedata
2012-12-01
Full Text Available The path and the pilot study presented here come from a synergy between a pharmaceutical, universities and institutions in the area. The intervention evaluation wants to establish itself as a means of "re-thinking" youth intervention benefited: the assumption that the integration of knowledge can constitute an enrichment of the whole person, we believe that such enrichment is more likely to occur in group in comparison with others and the development of social skills and human resources.
Gorbenko, Anna; Popov, Vladimir
2017-07-01
Different planning problems for robotic remote laser welding are of considerable interest. In this paper, we consider the problem of integrated task sequencing and path planning for robotic remote laser welding. We propose an efficient approach to solve the problem. In particular, we consider an explicit reduction from the decision version of the problem to the satisfiability problem. We present the results of computational experiments for different satisfiability algorithms.
Polyakov's spin factor for a classical spinning particle via BRST invariant path integral
Cho, J; Lee, H; Jin-Ho Cho; Seungjoon Hyun; Hyuk-Jae Lee
1994-01-01
For the "classical" formulation of a massive spinning particle, the propagator is obtained along with the spin factor. We treat the system with two kinds of constraints that were recently shown to be concerned with the reparametrization invariance and "quasi-supersymmetry". In the path integral, the BRST invariant Lagrangian is used and the same spin factor is obtained as in the pseudo-classical formulation.
Integrating cell on chip—Novel waveguide platform employing ultra-long optical paths
Fohrmann, Lena Simone; Sommer, Gerrit; Pitruzzello, Giampaolo; Krauss, Thomas F.; Petrov, Alexander Yu.; Eich, Manfred
2017-09-01
Optical waveguides are the most fundamental building blocks of integrated optical circuits. They are extremely well understood, yet there is still room for surprises. Here, we introduce a novel 2D waveguide platform which affords a strong interaction of the evanescent tail of a guided optical wave with an external medium while only employing a very small geometrical footprint. The key feature of the platform is its ability to integrate the ultra-long path lengths by combining low propagation losses in a silicon slab with multiple reflections of the guided wave from photonic crystal (PhC) mirrors. With a reflectivity of 99.1% of our tailored PhC-mirrors, we achieve interaction paths of 25 cm within an area of less than 10 mm2. This corresponds to 0.17 dB/cm effective propagation which is much lower than the state-of-the-art loss of approximately 1 dB/cm of single mode silicon channel waveguides. In contrast to conventional waveguides, our 2D-approach leads to a decay of the guided wave power only inversely proportional to the optical path length. This entirely different characteristic is the major advantage of the 2D integrating cell waveguide platform over the conventional channel waveguide concepts that obey the Beer-Lambert law.
Calculation of heat capacities of light and heavy water by path-integral molecular dynamics
Shiga, Motoyuki; Shinoda, Wataru
2005-10-01
As an application of atomistic simulation methods to heat capacities, path-integral molecular dynamics has been used to calculate the constant-volume heat capacities of light and heavy water in the gas, liquid, and solid phases. While the classical simulation based on conventional molecular dynamics has estimated the heat capacities too high, the quantum simulation based on path-integral molecular dynamics has given reasonable results based on the simple point-charge/flexible potential model. The calculated heat capacities (divided by the Boltzmann constant) in the quantum simulation are 3.1 in the vapor H2O at 300 K, 6.9 in the liquid H2O at 300 K, and 4.1 in the ice IhH2O at 250 K, respectively, which are comparable to the experimental data of 3.04, 8.9, and 4.1, respectively. The quantum simulation also reproduces the isotope effect. The heat capacity in the liquid D2O has been calculated to be 10% higher than that of H2O, while it is 13% higher in the experiment. The results demonstrate that the path-integral simulation is a promising approach to quantitatively evaluate the heat capacities for molecular systems, taking account of quantum-mechanical vibrations as well as strongly anharmonic motions.
i-PI: A Python interface for ab initio path integral molecular dynamics simulations
Ceriotti, Michele; More, Joshua; Manolopoulos, David E.
2014-03-01
Recent developments in path integral methodology have significantly reduced the computational expense of including quantum mechanical effects in the nuclear motion in ab initio molecular dynamics simulations. However, the implementation of these developments requires a considerable programming effort, which has hindered their adoption. Here we describe i-PI, an interface written in Python that has been designed to minimise the effort required to bring state-of-the-art path integral techniques to an electronic structure program. While it is best suited to first principles calculations and path integral molecular dynamics, i-PI can also be used to perform classical molecular dynamics simulations, and can just as easily be interfaced with an empirical forcefield code. To give just one example of the many potential applications of the interface, we use it in conjunction with the CP2K electronic structure package to showcase the importance of nuclear quantum effects in high-pressure water. Catalogue identifier: AERN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AERN_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 138626 No. of bytes in distributed program, including test data, etc.: 3128618 Distribution format: tar.gz Programming language: Python. Computer: Multiple architectures. Operating system: Linux, Mac OSX, Windows. RAM: Less than 256 Mb Classification: 7.7. External routines: NumPy Nature of problem: Bringing the latest developments in the modelling of nuclear quantum effects with path integral molecular dynamics to ab initio electronic structure programs with minimal implementational effort. Solution method: State-of-the-art path integral molecular dynamics techniques are implemented in a Python interface. Any electronic structure code can be patched to receive the atomic
Yu, Pengfei; Wang, Hailong; Chen, Jianyong; Shen, Shengping
2017-07-01
In this study, the conservation laws οf dissipative mechanical-diffusion-electrochemical reaction system are systematically obtained based on Noether's theorem. According to linear, irreversible thermodynamics, dissipative phenomena can be described by an irreversible force and an irreversible flow. Additionally, the Lagrange function, L and the generalized Hamilton least-action principle are proposed to be used to obtain the conservation integrals. A group of these integrals, including the J-, M-, and L-integrals, can be then obtained using the classical Noether approach for dissipative processes. The relation between the J-integral and the energy release rate is illustrated. The path-independence of the J-integral is then proven. The J-integral, derived based on Noether's theorem, is a line integral, contrary to the propositions of existing published works that describe it both as a line and an area integral. Herein, we prove that the outcomes are identical, and identify the physical meaning of the area integral, a concept that was not explained previously. To show that the J-integral can dominate the distribution of the corresponding field quantities, an example of a partial, stress-diffusion coupling process is disscussed.
Imaoka, Haruna; Kinugawa, Kenichi
2017-03-01
Thermal conductivity, shear viscosity, and bulk viscosity of normal liquid 4He at 1.7-4.0 K are calculated using path integral centroid molecular dynamics (CMD) simulations. The calculated thermal conductivity and shear viscosity above lambda transition temperature are on the same order of magnitude as experimental values, while the agreement of shear viscosity is better. Above 2.3 K the CMD well reproduces the temperature dependences of isochoric shear viscosity and of the time integral of the energy current and off-diagonal stress tensor correlation functions. The calculated bulk viscosity, not known in experiments, is several times larger than shear viscosity.
NLOS UV Channel Modeling Using Numerical Integration and an Approximate Closed-Form Path Loss Model
Gupta, Ankit; Brandt-Pearce, Maïté
2012-01-01
In this paper we propose a simulation method using numerical integration, and develop a closed-form link loss model for physical layer channel characterization for non-line of sight (NLOS) ultraviolet (UV) communication systems. The impulse response of the channel is calculated by assuming both uniform and Gaussian profiles for transmitted beams and different geometries. The results are compared with previously published results. The accuracy of the integration approach is compared to the Monte Carlo simulation. Then the path loss using the simulation method and the suggested closed-form expression are presented for different link geometries. The accuracies are evaluated and compared to the results obtained using other methods.
Hamiltonian Algorithm Sound Synthesis
大矢, 健一
2013-01-01
Hamiltonian Algorithm (HA) is an algorithm for searching solutions is optimization problems. This paper introduces a sound synthesis technique using Hamiltonian Algorithm and shows a simple example. "Hamiltonian Algorithm Sound Synthesis" uses phase transition effect in HA. Because of this transition effect, totally new waveforms are produced.
Bravetti, Alessandro, E-mail: alessandro.bravetti@iimas.unam.mx [Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, A. P. 70543, México, DF 04510 (Mexico); Cruz, Hans, E-mail: hans@ciencias.unam.mx [Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. P. 70543, México, DF 04510 (Mexico); Tapias, Diego, E-mail: diego.tapias@nucleares.unam.mx [Facultad de Ciencias, Universidad Nacional Autónoma de México, A.P. 70543, México, DF 04510 (Mexico)
2017-01-15
In this work we introduce contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and show that it is a natural candidate for a geometric description of non-dissipative and dissipative systems. For this purpose we review in detail the major features of standard symplectic Hamiltonian dynamics and show that all of them can be generalized to the contact case.
Classical mechanics Hamiltonian and Lagrangian formalism
Deriglazov, Alexei
2016-01-01
This account of the fundamentals of Hamiltonian mechanics also covers related topics such as integral invariants and the Noether theorem. With just the elementary mathematical methods used for exposition, the book is suitable for novices as well as graduates.
Bustamante, Miguel D [Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Chile); Hojman, Sergio A [Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Chile)
2003-01-10
In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behaviour has not been verified up to now. The Euler-Lagrange equations we get for these systems usually present 'time reparametrization' invariance, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion.
Orthogonal separable Hamiltonian systems on T2
无
2007-01-01
In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T2 for a given metric, and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T2 can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way.
An extended phase-space stochastic quantization of constrained Hamiltonian systems
Ter-Kazarian, G T [Byurakan Astrophysical Observatory, Byurakan 378433, Aragatsotn District (Armenia); Sobouti, Y [Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan, PO Box 45195-159 (Iran, Islamic Republic of)], E-mail: gago-50@yahoo.com, E-mail: sobouti@iasbs.ac.ir
2008-08-08
Having gained some insight into the concept of 'actual and virtual paths' in a phase-space formalism (Sobouti and Nasiri 1993 Int. J. Mod. Phys. B 7 3255, Nasiri et al 2006 J. Math. Phys. 47 092106), in the present paper we address the question of 'extended' phase-space stochastic quantization of Hamiltonian systems with first class holonomic constraints. We present the appropriate Langevin equations, which quantize such constrained systems, and prove the equivalence of the stochastic quantization method with the conventional path-integral gauge measure of Faddeev-Popov quantization.
Geng, Hua Y., E-mail: huay.geng@gmail.com [National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, CAEP, P.O. Box 919-102, Mianyang, Sichuan, 621900 (China); Department of Chemistry and Chemical Biology, Cornell University, Baker Laboratory, Ithaca, NY 14853 (United States)
2015-02-15
A multilevel approach to sample the potential energy surface in a path integral formalism is proposed. The purpose is to reduce the required number of ab initio evaluations of energy and forces in ab initio path integral molecular dynamics (AI-PIMD) simulation, without compromising the overall accuracy. To validate the method, the internal energy and free energy of an Einstein crystal are calculated and compared with the analytical solutions. As a preliminary application, we assess the performance of the method in a realistic model—the FCC phase of dense atomic hydrogen, in which the calculated result shows that the acceleration rate is about 3 to 4-fold for a two-level implementation, and can be increased up to 10 times if extrapolation is used. With only 16 beads used for the ab initio potential sampling, this method gives a well converged internal energy. The residual error in pressure is just about 3 GPa, whereas it is about 20 GPa for a plain AI-PIMD calculation with the same number of beads. The vibrational free energy of the FCC phase of dense hydrogen at 300 K is also calculated with an AI-PIMD thermodynamic integration method, which gives a result of about 0.51 eV/proton at a density of r{sub s}=0.912.
Geng, Hua Y.
2015-02-01
A multilevel approach to sample the potential energy surface in a path integral formalism is proposed. The purpose is to reduce the required number of ab initio evaluations of energy and forces in ab initio path integral molecular dynamics (AI-PIMD) simulation, without compromising the overall accuracy. To validate the method, the internal energy and free energy of an Einstein crystal are calculated and compared with the analytical solutions. As a preliminary application, we assess the performance of the method in a realistic model-the FCC phase of dense atomic hydrogen, in which the calculated result shows that the acceleration rate is about 3 to 4-fold for a two-level implementation, and can be increased up to 10 times if extrapolation is used. With only 16 beads used for the ab initio potential sampling, this method gives a well converged internal energy. The residual error in pressure is just about 3 GPa, whereas it is about 20 GPa for a plain AI-PIMD calculation with the same number of beads. The vibrational free energy of the FCC phase of dense hydrogen at 300 K is also calculated with an AI-PIMD thermodynamic integration method, which gives a result of about 0.51 eV/proton at a density of rs = 0.912.
Bustamante, M D
2003-01-01
In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behaviour has not been verified up to now. The Euler-Lagrange equations we get for these systems usually present 'time reparametrization' invariance, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, w...
A path integral formula with applications to quantum random walks in Z{sup d}
Yang Weishih [Department of Mathematics, Temple University, Philadelphia, PA 19122 (United States); Liu, Chaobin [Department of Mathematics, Bowie State University, Bowie, MD 20715 (United States); Zhang Kai [Department of Mathematics, Temple University, Philadelphia, PA 19122 (United States)
2007-07-20
We consider general quantum random walks in a d-dimensional half-space. We first obtain a path integral formula for general quantum random walks in a d-dimensional space. Our path integral formula is valid for general quantum random walks on Cayley graphs as well. Then the path integral formula is applied to obtain the scaling limit of the exit distribution, the expectation of exit time and the asymptotic behaviour of the exit probabilities, for general quantum random walks in a half-space under some conditions on amplitude functions. The conditions are shown to be satisfied by both the Hadamard and Grover quantum random walks in two-dimensional half-spaces. For the two-dimensional case, we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero, i.e. the natural magnitude of the hitting position is of order O(1) if the lattice spacing is set to be 1/n. We also show that the rate of convergence of the total hitting probability has lower bound n{sup -2} and upper bound n{sup -2+{epsilon}} for any {epsilon} > 0. For a quantum random walk with a fixed starting point, we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk. In both one and two dimensions, given the event of a hit, the conditional expectation of hitting times is finite, in contrast to being infinite for the classical case. In the one-dimensional case, we also obtain an exact order of the probability distribution of the hitting time at 0.
Liu, Jian, E-mail: jianliupku@pku.edu.cn [Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871 (China); State Key Joint Laboratory of Environmental Simulation and Pollution Control, College of Environmental Sciences and Engineering, Peking University, Beijing 100871 (China); Zhang, Zhijun [Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871 (China)
2016-01-21
Path integral Liouville dynamics (PILD) is applied to vibrational dynamics of several simple but representative realistic molecular systems (OH, water, ammonia, and methane). The dipole-derivative autocorrelation function is employed to obtain the infrared spectrum as a function of temperature and isotopic substitution. Comparison to the exact vibrational frequency shows that PILD produces a reasonably accurate peak position with a relatively small full width at half maximum. PILD offers a potentially useful trajectory-based quantum dynamics approach to compute vibrational spectra of molecular systems.
Garberoglio, Giovanni, E-mail: garberoglio@fbk.eu [Interdisciplinary Laboratory for Computational Science (LISC), FBK-CMM and University of Trento, via Sommarive 18, I-38123 Povo (Italy); Jankowski, Piotr [Department of Quantum Chemistry, Faculty of Chemistry, Nicolaus Copernicus University, Gagarina 7, PL-87-100 Toruń (Poland); Szalewicz, Krzysztof [Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716 (United States); Harvey, Allan H. [Applied Chemicals and Materials Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305-3337 (United States)
2014-07-28
We present a path-integral Monte Carlo procedure for the fully quantum calculation of the second molecular virial coefficient accounting for intramolecular flexibility. This method is applied to molecular hydrogen (H{sub 2}) and deuterium (D{sub 2}) in the temperature range 15–2000 K, showing that the effect of molecular flexibility is not negligible. Our results are in good agreement with experimental data, as well as with virials given by recent empirical equations of state, although some discrepancies are observed for H{sub 2} between 100 and 200 K.
Krajewski, Florian R.; Müser, Martin H.
2005-07-01
The spectral density of quantum mechanical Frenkel Kontorova chains moving in disordered, external potentials is investigated by means of path-integral molecular dynamics. If the second moment of the embedding potential is well defined (roughness exponent H=0), there is one regime in which the chain is pinned (large masses m of chain particles) and one in which it is unpinned (small m). If the embedding potential can be classified as a random walk on large length scales ( H=1/2), then the chain is always pinned irrespective of the value of m. For H=1/2, two phonon-like branches appear in the spectra.
Ab initio Path Integral Molecular Dynamics Based on Fragment Molecular Orbital Method
Fujita, Takatoshi; Watanabe, Hirofumi; Tanaka, Shigenori
2009-10-01
We have developed an ab initio path integral molecular dynamics method based on the fragment molecular orbital method. This “FMO-PIMD” method can treat both nuclei and electrons quantum mechanically, and is useful to simulate large hydrogen-bonded systems with high accuracy. After a benchmark calculation for water monomer, water trimer and glycine pentamer have been studied using the FMO-PIMD method to investigate nuclear quantum effects on structure and molecular interactions. The applicability of the present approach is demonstrated through a number of test calculations.
Formation of bound states in expanded metal studied via path integral molecular dynamics
Deymier, P. A.; Oh, Ki-Dong
2004-03-01
The usefulness of the restricted path integral molecular dynamics method for the study of strongly correlated electrons is demonstrated by studying the formation of bound electronic states in a half-filled expanded three-dimensional hydrogenoid body-centred cubic lattice at finite temperature. Starting from a metallic state with one-component plasma character, we find that bound electrons form upon expansion of the lattice. The bound electrons are spatially localized with their centre for the motion of gyration located at ionic positions. The number of bound electrons increases monotonically with decreasing density.
Path Integral Molecular Dynamics for Hydrogen with Orbital-Free Density Functional Theory
Runge, Keith; Karasiev, Valentin; Deymier, Pierre
2014-03-01
The computational bottleneck for performing path-integral molecular dynamics (PIMD) for nuclei on a first principles electronic potential energy surface has been the speed with which forces from the electrons can be generated. Recent advances in orbital-free density functional theory (OF-DFT) not only allow for faster generation of first principles forces but also include the effects of temperature on the electron density. We will present results of calculations on hydrogen in warm dense matter conditions where the protons are described by PIMD and the electrons by OF-DFT. Work supported by U.S. Dept. of Energy, grant DE-SC0002139.
Kapila, Vivek; Deymier, Pierre; Runge, Keith
2011-10-01
Several areas of study including heavy ion beam, large scale laser, and high pressure or Thomson scattering studies necessitate a fundamental understanding of warm dense matter (WDM) i.e. matter at high temperature and high density. The WDM regime, however, lacks any adequate highly developed class of simulation methods. Recent progress to address this deficit has been the development of orbital-free Density Functional Theory (ofDFT). However, scant benchmark information is available on temperature and pressure dependence of simple but realistic models in WDM regime. The present work aims to fill this critical gap using the restricted path-integral molecular dynamics (rPIMD) method. Within the discrete path integral representation, electrons are described as harmonic necklaces. Quantum exchange takes the form of cross linking between electron necklaces. The fermion sign problem is addressed by restricting the density matrix to positive values. The molecular dynamics algorithm is employed to sample phase space. Here, we focus on the behavior of strongly correlated electron plasmas under WDM conditions. We compute the kinetic and potential energies and compare them to those obtained with the ofDFT method. Several areas of study including heavy ion beam, large scale laser, and high pressure or Thomson scattering studies necessitate a fundamental understanding of warm dense matter (WDM) i.e. matter at high temperature and high density. The WDM regime, however, lacks any adequate highly developed class of simulation methods. Recent progress to address this deficit has been the development of orbital-free Density Functional Theory (ofDFT). However, scant benchmark information is available on temperature and pressure dependence of simple but realistic models in WDM regime. The present work aims to fill this critical gap using the restricted path-integral molecular dynamics (rPIMD) method. Within the discrete path integral representation, electrons are described as
Path-integral molecular dynamics simulations for water anion clusters (HO)5- and (DO)5-
Takayanagi, Toshiyuki; Yoshikawa, Takehiro; Motegi, Haruki; Shiga, Motoyuki
2009-11-01
Quantum path-integral molecular dynamics simulations have been performed for the (HO)5- and (DO)5- anion clusters on the basis of a semiempirical one-electron pseudopotential-polarization model. Due to larger zero-point vibrational amplitudes for H atoms than that of D atoms, hydrogen-bond lengths in the (HO)5- cluster are slightly larger than those in (DO)5-. The distribution of the vertical detachment energies for (HO)5- also show a broader feature than that for (DO)5-. The present PIMD simulations thus demonstrate the importance of nuclear quantum effects in water anion clusters.
Artoun, Ojenie; David-Rus, Diana; Emmett, Matthew; Fishman, Lou; Fital, Sandra; Hogan, Chad; Lim, Jisun; Lushi, Enkeleida; Marinov, Vesselin
2006-05-01
In this report we summarize an extension of Fourier analysis for the solution of the wave equation with a non-constant coefficient corresponding to an inhomogeneous medium. The underlying physics of the problem is exploited to link pseudodifferential operators and phase space path integrals to obtain a marching algorithm that incorporates the backward scattering into the evolution of the wave. This allows us to successfully apply single-sweep, one-way marching methods in inherently two-way environments, which was not achieved before through other methods for this problem.
Path integral representation of spin foam models of 4d gravity
Conrady, Florian
2008-01-01
We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's and Livine's SO(4) projected states. For 0 <= gamma < 1, the states can be restricted to SU(2) spin networks.
Quantum Brownian Motions and Navier-Stokes Weakly Turbulence — a Path Integral Study
Botelho, Luiz C. L.
In this paper, we present a new method to solve exactly the Schrödinger Harmonic oscillator wave equation in the presence of time-dependent parameter. We also apply such technique to solve exactly the problem of random frequency averaged quantum propagator of a harmonic oscillator with white-noise statistics frequency. We still apply our technique to solve exactly the Brownian Quantum Oscillator in the presence of an electric field. Finally, we use these quantum mechanic techniques to solve exactly the Statistical-Turbulence of the Navier-Stokes in a region of fluid random stirring weakly (analytical) coupling through time-dependent Euclidean-Quantum oscillators path-integrals.
Unified path integral approach to theories of diffusion-influenced reactions
Prüstel, Thorsten; Meier-Schellersheim, Martin
2017-08-01
Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we develop a general approach for computational modeling of diffusion-influenced reactions that is capable of capturing not only the classical Smoluchowski picture but also alternative theories, as is here exemplified by a volume reactivity model. In particular, we prove the path decomposition expansion of various Green's functions describing the irreversible and reversible reaction of an isolated pair of molecules. To this end, we exploit a connection between boundary value and interaction potential problems with δ - and δ'-function perturbation. We employ a known path-integral-based summation of a perturbation series to derive a number of exact identities relating propagators and survival probabilities satisfying different boundary conditions in a unified and systematic manner. Furthermore, we show how the path decomposition expansion represents the propagator as a product of three factors in the Laplace domain that correspond to quantities figuring prominently in stochastic spatially resolved simulation algorithms. This analysis will thus be useful for the interpretation of current and the design of future algorithms. Finally, we discuss the relation between the general approach and the theory of Brownian functionals and calculate the mean residence time for the case of irreversible and reversible reactions.
The most likely voltage path and large deviations approximations for integrate-and-fire neurons.
Paninski, Liam
2006-08-01
We develop theory and numerical methods for computing the most likely subthreshold voltage path of a noisy integrate-and-fire (IF) neuron, given observations of the neuron's superthreshold spiking activity. This optimal voltage path satisfies a second-order ordinary differential (Euler-Lagrange) equation which may be solved analytically in a number of special cases, and which may be solved numerically in general via a simple "shooting" algorithm. Our results are applicable for both linear and nonlinear subthreshold dynamics, and in certain cases may be extended to correlated subthreshold noise sources. We also show how this optimal voltage may be used to obtain approximations to (1) the likelihood that an IF cell with a given set of parameters was responsible for the observed spike train; and (2) the instantaneous firing rate and interspike interval distribution of a given noisy IF cell. The latter probability approximations are based on the classical Freidlin-Wentzell theory of large deviations principles for stochastic differential equations. We close by comparing this most likely voltage path to the true observed subthreshold voltage trace in a case when intracellular voltage recordings are available in vitro.
Horwitz, Lawrence; Zion, Yossi Ben; Lewkowicz, Meir;
2007-01-01
The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce ...... results in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions....
The Klauder-Daubechies Construction of the Phase Space Path Integral and the Harmonic Oscillator
Govaerts, Jan; Mattelaer, Olivier
2009-01-01
The canonical operator quantisation formulation corresponding to the Klauder-Daubechies construction of the phase space path integral is considered. This formulation is explicitly applied and solved in the case of the harmonic oscillator, thereby illustrating in a manner complementary to Klauder and Daubechies' original work some of the promising features offered by their construction of a quantum dynamics. The Klauder-Daubechies functional integral involves a regularisation parameter eventually taken to vanish, which defines a new physical time scale. When extrapolated to the field theory context, besides providing a new regularisation of short distance divergences, keeping a finite value for that time scale offers some tantalising prospects when it comes to strong gravitational quantum systems.
A reductionist perspective on quantum statistical mechanics: Coarse-graining of path integrals
Sinitskiy, Anton V.; Voth, Gregory A., E-mail: gavoth@uchicago.edu [Department of Chemistry, James Franck Institute, Institute for Biophysical Dynamics, and Computation Institute, The University of Chicago, 5735 S. Ellis Ave., Chicago, Illinois 60637 (United States)
2015-09-07
Computational modeling of the condensed phase based on classical statistical mechanics has been rapidly developing over the last few decades and has yielded important information on various systems containing up to millions of atoms. However, if a system of interest contains important quantum effects, well-developed classical techniques cannot be used. One way of treating finite temperature quantum systems at equilibrium has been based on Feynman’s imaginary time path integral approach and the ensuing quantum-classical isomorphism. This isomorphism is exact only in the limit of infinitely many classical quasiparticles representing each physical quantum particle. In this work, we present a reductionist perspective on this problem based on the emerging methodology of coarse-graining. This perspective allows for the representations of one quantum particle with only two classical-like quasiparticles and their conjugate momenta. One of these coupled quasiparticles is the centroid particle of the quantum path integral quasiparticle distribution. Only this quasiparticle feels the potential energy function. The other quasiparticle directly provides the observable averages of quantum mechanical operators. The theory offers a simplified perspective on quantum statistical mechanics, revealing its most reductionist connection to classical statistical physics. By doing so, it can facilitate a simpler representation of certain quantum effects in complex molecular environments.
Geng, Hua Y
2014-01-01
A multilevel approach to sample the potential energy surface in a path integral formalism is proposed. The purpose is to reduce the required number of ab initio evaluations of energy and forces in ab initio path integral molecular dynamics (AI-PIMD) simulation, without compromising the overall accuracy. To validate the method, the internal energy and free energy of an Einstein crystal are calculated and compared with the analytical solutions. As a preliminary application, we assess the performance of the method in a realistic model, the FCC phase of dense atomic hydrogen, in which the calculated result shows that the acceleration rate is about 3 to 4 fold for a two-level implementation, and can be increased to 10 times if extrapolation is used. With only 16 beads used for the ab initio potential sampling, this method gives a well converged internal energy. The residual error in pressure is just about 3 GPa, whereas it is about 20 GPa for a plain AI-PIMD calculation with the same number of beads. The vibration...
Transport properties of liquid para-hydrogen: The path integral centroid molecular dynamics approach
Yonetani, Yoshiteru; Kinugawa, Kenichi
2003-11-01
Several fundamental transport properties of a quantum liquid para-hydrogen (p-H2) at 17 K have been numerically evaluated by means of the quantum dynamics simulation called the path integral centroid molecular dynamics (CMD). For comparison, classical molecular dynamics (MD) simulations have also been performed under the same condition. In accordance with the previous path integral simulations, the calculated static properties of the liquid agree well with the experimental results. For the diffusion coefficient, thermal conductivity, and shear viscosity, the CMD predicts the values closer to the experimental ones though the classical MD results are far from the reality. The agreement of the CMD result with the experimental one is especially good for the shear viscosity with the difference less than 5%. The calculated diffusion coefficient and the thermal conductivity agree with the experimental values at least in the same order. We predict that the ratio of bulk viscosity to shear viscosity for liquid p-H2 is much larger than classical van der Waals simple liquids such as rare gas liquids.
Path integral approach for electron transport in disturbed magnetic field lines
Kanno, Ryutaro; Nakajima, Noriyoshi; Takamaru, Hisanori
2002-05-01
A path integral method is developed to investigate statistical property of an electron transport described as a Langevin equation in a statically disturbed magnetic field line structure; especially a transition probability of electrons strongly tied to field lines is considered. The path integral method has advantages that 1) it does not include intrinsically a growing numerical error of an orbit, which is caused by evolution of the Langevin equation under a finite calculation accuracy in a chaotic field line structure, and 2) it gives a method of understanding the qualitative content of the Langevin equation and assists to expect statistical property of the transport. Monte Carlo calculations of the electron distributions under both effects of chaotic field lines and collisions are demonstrated to comprehend above advantages through some examples. The mathematical techniques are useful to study statistical properties of various phenomena described as Langevin equations in general. By using parallel generators of random numbers, the Monte Carlo scheme to calculate a transition probability can be suitable for a parallel computation. (author)
Gary Raucher
2013-09-01
Full Text Available This paper examines, from an emic stance, a strand of Western esoteric wisdom that offers a particular perspective on psycho-spiritual development in relation to spiritual emergence, the mutually interdependent evolution of consciousness and substance, and the functional role of human incarnation within our planetary life. The writings of Alice A. Bailey (1880-1949 and Lucille Cedercrans (1921-1984 serve as significant reference points in this effort. These teachings hold an integral view of human development in which a person’s awareness and self-identification progress from polarization in physical matter and sensation through progressively subtler gradients of emotional and mental experience, culminating in “The Path of Initiation,” a phase of psychological and spiritual expansions into deepening levels of transcendent, supramental consciousness and functioning. The esoteric teachings described here portray this path descriptively rather than prescriptively, and have significant parallels to Sri Aurobindo’s Integral vision. Both consider human life in form to be a vital and necessary phase within the larger cosmic evolution of consciousness and matter, and both are frameworks that expansively embrace the significance of the Divine as both immanent and transcendent presence. The important issue of epistemological methodology and the testing of esoteric assertions is also considered.
Path-integrated measurements of carbon dioxide in the urban canopy layer
Büns, Christian; Kuttler, Wilhelm
2012-01-01
Continuous CO 2 concentration measurements have been recorded within the city center of Essen, Germany, using a path-integrated measuring system above a typical urban area over the course of nine months (February-October 2010). Mean monthly urban CO 2 concentrations were 396 and 446 ppm in summer and winter, respectively, which were 8.5 % in average higher than at a nearby suburban measuring site. Urban-suburban differences mainly occur due to increased CO 2 emissions from traffic and industry within the urban area, as well as domestic heating in winter. Among the analyzed meteorological variables, low wind velocities increased CO 2 concentrations as well as high atmospheric stability within the urban boundary layer, respectively. The influence of wind direction reflects the heterogeneous distribution of local CO 2 sources at the recording sites, particularly industrial point sources. Other point sources in the vicinity of the urban site strongly influence the additional point measurements but show no significant effect on the measured CO 2 concentrations by the path-integrated measuring system. Within an eight-day case study, a significant positive correlation between CO 2 concentration and traffic count ( R = 0.26; p system provides CO 2 concentrations on a greater temporal and spatial scale than common point measurements, which can be influenced by strong adjacent local CO 2 sources.
A reductionist perspective on quantum statistical mechanics: Coarse-graining of path integrals.
Sinitskiy, Anton V; Voth, Gregory A
2015-09-07
Computational modeling of the condensed phase based on classical statistical mechanics has been rapidly developing over the last few decades and has yielded important information on various systems containing up to millions of atoms. However, if a system of interest contains important quantum effects, well-developed classical techniques cannot be used. One way of treating finite temperature quantum systems at equilibrium has been based on Feynman's imaginary time path integral approach and the ensuing quantum-classical isomorphism. This isomorphism is exact only in the limit of infinitely many classical quasiparticles representing each physical quantum particle. In this work, we present a reductionist perspective on this problem based on the emerging methodology of coarse-graining. This perspective allows for the representations of one quantum particle with only two classical-like quasiparticles and their conjugate momenta. One of these coupled quasiparticles is the centroid particle of the quantum path integral quasiparticle distribution. Only this quasiparticle feels the potential energy function. The other quasiparticle directly provides the observable averages of quantum mechanical operators. The theory offers a simplified perspective on quantum statistical mechanics, revealing its most reductionist connection to classical statistical physics. By doing so, it can facilitate a simpler representation of certain quantum effects in complex molecular environments.
Isotope dependence of the lattice parameter of germanium from path-integral Monte Carlo simulations
Noya, José C.; Herrero, Carlos P.; Ramírez, Rafael
1997-07-01
The dependence of the lattice parameter upon the isotope mass for five isotopically pure Ge crystals was studied by quantum path-integral Monte Carlo simulations. The interatomic interactions in the solid were described by an empirical potential of the Stillinger-Weber type. At 50 K the isotopic effect leads to an increase of 2.3×10-4 Å in the lattice parameter of 70Ge with respect to 76Ge. Comparison of the simulation results with available experimental data for 74Ge shows that the employed model provides a realistic description of this anharmonic effect. The path-integral results were compared to those derived from a quasiharmonic approximation of the crystal. Within this approximation, the calculated fractional change of the lattice parameter of 74Ge with respect to a crystal whose atoms have the average mass of natural Ge amounts to Δa/a=-9.2×10-6 at T=0 K. Some limitations of the quasiharmonic approximation are shown at temperatures above 200 K.
Relations between the EU and Republic of Kosovo - The path of Kosovo integration towards the EU
Arif Riza
2016-07-01
Full Text Available Almost all the European Union member states have surpassed various challenges toward their integration into the European family. Although all these challenges are special cases on their own, Kosovo’s journey differs from the above mentioned cases, because Kosovo has not been recognized as an independent state by some members of the European family. The other key element that differs Kosovo’s journey from other cases is the presence of international institutions such as: EULEX, ICO, UNMIK, KFOR etc. in Kosovo’s territory. These organizations were not present in other member states of the European Union and other countries which aim for European integration. This manuscript aims to analyze the Kosovo challenges in its path towards the European family, which is only possible if Kosovo can create sustainable politics and cause fundamental changes in all fields, whether in public or private institutions, in order to build the rule of law. In general, this article will discuss the presence of international institutions in Kosovo such as: EULEX, ICO, UNMIK, KFOR and other international organizations, their effects on the rule of law, economic development and the sustainability of institutions. Moreover, this paper will particularly analyze the influence of the above mentioned factors to ease Kosovo’s path, as an observed country, compared to other countries in the region.
Tansey, Timothy N; Iwanaga, Kanako; Bezyak, Jill; Ditchman, Nicole
2017-05-04
Individuals with disabilities are more likely to live in poverty, have more health issues, and be less likely to be employed than their same-aged peers. Although these issues may be attenuated by vocational rehabilitation services, amotivation and ambivalence to employment can limit the readiness of persons with disabilities to engage in these services. Drawing on self-efficacy, self-determination, and stages of change theories, the purpose of this study was to develop and test an integrated self-determined work motivation model for people with disabilities. Participants included 277 people with disabilities recruited through vocational rehabilitation agencies across 8 states. Path analysis was used to evaluate the contribution of functional disability, self-determination, and social efficacy variables in a hypothesized integrated self-determined work motivation model. Model estimations used maximum likelihood estimation and model-data fit was examined using several goodness-of-fit indices. The initial path analysis indicated a less than optimal fit between the model and the observed data. Post hoc model modifications were conducted based on examination of the critical ratios and modification indices and theoretical consideration. The respecified integrated self-determined work motivation model fit the data very well, χ2/df = 1.88, CFI = .99, and RMSEA = 0.056. The R2 for the endogenous variables in the model ranged from .19 to .54. Findings from this study support the integrated self-determined work motivation model in vocational rehabilitation as a useful framework for understanding the relationship among functioning levels, self-determination and self-efficacy factors, vocational rehabilitation engagement, and readiness for employment. (PsycINFO Database Record (c) 2017 APA, all rights reserved).
Hamiltonian mechanics of stochastic acceleration.
Burby, J W; Zhmoginov, A I; Qin, H
2013-11-08
We show how to find the physical Langevin equation describing the trajectories of particles undergoing collisionless stochastic acceleration. These stochastic differential equations retain not only one-, but two-particle statistics, and inherit the Hamiltonian nature of the underlying microscopic equations. This opens the door to using stochastic variational integrators to perform simulations of stochastic interactions such as Fermi acceleration. We illustrate the theory by applying it to two example problems.
Harmonic bath averaged Hamiltonian: an efficient tool to capture quantum effects of large systems.
Yang, Yonggang; Liu, Xiaomeng; Meuwly, Markus; Xiao, Liantuan; Jia, Suotang
2012-11-26
Starting from a reaction path Hamiltonian, a suitably reduced harmonic bath averaged Hamiltonian is derived by averaging over all the normal mode coordinates. Generalization of the harmonic bath averaged Hamiltonian to any dimensions are performed and the feasibility to use a linear reaction path/surface are investigated and discussed. By use of a harmonic bath averaged Hamiltonian, the tunneling splitting and proton transfer dynamics of malonaldehyde is briefly discussed and shows that the harmonic bath averaged Hamiltonian is an efficient tool to capture quantum effects in larger systems.
Reinhardt, Hugo [Tuebingen Univ. (Germany). Inst. fuer Theoretische Physik
2012-11-01
The first volume of this two-volume textbook gives a modern introduction to the quantum theory, which connects Feynman's path-integral formulation with the traditional operator formalism. In easily understandable form starting from the double-slit experiment the characteristic features and foundations of quantum theory are made accessible by means of the functional-integral approach. Just this approach makes a ''derivation'' of the Schroedinger equation from the principle of the interfering alternatives possible. In the following the author developes the traditional operator formulation of quantum mechanics, which is better suited for practical solution of elementary problems. However he then refers to the functional-integral approach, when this contributes to a better understanding. A further advance of this concept: The functional-integral approach facilitates essentially the later access to quantum field theory. The work is in like manner suited for the self-study as for the deepening accompanying of the course.
Maxwell's Optics Symplectic Hamiltonian
Kulyabov, D S; Sevastyanov, L A
2015-01-01
The Hamiltonian formalism is extremely elegant and convenient to mechanics problems. However, its application to the classical field theories is a difficult task. In fact, you can set one to one correspondence between the Lagrangian and Hamiltonian in the case of hyperregular Lagrangian. It is impossible to do the same in gauge-invariant field theories. In the case of irregular Lagrangian the Dirac Hamiltonian formalism with constraints is usually used, and this leads to a number of certain difficulties. The paper proposes a reformulation of the problem to the case of a field without sources. This allows to use a symplectic Hamiltonian formalism. The proposed formalism will be used by the authors in the future to justify the methods of vector bundles (Hamiltonian bundles) in transformation optics.
Diagonalization of Hamiltonian; Diagonalization of Hamiltonian
Garrido, L. M.; Pascual, P.
1960-07-01
We present a general method to diagonalized the Hamiltonian of particles of arbitrary spin. In particular we study the cases of spin 0,1/2, 1 and see that for spin 1/2 our transformation agrees with Foldy's and obtain the expression for different observables for particles of spin C and 1 in the new representation. (Author) 7 refs.
Zheng, Zewei; Zou, Yao
2016-11-01
This paper investigates the path following control problem for an unmanned airship in the presence of unknown wind and uncertainties. The backstepping technique augmented by a robust adaptive radial basis function neural network (RBFNN) is employed as the main control framework. Based on the horizontal dynamic model of the airship, an improved adaptive integral line-of-sight (LOS) guidance law is first proposed, which suits any parametric paths. The guidance law calculates the desired yaw angle and estimates the wind. Then the controller is extended to cope with the airship yaw tracking and velocity control by resorting to the augmented backstepping technique. The uncertainties of the dynamics are compensated by using the robust RBFNNs. Each robust RBFNN utilizes an nth-order smooth switching function to combine a conventional RBFNN with a robust control. The conventional RBFNN dominates in the neural active region, while the robust control retrieves the transient outside the active region, so that the stability range can be widened. Stability analysis shows that the controlled closed-loop system is globally uniformly ultimately bounded. Simulations are provided to validate the effectiveness of the proposed control approach. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Aristeidis Antonakis
2017-04-01
Full Text Available In this article, a new multi-objective approach to the aircraft climb path optimization problem, based on the Particle Swarm Optimization algorithm, is introduced to be used for aircraft–engine integration studies. This considers a combination of a simulation with a traditional Energy approach, which incorporates, among others, the use of a proposed path-tracking scheme for guidance in the Altitude–Mach plane. The adoption of population-based solver serves to simplify case setup, allowing for direct interfaces between the optimizer and aircraft/engine performance codes. A two-level optimization scheme is employed and is shown to improve search performance compared to the basic PSO algorithm. The effectiveness of the proposed methodology is demonstrated in a hypothetic engine upgrade scenario for the F-4 aircraft considering the replacement of the aircraft’s J79 engine with the EJ200; a clear advantage of the EJ200-equipped configuration is unveiled, resulting, on average, in 15% faster climbs with 20% less fuel.
Greybody factors for Schwarzschild black holes: Path-ordered exponentials and product integrals
Gray, Finnian
2015-01-01
In recent work concerning the sparsity of the Hawking flux [arXiv:1506.03975v2], we found it necessary to re-examine what is known regarding the greybody factors of black holes, with a view to extending and expanding on some old results from the 1970s. Focussing specifically on Schwarzschild black holes, we re-calculated and re-assessed the greybody factors using a path-ordered-exponential approach, a technique which has the virtue of providing a semi-explicit formula for the relevant Bogoliubov coefficients. These path-ordered-exponentials, (being based on a "transfer matrix" formalism), are closely related to so-called "product integrals", leading to quite straightforward and direct numerical evaluation, while avoiding any need for numerically solving differential equations. Furthermore, while considerable analytic information is already available regarding both the high-frequency and low-frequency asymptotics of these greybody factors, numerical approaches seem better adapted to finding suitable "global mo...
Path integral evaluation of non-abelian anomaly and Pauli-Villars-Gupta regularization
Okuyama, K; Okuyama, Kiyoshi; Suzuki, Hiroshi
1996-01-01
When the path integral method of anomaly evaluation is applied to chiral gauge theories, two different types of gauge anomaly, i.e., the consistent form and the covariant form, appear depending on the regularization scheme for the Jacobian factor. We clarify the relation between the regularization scheme and the Pauli--Villars--Gupta (PVG) type Lagrangian level regularization. The conventional PVG, being non-gauge invariant for chiral gauge theories, in general corresponds to the consistent regularization scheme. The covariant regularization scheme, on the other hand, is realized by the generalized PVG Lagrangian recently proposed by Frolov and Slavnov. These correspondences are clarified by reformulating the PVG method as a regularization of the composite gauge current operator.
Gaussian white noise analysis and its application to Feynman path integral
Suryawan, Herry Pribawanto
2016-02-01
In applied science, Gaussian white noise (the time derivative of Brownian motion) is often chosen as a mathematical idealization of phenomena involving sudden and extremely large fluctuations. It is also possible to define and study Gaussian white noise in a mathematically rigorous framework. In this survey paper we review the Gaussian white noise as an object in an infinite dimensional topological vector space. A brief construction of Gaussian white noise space and Gaussian white noise distributions will be presented. Gaussian white noise analysis provides a framework which offers various generalization of concept known from finite dimensional analysis to the infinite dimensional case, among them are differential operators, Fourier transform, and distribution theory. We will also present some recent developments and results on the application of Gaussian white noise theory to Feynman's path integral approach for quantum mechanics.
Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
Behtash, Alireza; Schaefer, Thomas; Sulejmanpasic, Tin; Unsal, Mithat
2015-01-01
We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on ${\\mathbb R}^3\\times S^1$, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the groun...
Lee, Mi Kyung; Huo, Pengfei; Coker, David F.
2016-05-01
This article reviews recent progress in the theoretical modeling of excitation energy transfer (EET) processes in natural light harvesting complexes. The iterative partial linearized density matrix path-integral propagation approach, which involves both forward and backward propagation of electronic degrees of freedom together with a linearized, short-time approximation for the nuclear degrees of freedom, provides an accurate and efficient way to model the nonadiabatic quantum dynamics at the heart of these EET processes. Combined with a recently developed chromophore-protein interaction model that incorporates both accurate ab initio descriptions of intracomplex vibrations and chromophore-protein interactions treated with atomistic detail, these simulation tools are beginning to unravel the detailed EET pathways and relaxation dynamics in light harvesting complexes.
Path-integral simulation of ice VII: Pressure and temperature effects
Herrero, Carlos P
2015-01-01
Effects of pressure and temperature on structural and thermodynamic properties of ice VII have been studied by using path-integral molecular dynamics (PIMD) simulations. Temperatures between 25 and 450 K, as well as pressures up to 12 GPa were considered. Interatomic interactions were modeled by using the effective q-TIP4P/F potential for flexible water. We analyze the pressure dependence of the molar volume, bulk modulus, interatomic distances, kinetic energy, and atomic delocalization at various temperatures. Results of PIMD simulations are compared with those derived from a quasi-harmonic approximation (QHA) of vibrational modes, which helps to assess the importance of anharmonic effects, as well as the influence of the different modes on the properties of ice VII. The accuracy of the QHA for describing this high-pressure phase decreases for rising temperature, but this approximation becomes more reliable as pressure grows, since anharmonicity becomes less relevant. Comparisons with low-pressure cubic ice ...
PATH INTEGRAL SOLUTION OF NONLINEAR DYNAMIC BEHAVIOR OF STRUCTURE UNDER WIND EXCITATION
无
2005-01-01
A numerical scheme for the nonlinear behavior of structure under wind excitation is investigated. With the white noise filter of turbulent-wind fluctuations, the nonlinear motion equation of structures subjected to wind load was modeled as the Ito' s stochastic differential equation. The state vector associated with such a model is a diffusion process. A continuous linearization strategy in the time-domain was adopted.Based on the solution series of its stochastic linearization equations, the formal probabilistic density of the structure response was developed by the path integral technique. It is shown by the numerical example of a guyed mast that compared with the frequency-domain method and the time-domain nonlinear analysis, the proposed approach is highlighted by high accuracy and effectiveness. The influence of the structure non-linearity on the dynamic reliability assessment is also analyzed in the example.
A New Perspective on Path Integral Quantum Mechanics in Curved Space-Time
Singh Dinesh
2013-09-01
Full Text Available Abstract. A new approach to path integral quantum mechanics in curved space-time is presented for scalar particle propagation, expressed in terms of Lie transport and Fermi or Riemann normal co-ordinates to describe local curvature. While the presence of local curvature results in a strictly non-unitary representation of local time translation, the formalism nevertheless correctly recovers the free-particle Lagrangian in curved space-time, along with new terms that predict a simultaneous breakdown of time-reversal symmetry and a quantum violation of the weak equivalence principle at the particle’s Compton wavelength scale. Furthermore, the formalism reveals the prediction of a gauge-invariant phase factor interpreted as the gravitational Aharonov-Bohm effect and Berry’s phase.
Path integral centroid molecular dynamics simulation of para-hydrogen sandwiched by graphene sheets
Minamino, Yuki; Kinugawa, Kenichi
2016-11-01
The carbon-hydrogen composite systems of para-hydrogen (p-H2) sandwiched by a couple of graphene sheets have been investigated by means of path integral centroid molecular dynamics simulations at 17 K. It has been shown that sandwiched hydrogen is liquid-like but p-H2 molecules are preferably adsorbed onto the graphene sheets because of attractive graphene-hydrogen interaction. The diffusion coefficient of p-H2 molecules in the direction parallel to the graphene sheets is comparable to that in pure liquid p-H2. There exists a characteristic mode of 140 cm-1 of the p-H2 molecules, attributed to adsorption-binding motion perpendicular to the graphene sheets.
Isotope effects in water as investigated by neutron diffraction and path integral molecular dynamics
Zeidler, Anita [University of Bath; Salmon, Phil [University of Bath; Fischer, Henry E [Institut Laue-Langevin (ILL); Neuefeind, Joerg C [ORNL; Simonson, J Michael {Mike} [ORNL; Markland, Thomas [Columbia University
2012-01-01
The structure of heavy and light water at 300 K was investigated by using a joint approach in which the method of neutron di raction with oxygen isotope substitution was combined with path integral molecular dynamics simulations. The di raction results, which give intra-molecular O-D and O-H bond distances of 0.985(5) and 0.990(5) A, were found to be in best agreement with those obtained by using the exible anharmonic TTM3-F water model. Both techniques show a di erence of '0.5% between the O-D and O-H intra-molecular bond lengths and the results support a competing quantum e ects model for water in which its structural and dynamical properties are governed by an o set between intra-molecular and inter-molecular quantum contributions. Further consideration of the O-O correlations is needed in order to improve agreement with experiment.
Kinugawa, Kenichi [Nara Women`s Univ., Nara (Japan). Dept. of Chemistry
1998-10-01
It has been unsuccessful to solve a set of time-dependent Schroedinger equations numerically for many-body quantum systems which involve, e.g., a number of hydrogen molecules, protons, and excess electrons at a low temperature, where quantum effect evidently appears. This undesirable situation is fatal for the investigation of real low-temperature chemical systems because they are essentially composed of many quantum degrees of freedom. However, if we use a new technique called `path integral centroid molecular dynamics (CMD) simulation` proposed by Cao and Voth in 1994, the real-time semi-classical dynamics of many degrees of freedom can be computed by utilizing the techniques already developed in the traditional classical molecular dynamics (MD) simulations. Therefore, the CMD simulation is expected to be very powerful tool for the quantum dynamics studies or real substances. (J.P.N.)
Wang, Qi; Suzuki, Kimichi; Nagashima, Umpei; Tachikawa, Masanori; Yan, Shiwei
2013-11-01
The geometric isotope effects on the structures of hydrated chloride ionic hydrogen bonded clusters are explored by carrying out path integral molecular dynamics simulations. First, an outer shell coordinate is selected to display the rearrangement of single and multi hydration shell cluster structures. Next, to show the competition of intramolecular and intermolecular nuclear quantum effects, the intramolecular OH∗ stretching and intermolecular ion-water wagging motions are studied for single and multi shell structures, respectively. The results indicate that the intermolecular nuclear quantum effects stabilize the ionic hydrogen bonds in single shell structures, while they are destabilized through the competition with intramolecular nuclear quantum effects in multi shell structures. In addition, the correlations between ion-water stretching motion and other cluster vibrational coordinates are discussed. The results indicate that the intermolecular nuclear quantum effects on the cluster structures are strongly related to the cooperation of the water-water hydrogen bond interactions.
Reilly, Anthony M.; Habershon, Scott; Morrison, Carole A.; Rankin, David W. H.
2010-03-01
Path-integral molecular dynamics (PIMD) simulations with an empirical interaction potential have been used to determine the experimental equilibrium structure of solid nitromethane at 4.2 and 15 K. By comparing the time-averaged molecular structure determined in a PIMD simulation to the calculated minimum-energy (zero-temperature) molecular structure, we have derived structural corrections that describe the effects of thermal motion. These corrections were subsequently used to determine the equilibrium structure of nitromethane from the experimental time-averaged structure. We find that the corrections to the intramolecular and intermolecular bond distances, as well as to the torsion angles, are quite significant, particularly for those atoms participating in the anharmonic motion of the methyl group. Our results demonstrate that simple harmonic models of thermal motion may not be sufficiently accurate, even at low temperatures, while molecular simulations employing more realistic potential-energy surfaces can provide important insight into the role and magnitude of anharmonic atomic motions.
Path Integral Monte Carlo and Density Functional Molecular Dynamics Simulations of Warm Dense Matter
Militzer, Burkhard; Driver, Kevin
2011-10-01
We analyze the applicability of two first-principles simulation techniques, path integral Monte Carlo (PIMC) and density functional molecular dynamics (DFT-MD), to study the regime of warm dense matter. We discuss the advantages as well as the limitations of each method and propose directions for future development. Results for dense, liquid helium, where both methods have been applied, demonstrate the range of each method's applicability. Comparison of the equations of state from simulations with analytical theories and free energy models show that DFT is useful for temperatures below 100000 K and then PIMC provides accurate results for all higher temperatures. We characterize the structure of the liquid in terms of pair correlation functions and study the closure of the band gap with increasing density and temperature. Finally, we discuss simulations of heavier elements and demonstrate the reliability are both methods in such cases with preliminary results.
Ananth, Nandini
2013-09-01
We introduce mapping-variable ring polymer molecular dynamics (MV-RPMD), a model dynamics for the direct simulation of multi-electron processes. An extension of the RPMD idea, this method is based on an exact, imaginary time path-integral representation of the quantum Boltzmann operator using continuous Cartesian variables for both electronic states and nuclear degrees of freedom. We demonstrate the accuracy of the MV-RPMD approach in calculations of real-time, thermal correlation functions for a range of two-state single-mode model systems with different coupling strengths and asymmetries. Further, we show that the ensemble of classical trajectories employed in these simulations preserves the Boltzmann distribution and provides a direct probe into real-time coupling between electronic state transitions and nuclear dynamics.
Low-temperature metallic liquid hydrogen: an ab-initio path-integral molecular dynamics perspective
Chen, Ji; Li, Xin-Zheng; Zhang, Qianfan; Probert, Matthew; Pickard, Chris; Needs, Richard; Michaelides, Angelos; Wang, Enge
2013-03-01
Experiments and computer simulations have shown that the melting temperature of solid hydrogen drops with pressure above about 65 GPa, suggesting that a low temperature liquid state might exist. It has also been suggested that this liquid state might be non-molecular and metallic, although evidence for such behaviour is lacking. Using a combination of ab initio path-integral molecular dynamics and the two-phase methods, we have simulated the melting of solid hydrogen under finite temperatures. We found an atomic solid phase from 500 to 800 GPa which melts at < 200 K. Beyond this and up to pressures of 1,200 GPa a metallic atomic liquid is stable at temperatures as low as 50 K. The quantum motion of the protons is critical to the low melting temperature in this system as ab initio simulations with classical nuclei lead to a considerably higher melting temperature of ~300 K across the entire pressure range considered.
Hydrogen and muonium in diamond: A path-integral molecular dynamics simulation
Herrero, Carlos P.; Ramírez, Rafael; Hernández, Eduardo R.
2006-06-01
Isolated hydrogen, deuterium, and muonium in diamond have been studied by path-integral molecular dynamics simulations in the canonical ensemble. Finite-temperature properties of these point defects were analyzed in the range from 100 to 800K . Interatomic interactions were modeled by a tight-binding potential fitted to density-functional calculations. The most stable position for these hydrogenic impurities is found at the C-C bond center. Vibrational frequencies have been obtained from a linear-response approach, based on correlations of atom displacements at finite temperatures. The results show a large anharmonic effect in impurity vibrations at the bond center site, which hardens the vibrational modes with respect to a harmonic approximation. Zero-point motion causes an appreciable shift of the defect level in the electronic gap, as a consequence of electron-phonon interaction. This defect level goes down by 70meV when replacing hydrogen by muonium.
Witt, Alexander; Ivanov, Sergei D.; Shiga, Motoyuki; Forbert, Harald; Marx, Dominik
2009-05-01
Centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) are two conceptually distinct extensions of path integral molecular dynamics that are able to generate approximate quantum dynamics of complex molecular systems. Both methods can be used to compute quasiclassical time correlation functions which have direct application in molecular spectroscopy; in particular, to infrared spectroscopy via dipole autocorrelation functions. The performance of both methods for computing vibrational spectra of several simple but representative molecular model systems is investigated systematically as a function of temperature and isotopic substitution. In this context both CMD and RPMD feature intrinsic problems which are quantified and investigated in detail. Based on the obtained results guidelines for using CMD and RPMD to compute infrared spectra of molecular systems are provided.
Isotope effects in water as investigated by neutron diffraction and path integral molecular dynamics
Zeidler, Anita; Salmon, Philip S.; Fischer, Henry E.; Neuefeind, Jörg C.; Simonson, J. Mike; Markland, Thomas E.
2012-07-01
The structures of heavy and light water at 300 K were investigated by using a joint approach in which the method of neutron diffraction with oxygen isotope substitution was complemented by path integral molecular dynamics simulations. The diffraction results, which give intra-molecular O-D and O-H bond distances of 0.985(5) and 0.990(5) Å, were found to be in best agreement with those obtained by using the flexible anharmonic TTM3-F water model. Both techniques show a difference of ≃ 0.5% between the O-D and O-H intra-molecular bond lengths, and the results support a competing quantum effects model for water in which its structural and dynamical properties are governed by an offset between intra-molecular and inter-molecular quantum contributions. Further consideration of the O-O correlations is needed in order to improve agreement with experiment.
Yoshikawa, Takehiro; Sugawara, Shuichi; Takayanagi, Toshiyuki; Shiga, Motoyuki; Tachikawa, Masanori
2012-02-01
Path-integral molecular dynamics simulations have been performed for porphycene and its isotopic variants in order to understand the effect of isotopic substitution of inner protons on the double proton transfer mechanism. We have used an on-the-fly direct dynamics technique at the semiempirical PM6 level combined with specific reaction parameterization. Our quantum simulations show that double proton transfer of the unsubstituted porphycene at T = 300 K mainly occurs via a so-called concerted mechanism through the D2h second-order saddle point. In addition, we found that both isotopic substitution and temperature significantly affect the double proton transfer mechanism. For example, the contribution of the stepwise mechanism increases with a temperature increase. We have also carried out hypothetical simulations with the porphycene configurations being completely planar. It has been found that out-of-plane vibrational motions significantly decrease the contribution of the concerted proton transfer mechanism.
The canonical versrus path integral quantization approach to generalized Kodama states (Part I)
Ita, Eyo Eyo
2007-01-01
This is the fifth paper in the series outlining an algorithm to consistently quantize four-dimensional gravity. We derive the pure Kodama state in analogy to the no-boundary proposal for constructing quantum gravitational wavefunctions, checking at each stage of the process the equivalence of the canonical and path integral approaches. A family of additional pure Kodama states is identified via the canonical approach and a criterion for their suitability as a basis of states is examined. We provide an interpretation for the problem of time within the context of the generalized Kodama states and propose a possible method of resolution. We also develop different techniques for solving the Gauss' law constraintsd at the kinematical level, in preparation for future work in this series.
Path integral polymer propagator of relativistic and non-relativistic particles
Morales-Técotl, Hugo A; Ruelas, Juan C
2016-01-01
A recent proposal to connect the loop quantization with the spin foam model for cosmology via the path integral is hereby adapted to the case of mechanical systems within the framework of the so called polymer quantum mechanics. The mechanical models we consider are deparametrized and thus the group averaging technique is used to deal with the corresponding constraints. The transition amplitudes are written in a vertex expansion form used in the spin foam models, where here a vertex is actually a jump in position. Polymer Propagators previously obtained by spectral methods for a nonrelativistic polymer particle, both free and in a box, are regained with this method. Remarkably, the approach is also shown to yield the polymer propagator of the relativistic particle. This reduces to the standard form in the continuum limit for which the length scale parameter of the polymer quantization is taken to be small. Some possible future developments are commented upon.
Quantum Mechanical Single Molecule Partition Function from PathIntegral Monte Carlo Simulations
Chempath, Shaji; Bell, Alexis T.; Predescu, Cristian
2006-10-01
An algorithm for calculating the partition function of a molecule with the path integral Monte Carlo method is presented. Staged thermodynamic perturbation with respect to a reference harmonic potential is utilized to evaluate the ratio of partition functions. Parallel tempering and a new Monte Carlo estimator for the ratio of partition functions are implemented here to achieve well converged simulations that give an accuracy of 0.04 kcal/mol in the reported free energies. The method is applied to various test systems, including a catalytic system composed of 18 atoms. Absolute free energies calculated by this method lead to corrections as large as 2.6 kcal/mol at 300 K for some of the examples presented.
Topics in mode conversion theory and the group theoretical foundations of path integrals
Richardson, Andrew Stephen
discrete Beisenberg-Wey1 group to construct the symbol of a matrix. We then go on to show how the path integral arises when calculating the symbol of a function of an operator. We also show how the phase space and configuration space path integrals arise when considering reductions of the regular representation of the Heisenberg-Wey1 group to the primary representations and irreducible representations, respectively. We also show how the path integral can be interpreted as a Fourier transform on the space of measures, opening up the possibility of using tools from statistical mechanics (such as maximum entropy techniques) to analyze the path integral. We conclude with a survey of ideas for future research and describe several potential applications of this group theoretical perspective to problems in mode conversion.
Accelerating the convergence of path integral dynamics with a generalized Langevin equation.
Ceriotti, Michele; Manolopoulos, David E; Parrinello, Michele
2011-02-28
The quantum nature of nuclei plays an important role in the accurate modelling of light atoms such as hydrogen, but it is often neglected in simulations due to the high computational overhead involved. It has recently been shown that zero-point energy effects can be included comparatively cheaply in simulations of harmonic and quasiharmonic systems by augmenting classical molecular dynamics with a generalized Langevin equation (GLE). Here we describe how a similar approach can be used to accelerate the convergence of path integral (PI) molecular dynamics to the exact quantum mechanical result in more strongly anharmonic systems exhibiting both zero point energy and tunnelling effects. The resulting PI-GLE method is illustrated with applications to a double-well tunnelling problem and to liquid water.
Path-integral action of a particle in the noncommutative plane.
Gangopadhyay, Sunandan; Scholtz, Frederik G
2009-06-19
Noncommutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on noncommutative configuration space. Taking this as a departure point, we formulate a coherent state approach to the path-integral representation of the transition amplitude. From this we derive an action for a particle moving in the noncommutative plane and in the presence of an arbitrary potential. We find that this action is nonlocal in time. However, this nonlocality can be removed by introducing an auxilary field, which leads to a second class constrained system that yields the noncommutative Heisenberg algebra upon quantization. Using this action, the propagator of the free particle and harmonic oscillator are computed explicitly.
Path-integral Monte Carlo method for Rényi entanglement entropies.
Herdman, C M; Inglis, Stephen; Roy, P-N; Melko, R G; Del Maestro, A
2014-07-01
We introduce a quantum Monte Carlo algorithm to measure the Rényi entanglement entropies in systems of interacting bosons in the continuum. This approach is based on a path-integral ground state method that can be applied to interacting itinerant bosons in any spatial dimension with direct relevance to experimental systems of quantum fluids. We demonstrate how it may be used to compute spatial mode entanglement, particle partitioned entanglement, and the entanglement of particles, providing insights into quantum correlations generated by fluctuations, indistinguishability, and interactions. We present proof-of-principle calculations and benchmark against an exactly soluble model of interacting bosons in one spatial dimension. As this algorithm retains the fundamental polynomial scaling of quantum Monte Carlo when applied to sign-problem-free models, future applications should allow for the study of entanglement entropy in large-scale many-body systems of interacting bosons.
Correct Path-Integral Formulation of Quantum Thermal Field Theory in Coherent State Representation
SU Jun-Chen; ZHENG Fu-Hou
2005-01-01
The path-integral quantization of thermal scalar, vector, and spinor fields is performed newly in the coherent-state representation. In doing this, we choose the thermal electrodynamics and ψ4 theory as examples. By this quantization, correct expressions of the partition functions and the generating functionals for the quantum thermal electrodynamics and ψ4 theory are obtained in the coherent-state representation. These expressions allow us to perform analytical calculations of the partition functions and generating functionals and therefore are useful in practical applications. Especially, the perturbative expansions of the generating functionals are derived specifically by virtue of the stationary-phase method. The generating functionals formulated in the position space are re-derived from the ones given in the coherent-state representation.
Finite Size Effect in Path Integral Monte Carlo Simulations of 4He Systems
ZHAO Xing-Wen; CHENG Xin-Lu
2008-01-01
Path integral Monte Carlo (PIMC) simulations are a powerful computational method to study interacting quantum systems at finite temperatures. In this work, PIMC has been applied to study the finite size effect of the simulated systems of 4He. We determine the energy as a function of temperature at saturated-vapor-pressure (SVP) conditions in the temperature range of T ∈ [1.0 K,4.0 K], and the equation of state (EOS) in the ground state for systems consisted of 32, 64 and 128 4He atoms, respectively. We find that the energy at SVP is influenced significantly by the size of the simulated system in the temperature range of T ∈ [2.1 K, 3.0 K] and the larger the system is, the better results are obtained in comparison with the experimental values; while the EOS appeared to be unrelated to it.
Thermally assisted tunneling of hydrogen in silicon: A path-integral Monte Carlo study
Herrero, Carlos P.
1997-04-01
Quantum transition-state theory, based on the path-integral formalism, has been applied to study the jump rate of atomic hydrogen and deuterium in crystalline silicon. This technique provides a methodology to study the influence of vibrational mode quantization and quantum tunneling on the impurity jump rate. The atomic interactions were modeled by effective potentials, fitted to earlier ab initio pseudopotential calculations. Silicon nuclei were treated as quantum particles up to second-nearest neighbors of the impurity. The hydrogen jump rate follows an Arrhenius law, describable with classical transition-state theory, at temperatures higher than 100 K. At ~80 K, a change in the slope of the Arrhenius plot is obtained for hydrogen, as expected for the onset of a diffusion regime controlled by phonon-assisted tunneling of the impurity. For deuterium, no change of slope is observed in the studied temperature range (down to 40 K).
A PATH INTEGRAL FORMULATION OF THE WRIGHT-FISHER PROCESS WITH GENIC SELECTION
SCHRAIBER, JOSHUA G.
2014-01-01
The Wright-Fisher process with selection is an important tool in population genetics theory. Traditional analysis of this process relies on the diffusion approximation. The diffusion approximation is usually studied in a partial differential equations framework. In this paper, I introduce a path integral formalism to study the Wright-Fisher process with selection and use that formalism to obtain a simple perturbation series to approximate the transition density. The perturbation series can be understood in terms of Feynman diagrams, which have a simple probabilistic interpretation in terms of selective events. The perturbation series proves to be an accurate approximation of the transition density for weak selection and is shown to be arbitrarily accurate for any selection coefficient. PMID:24269333
Relaxation of two coupled quantum oscillators to quasi-equilibrium states based on path integrals
Dorofeyev, Illarion
2013-01-01
The paper addresses the problem of relaxation of open quantum systems. Using the path integral methods we found an analytical expression for time-dependent density matrix of two coupled quantum oscillators interacting with different baths of oscillators. The expression for density matrix was found in the linear regime with respect to the coupling constant between selected oscillators. Time-dependent spatial variances and covariance were investigated analytically and numerically. It was shown that asymptotic variances in the long-time limit are always in accordance with the fluctuation dissipation theorem despite on their initial values. In the weak coupling approach there is good reason to believe that subsystems asymptotically in equilibrium at their own temperatures even despite of the arbitrary difference in temperatures within the whole system.
Singh, Upendra N.; Yu, Jirong; Petros, Mulugeta; Refaat, Tamer F.; Remus, Ruben G.; Fay, James J.; Reithmaier, Karl
2014-01-01
Double-pulse 2-micron lasers have been demonstrated with energy as high as 600 millijouls and up to 10 Hz repetition rate. The two laser pulses are separated by 200 microseconds and can be tuned and locked separately. Applying double-pulse laser in DIAL system enhances the CO2 measurement capability by increasing the overlap of the sampled volume between the on-line and off-line. To avoid detection complicity, integrated path differential absorption (IPDA) lidar provides higher signal-to-noise ratio measurement compared to conventional range-resolved DIAL. Rather than weak atmospheric scattering returns, IPDA rely on the much stronger hard target returns that is best suited for airborne platforms. In addition, the IPDA technique measures the total integrated column content from the instrument to the hard target but with weighting that can be tuned by the transmitter. Therefore, the transmitter could be tuned to weight the column measurement to the surface for optimum CO2 interaction studies or up to the free troposphere for optimum transport studies. Currently, NASA LaRC is developing and integrating a double-Pulsed 2-micron direct detection IPDA lidar for CO2 column measurement from an airborne platform. The presentation will describe the development of the 2-micron IPDA lidar system and present the airborne measurement of column CO2 and will compare to in-situ measurement for various ground target of different reflectivity.
Maintaining a cognitive map in darkness: the need to fuse boundary knowledge with path integration.
Cheung, Allen; Ball, David; Milford, Michael; Wyeth, Gordon; Wiles, Janet
2012-01-01
Spatial navigation requires the processing of complex, disparate and often ambiguous sensory data. The neurocomputations underpinning this vital ability remain poorly understood. Controversy remains as to whether multimodal sensory information must be combined into a unified representation, consistent with Tolman's "cognitive map", or whether differential activation of independent navigation modules suffice to explain observed navigation behaviour. Here we demonstrate that key neural correlates of spatial navigation in darkness cannot be explained if the path integration system acted independently of boundary (landmark) information. In vivo recordings demonstrate that the rodent head direction (HD) system becomes unstable within three minutes without vision. In contrast, rodents maintain stable place fields and grid fields for over half an hour without vision. Using a simple HD error model, we show analytically that idiothetic path integration (iPI) alone cannot be used to maintain any stable place representation beyond two to three minutes. We then use a measure of place stability based on information theoretic principles to prove that featureless boundaries alone cannot be used to improve localization above chance level. Having shown that neither iPI nor boundaries alone are sufficient, we then address the question of whether their combination is sufficient and--we conjecture--necessary to maintain place stability for prolonged periods without vision. We addressed this question in simulations and robot experiments using a navigation model comprising of a particle filter and boundary map. The model replicates published experimental results on place field and grid field stability without vision, and makes testable predictions including place field splitting and grid field rescaling if the true arena geometry differs from the acquired boundary map. We discuss our findings in light of current theories of animal navigation and neuronal computation, and elaborate on
Maintaining a cognitive map in darkness: the need to fuse boundary knowledge with path integration.
Allen Cheung
Full Text Available Spatial navigation requires the processing of complex, disparate and often ambiguous sensory data. The neurocomputations underpinning this vital ability remain poorly understood. Controversy remains as to whether multimodal sensory information must be combined into a unified representation, consistent with Tolman's "cognitive map", or whether differential activation of independent navigation modules suffice to explain observed navigation behaviour. Here we demonstrate that key neural correlates of spatial navigation in darkness cannot be explained if the path integration system acted independently of boundary (landmark information. In vivo recordings demonstrate that the rodent head direction (HD system becomes unstable within three minutes without vision. In contrast, rodents maintain stable place fields and grid fields for over half an hour without vision. Using a simple HD error model, we show analytically that idiothetic path integration (iPI alone cannot be used to maintain any stable place representation beyond two to three minutes. We then use a measure of place stability based on information theoretic principles to prove that featureless boundaries alone cannot be used to improve localization above chance level. Having shown that neither iPI nor boundaries alone are sufficient, we then address the question of whether their combination is sufficient and--we conjecture--necessary to maintain place stability for prolonged periods without vision. We addressed this question in simulations and robot experiments using a navigation model comprising of a particle filter and boundary map. The model replicates published experimental results on place field and grid field stability without vision, and makes testable predictions including place field splitting and grid field rescaling if the true arena geometry differs from the acquired boundary map. We discuss our findings in light of current theories of animal navigation and neuronal computation
Agarwal, Animesh
2015-01-01
Quantum effects due to the spatial delocalization of light atoms are treated in molecular simulation via the path integral technique. Among several methods, Path Integral (PI) Molecular Dynamics (MD) is nowadays a powerful tool to investigate properties induced by spatial delocalization of atoms; however computationally this technique is very demanding. The abovementioned limitation implies the restriction of PIMD applications to relatively small systems and short time scales. One possible solution to overcome size and time limitation is to introduce PIMD algorithms into the Adaptive Resolution Simulation Scheme (AdResS). AdResS requires a relatively small region treated at path integral level and embeds it into a large molecular reservoir consisting of generic spherical coarse grained molecules. It was previously shown that the realization of the idea above, at a simple level, produced reasonable results for toy systems or simple/test systems like liquid parahydrogen. Encouraged by previous results, in this ...
Hamiltonian Dynamics of Cosmological Quintessence Models
Ivanov, Rossen I
2016-01-01
The time-evolution dynamics of two nonlinear cosmological real gas models has been reexamined in detail with methods from the theory of Hamiltonian dynamical systems. These examples are FRWL cosmologies, one based on a gas, satisfying the van der Waals equation and another one based on the virial expansion gas equation. The cosmological variables used are the expansion rate, given by the Hubble parameter, and the energy density. The analysis is aided by the existence of global first integral as well as several special (second) integrals in each case. In addition, the global first integral can serve as a Hamiltonian for a canonical Hamiltonian formulation of the evolution equations. The conserved quantities lead to the existence of stable periodic solutions (closed orbits) which are models of a cyclic Universe. The second integrals allow for explicit solutions as functions of time on some special trajectories and thus for a deeper understanding of the underlying physics. In particular, it is shown that any pos...
Scattering from the Potential Barrier $V=cosh^{-2} \\omega x$ from the Path Integration over SO(1,2)
Ahmedov, H
1996-01-01
Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and noncompact basis and the new addition theorems are derived which are required in path integral applications involving the positively signed potential. The Green function for the potential barrier $V=cosh^{-2}\\omega x$ is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup.The transition and the reflection coefficients are given.Results for the moving barrier $V=cosh^{-2}\\omega (x-g_0t)$ are also presented.
Song, Linze; Shi, Qiang
2015-05-07
We present a new non-perturbative method to calculate the charge carrier mobility using the imaginary time path integral approach, which is based on the Kubo formula for the conductivity, and a saddle point approximation to perform the analytic continuation. The new method is first tested using a benchmark calculation from the numerical exact hierarchical equations of motion method. Imaginary time path integral Monte Carlo simulations are then performed to explore the temperature dependence of charge carrier delocalization and mobility in organic molecular crystals (OMCs) within the Holstein and Holstein-Peierls models. The effects of nonlocal electron-phonon interaction on mobility in different charge transport regimes are also investigated.
A Hamiltonian theory of adaptive resolution simulations of classical and quantum models of nuclei
Kreis, Karsten; Donadio, Davide; Kremer, Kurt; Potestio, Raffaello
2015-03-01
Quantum delocalization of atomic nuclei strongly affects the physical properties of low temperature systems, such as superfluid helium. However, also at room temperature nuclear quantum effects can play an important role for molecules composed by light atoms. An accurate modeling of these effects is possible making use of the Path Integral formulation of Quantum Mechanics. In simulations, this numerically expensive description can be restricted to a small region of space, while modeling the remaining atoms as classical particles. In this way the computational resources required can be significantly reduced. In the present talk we demonstrate the derivation of a Hamiltonian formulation for a bottom-up, theoretically solid coupling between a classical model and a Path Integral description of the same system. The coupling between the two models is established with the so-called Hamiltonian Adaptive Resolution Scheme, resulting in a fully adaptive setup in which molecules can freely diffuse across the classical and the Path Integral regions by smoothly switching their description on the fly. Finally, we show the validation of the approach by means of adaptive resolution simulations of low temperature parahydrogen. Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany.
An intuitive Hamiltonian for quantum search
Fenner, S A
2000-01-01
We present new intuition behind Grover's quantum search algorithm by means of a Hamiltonian. Given a black-box Boolean function f mapping strings of length n into {0,1} such that f(w) = 1 for exactly one string w, L. K. Grover describes a quantum algorithm that finds w in O(2^{n/2}) time. Farhi & Gutmann show that w can also be found in the same amount time by letting the quantum system evolve according to a simple Hamiltonian depending only on f. Their system evolves along a path far from that taken by Grover's original algorithm, however. The current paper presents an equally simple Hamiltonian matching Grover's algorithm step for step. The new Hamiltonian is similar in appearance from that of Farhi & Gutmann, but has some important differences, and provides new intuition for Grover's algorithm itself. This intuition both contrasts with and supplements other explanations of Grover's algorithm as a rotation in two dimensions, and suggests that the Hamiltonian-based approach to quantum algorithms can ...
The Effect of Learning in Virtual Path Integration%虚拟路径整合的学习效应
过继成思; 宛小昂
2015-01-01
Path integration is one type of navigations in which navigators integrate self-motion information to update their current position and orientation relative to the origin of their travel. Human path integration ability is often measured in the path completion task. In this task, participants travel along several segments, make several turns at the intersections of each two segments, and arrive at the end of the outbound path. Then they are asked to directly return to the origin of the outbound path. Previous studies have revealed that athletes showed better path completion performance than general population. The purpose of the present study was to examine whether the path integration ability of general population can be improved if they are repeatedly exposed to outbound paths with the same configurations. In two experiments, we used the Head-Mounted Display Virtual Reality to present hallway mazes, and each outbound path consisted of 5 segments. Participants pressed a button on the gamepad to travel along a segment, so the information about transition was based on optical flow. By contrast, they were asked to actually rotate their bodies at the intersections, so the information about rotation came from both optical flow and body senses. Each participant completed 4 blocks, 6 trials of each. Within each block, they performed the path completion task on 6 different outbound paths. From one block to the next, they performed the path completion task on outbound paths with the same configurations. In Experiment 1, all the 5 segments within each outbound path had the same lengths, and the turning angle at each interaction was always 60 degree, clockwise or counterclockwise. When the participants repeatedly performed the path completion task on these outbound paths with the same configurations, they showed reduced position errors, direction errors, and RTs. By contrast, more complicated path configurations were used in Experiment 2. Specifically, within each outbound
International Organization for Standardization. Geneva
2003-01-01
Information technology - Telecommunications and information exchange between systems - Private integrated services network - Inter-exchange signalling protocol - Path replacement additional network feature
Running Couplings in Hamiltonians
Glazek, S D
2000-01-01
We describe key elements of the perturbative similarity renormalization group procedure for Hamiltonians using two, third-order examples: phi^3 interaction term in the Hamiltonian of scalar field theory in 6 dimensions and triple-gluon vertex counterterm in the Hamiltonian of QCD in 4 dimensions. These examples provide insight into asymptotic freedom in Hamiltonian approach to quantum field theory. The renormalization group procedure also suggests how one may obtain ultraviolet-finite effective Schrödinger equations that correspond to the asymptotically free theories, including transition from quark and gluon to hadronic degrees of freedom in case of strong interactions. The dynamics is invariant under boosts and allows simultaneous analysis of bound state structure in the rest and infinite momentum frames.
PathText: a text mining integrator for biological pathway visualizations
Kemper, Brian; Matsuzaki, Takuya; Matsuoka, Yukiko; Tsuruoka, Yoshimasa; Kitano, Hiroaki; Ananiadou, Sophia; Tsujii, Jun'ichi
2010-01-01
Motivation: Metabolic and signaling pathways are an increasingly important part of organizing knowledge in systems biology. They serve to integrate collective interpretations of facts scattered throughout literature. Biologists construct a pathway by reading a large number of articles and interpreting them as a consistent network, but most of the models constructed currently lack direct links to those articles. Biologists who want to check the original articles have to spend substantial amounts of time to collect relevant articles and identify the sections relevant to the pathway. Furthermore, with the scientific literature expanding by several thousand papers per week, keeping a model relevant requires a continuous curation effort. In this article, we present a system designed to integrate a pathway visualizer, text mining systems and annotation tools into a seamless environment. This will enable biologists to freely move between parts of a pathway and relevant sections of articles, as well as identify relevant papers from large text bases. The system, PathText, is developed by Systems Biology Institute, Okinawa Institute of Science and Technology, National Centre for Text Mining (University of Manchester) and the University of Tokyo, and is being used by groups of biologists from these locations. Contact: brian@monrovian.com. PMID:20529930
Ab initio path-integral molecular dynamics and the quantum nature of hydrogen bonds
Yexin, Feng; Ji, Chen; Xin-Zheng, Li; Enge, Wang
2016-01-01
The hydrogen bond (HB) is an important type of intermolecular interaction, which is generally weak, ubiquitous, and essential to life on earth. The small mass of hydrogen means that many properties of HBs are quantum mechanical in nature. In recent years, because of the development of computer simulation methods and computational power, the influence of nuclear quantum effects (NQEs) on the structural and energetic properties of some hydrogen bonded systems has been intensively studied. Here, we present a review of these studies by focussing on the explanation of the principles underlying the simulation methods, i.e., the ab initio path-integral molecular dynamics. Its extension in combination with the thermodynamic integration method for the calculation of free energies will also be introduced. We use two examples to show how this influence of NQEs in realistic systems is simulated in practice. Project supported by the National Natural Science Foundation of China (Grant Nos. 11275008, 91021007, and 10974012) and the China Postdoctoral Science Foundation (Grant No. 2014M550005).
Fishman, Louis
2000-11-01
The role of mathematical modeling in the physical sciences will be briefly addressed. Examples will focus on computational acoustics, with applications to underwater sound propagation, electromagnetic modeling, optics, and seismic inversion. Direct and inverse wave propagation problems in both the time and frequency domains will be considered. Focusing on fixed-frequency (elliptic) wave propagation problems, the usual, two-way, partial differential equation formulation will be exactly reformulated, in a well-posed manner, as a one-way (marching) problem. This is advantageous for both direct and inverse considerations, as well as stochastic modeling problems. The reformulation will require the introduction of pseudodifferential operators and their accompanying phase space analysis (calculus), in addition to path integral representations for the fundamental solutions and their subsequent computational algorithms. Unlike the more traditional, purely numerical applications of, for example, finite-difference and finite-element methods, this approach, in effect, writes the exact, or, more generally, the asymptotically correct, answer as a functional integral and, subsequently, computes it directly. The overall computational philosophy is to combine analysis, asymptotics, and numerical methods to attack complicated, real-world problems. Exact and asymptotic analysis will stress the complementary nature of the direct and inverse formulations, as well as indicating the explicit structural connections between the time- and frequency-domain solutions.
Covariant Hamiltonian field theory
Giachetta, G; Sardanashvily, G
1999-01-01
We study the relationship between the equations of first order Lagrangian field theory on fiber bundles and the covariant Hamilton equations on the finite-dimensional polysymplectic phase space of covariant Hamiltonian field theory. The main peculiarity of these Hamilton equations lies in the fact that, for degenerate systems, they contain additional gauge fixing conditions. We develop the BRST extension of the covariant Hamiltonian formalism, characterized by a Lie superalgebra of BRST and anti-BRST symmetries.
Semi-classical Locality for the Non-relativistic Path Integral in Configuration Space
Gomes, Henrique
2017-09-01
In an accompanying paper Gomes (arXiv:1504.02818, 2015), we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe M, existing on timeless configuration space Q of some field over M. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, M-O, if the projection of extremal curves on Q onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to e^{i\\hat{H}t}circ prO= prOcirc e^{i\\hat{H}t}, where prO:Q→ Q_O^{partial O} is a linear projection. This criterion for locality can be made approximate—an impossible feat had it been already postulated—and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality.
Wong, Kin-Yiu; Xu, Yuqing; Xu, Liang
2015-11-01
Enzymatic reactions are integral components in many biological functions and malfunctions. The iconic structure of each reaction path for elucidating the reaction mechanism in details is the molecular structure of the rate-limiting transition state (RLTS). But RLTS is very hard to get caught or to get visualized by experimentalists. In spite of the lack of explicit molecular structure of the RLTS in experiment, we still can trace out the RLTS unique "fingerprints" by measuring the isotope effects on the reaction rate. This set of "fingerprints" is considered as a most direct probe of RLTS. By contrast, for computer simulations, oftentimes molecular structures of a number of TS can be precisely visualized on computer screen, however, theoreticians are not sure which TS is the actual rate-limiting one. As a result, this is an excellent stage setting for a perfect "marriage" between experiment and theory for determining the structure of RLTS, along with the reaction mechanism, i.e., experimentalists are responsible for "fingerprinting", whereas theoreticians are responsible for providing candidates that match the "fingerprints". In this Review, the origin of isotope effects on a chemical reaction is discussed from the perspectives of classical and quantum worlds, respectively (e.g., the origins of the inverse kinetic isotope effects and all the equilibrium isotope effects are purely from quantum). The conventional Bigeleisen equation for isotope effect calculations, as well as its refined version in the framework of Feynman's path integral and Kleinert's variational perturbation (KP) theory for systematically incorporating anharmonicity and (non-parabolic) quantum tunneling, are also presented. In addition, the outstanding interplay between theory and experiment for successfully deducing the RLTS structures and the reaction mechanisms is demonstrated by applications on biochemical reactions, namely models of bacterial squalene-to-hopene polycyclization and RNA 2'-O
Hamiltonian tomography of photonic lattices
Ma, Ruichao; Owens, Clai; LaChapelle, Aman; Schuster, David I.; Simon, Jonathan
2017-06-01
In this paper we introduce an approach to Hamiltonian tomography of noninteracting tight-binding photonic lattices. To begin with, we prove that the matrix element of the low-energy effective Hamiltonian between sites α and β may be obtained directly from Sα β(ω ) , the (suitably normalized) two-port measurement between sites α and β at frequency ω . This general result enables complete characterization of both on-site energies and tunneling matrix elements in arbitrary lattice networks by spectroscopy, and suggests that coupling between lattice sites is a topological property of the two-port spectrum. We further provide extensions of this technique for measurement of band projectors in finite, disordered systems with good band flatness ratios, and apply the tool to direct real-space measurement of the Chern number. Our approach demonstrates the extraordinary potential of microwave quantum circuits for exploration of exotic synthetic materials, providing a clear path to characterization and control of single-particle properties of Jaynes-Cummings-Hubbard lattices. More broadly, we provide a robust, unified method of spectroscopic characterization of linear networks from photonic crystals to microwave lattices and everything in between.
Equivalence of two sets of deformed Calogero-Moser Hamiltonians
Gorbe, T F
2015-01-01
The equivalence of two complete sets of Poisson commuting Hamiltonians of the (super)integrable rational BC(n) Ruijsenaars-Schneider-van Diejen system is established. Specifically, the commuting Hamiltonians constructed by van Diejen are shown to be linear combinations of the Hamiltonians generated by the characteristic polynomial of the Lax matrix obtained recently by Pusztai, and the explicit formula of this invertible linear transformation is found.
Hector James Ingram Page
2015-02-01
Full Text Available Head direction cells fire to signal the direction in which an animal's head is pointing. They are able to track head direction using only internally-derived information (path integration. In this simulation study we investigate the factors that affect path integration accuracy. Specifically, two major limiting factors are identified: rise time, the time after stimulation it takes for a neuron to start firing, and the presence of symmetric non-offset within-layer recurrent collateral connectivity. On the basis of the latter, the important prediction is made that head direction cell regions directly involved in path integration will not contain this type of connectivity; giving a theoretical explanation for architectural observations. Increased neuronal rise time is found to slow path integration, and the slowing effect for a given rise time is found to be more severe in the context of short conduction delays. Further work is suggested on the basis of our findings, which represent a valuable contribution to understanding of the head direction cell system.
Kleinert, H; Zatloukal, V
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Wiese, Kay Jörg
2016-04-01
We derive and study two different formalisms used for nonequilibrium processes: the coherent-state path integral, and an effective, coarse-grained stochastic equation of motion. We first study the coherent-state path integral and the corresponding field theory, using the annihilation process A+A→A as an example. The field theory contains counterintuitive quartic vertices. We show how they can be interpreted in terms of a first-passage problem. Reformulating the coherent-state path integral as a stochastic equation of motion, the noise generically becomes imaginary. This renders it not only difficult to interpret, but leads to convergence problems at finite times. We then show how alternatively an effective coarse-grained stochastic equation of motion with real noise can be constructed. The procedure is similar in spirit to the derivation of the mean-field approximation for the Ising model, and the ensuing construction of its effective field theory. We finally apply our findings to stochastic Manna sandpiles. We show that the coherent-state path integral is inappropriate, or at least inconvenient. As an alternative, we derive and solve its mean-field approximation, which we then use to construct a coarse-grained stochastic equation of motion with real noise.
Shiga, Motoyuki; Tachikawa, Masanori; Miura, Shinichi
2000-12-01
We present an accurate calculational scheme for many-body systems composed of electrons and nuclei, by path integral molecular dynamics technique combined with the ab initio molecular orbital theory. Based upon the scheme, the simulation of a water molecule at room temperature is demonstrated, applying all-electron calculation at the Hartree-Fock level of theory.
Ould-Lahoucine, H. K.; Chetouani, L.
2012-07-01
Exact Green function for a Dirac particle subject to a couple of orthogonal plane wave fields is obtained throughout a path integral approach. In addition, a suitable representation of the Dirac matrices is deduced so that the initial problem becomes the one of a free particle.
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
Study of lower hybrid wave propagation in ionized gas by Hamiltonian theory
Casolari, Andrea
2013-01-01
In order to find an approximate solution to the Vlasov-Maxwell equation system describing the lower hybrid wave propagation in magnetic confined plasmas, the use of the WKB method leads to the ray tracing equations. The Hamiltonian character of the ray tracing equations is investigated analytically and numerically in order to deduce the physical properties of the wave propagating without absorption in the confined plasma. The consequences of the Hamiltonian character of the equations on the travelling wave, in particular, on the evolution of the parallel wavenumber along the propagation path have been accounted and the chaotic diffusion of the timeaveraged parallel wave-number towards higher values has been evaluated. Numerical analysis by means of a Runge-Kutta based algorithm implemented in a ray tracing code supplies the analytical considerations. A numerical tool based on the symplectic integration of the ray trajectories has been developed.
Study of lower hybrid wave propagation in ionized gas by Hamiltonian theory
Casolari, A. [Università di Pisa, Pisa (Italy); Cardinali, A. [Associazione Euratom-ENEA sulla Fusione, C.P. 65 - I-00044 - Frascati, Rome (Italy)
2014-02-12
In order to find an approximate solution to the Vlasov-Maxwell equation system describing the lower hybrid wave propagation in magnetic confined plasmas, the use of the WKB method leads to the ray tracing equations. The Hamiltonian character of the ray tracing equations is investigated analytically and numerically in order to deduce the physical properties of the wave propagating without absorption in the confined plasma. The consequences of the Hamiltonian character of the equations on the travelling wave, in particular, on the evolution of the parallel wavenumber along the propagation path have been accounted and the chaotic diffusion of the timeaveraged parallel wave-number towards higher values has been evaluated. Numerical analysis by means of a Runge-Kutta based algorithm implemented in a ray tracing code supplies the analytical considerations. A numerical tool based on the symplectic integration of the ray trajectories has been developed.
Sesé, Luis M.
This paper addresses several points of interest concerning the computation of the static structure factor of path-integral monatomic quantum fluids. First of all, the connection between the structure factor and the path-integral linear response pair radial correlation function is shown as its defining quantity by assuming a generalized Fermi's potential for the neutron- nuclei interactions, which is to be included in the general expression of the dynamic structure factor. Second, the possibilities of finding Ornstein-Zernike equations for full path-integral fluids, and also for the effective potential models of fluids derived from the path-integral formalism, are explored by working in the grand canonical ensemble. By so doing, the success and features for improvement of the weak-field approach used previously in this context of determining quantum static structure factors [SESE,L.M.,1996, Molec. Phys., 89, 1783; SESE, L.M., and LEDESMA,R., 1997, J. chem. Phys., 106, 1134] can be understood. New numerical applications are performed within this weak-field approach taking as probes the quantum hard-sphere fluid and dense fluid helium-4, the latter being described through LennardJones and Aziz-Slaman underlying interactions. The results show that the structure factors associated with the linear response and instantaneous path-integral pair radial correlation functions differ noticeably from each other with increasing quantum effects. In particular, the linear response description leads to more compressible fluids than the instantaneous one. Besides, the equality between the isothermal compressibilities fixed via the linear response and the quantum particle centre-of-gravity pair radial correlation functions does not hold beyond the situations that can be treated with the Gaussian Feynman-Hibbs effective potential picture. Comparison with experiment in the case of helium-4 (T = 4.2 K) reveals clearly that, under strong quantum conditions, an operative framework more
Quantum path-integral study of the phase diagram and isotope effects of neon
Ramirez, R; 10.1063/1.3023036
2011-01-01
The phase diagram of natural neon has been calculated for temperatures in the range 17-50 K and pressures between 0.01 and 2000 bar. The phase coexistence between solid, liquid, and gas phases has been determined by the calculation of the separate free energy of each phase as a function of temperature. Thus, for a given pressure, the coexistence temperature was obtained by the condition of equal free energy of coexisting phases. The free energy was calculated by using non-equilibrium techniques such as adiabatic switching and reversible scaling. The phase diagram obtained by classical Monte Carlo simulations has been compared to that obtained by quantum path-integral simulations. Quantum effects related to the finite mass of neon cause that coexistence lines are shifted towards lower temperatures when compared to the classical limit. The shift found in the triple point amounts to 1.5 K, i.e., about 6 % of the triple-point temperature. The triple-point isotope effect has been determined for 20Ne, 21Ne, 22Ne, a...
Dopieralski, Przemyslaw; Perrin, Charles L; Latajka, Zdzislaw
2011-11-08
The issue of the symmetry of short, low-barrier hydrogen bonds in solution is addressed here with advanced ab initio simulations of a hydrogen maleate anion in different environments, starting with the isolated anion, going through two crystal structures (sodium and potassium salts), then to an aqueous solution, and finally in the presence of counterions. By Car-Parrinello and path integral molecular dynamics simulations, it is demonstrated that the position of the proton in the intramolecular hydrogen bond of an aqueous hydrogen maleate anion is entirely related to the solvation pattern around the oxygen atoms of the intramolecular hydrogen bond. In particular, this anion has an asymmetric hydrogen bond, with the proton always located on the oxygen atom that is less solvated, owing to the instantaneous solvation environment. Simulations of water solutions of hydrogen maleate ion with two different counterions, K(+) and Na(+), surprisingly show that the intramolecular hydrogen-bond potential in the case of the Na(+) salt is always asymmetric, regardless of the hydrogen bonds to water, whereas for the K(+) salt, the potential for H motion depends on the location of the K(+). It is proposed that repulsion by the larger and more hydrated K(+) is weaker than that by Na(+) and competitive with solvation by water.
Neural Network-Based Solutions for Stochastic Optimal Control Using Path Integrals.
Rajagopal, Karthikeyan; Balakrishnan, Sivasubramanya Nadar; Busemeyer, Jerome R
2017-03-01
In this paper, an offline approximate dynamic programming approach using neural networks is proposed for solving a class of finite horizon stochastic optimal control problems. There are two approaches available in the literature, one based on stochastic maximum principle (SMP) formalism and the other based on solving the stochastic Hamilton-Jacobi-Bellman (HJB) equation. However, in the presence of noise, the SMP formalism becomes complex and results in having to solve a couple of backward stochastic differential equations. Hence, current solution methodologies typically ignore the noise effect. On the other hand, the inclusion of noise in the HJB framework is very straightforward. Furthermore, the stochastic HJB equation of a control-affine nonlinear stochastic system with a quadratic control cost function and an arbitrary state cost function can be formulated as a path integral (PI) problem. However, due to curse of dimensionality, it might not be possible to utilize the PI formulation for obtaining comprehensive solutions over the entire operating domain. A neural network structure called the adaptive critic design paradigm is used to effectively handle this difficulty. In this paper, a novel adaptive critic approach using the PI formulation is proposed for solving stochastic optimal control problems. The potential of the algorithm is demonstrated through simulation results from a couple of benchmark problems.
Heilmann, D.B.
2007-02-15
The two-plane HUBBARD model, which is a model for some electronic properties of undoped YBCO superconductors as well as displays a MOTT metal-to-insulator transition and a metal-to-band insulator transition, is studied within Dynamical Mean-Field Theory using HIRSCH-FYE Monte Carlo. In order to find the different transitions and distinguish the types of insulator, we calculate the single-particle spectral densities, the self-energies and the optical conductivities. We conclude that there is a continuous transition from MOTT to band insulator. In the second part, ground state properties of a diagonally disordered HUBBARD model is studied using a generalisation of Path Integral Renormalisation Group, a variational method which can also determine low-lying excitations. In particular, the distribution of antiferromagnetic properties is investigated. We conclude that antiferromagnetism breaks down in a percolation-type transition at a critical disorder, which is not changed appreciably by the inclusion of correlation effects, when compared to earlier studies. Electronic and excitation properties at the system sizes considered turn out to primarily depend on the geometry. (orig.)
The quantum nature of the hydrogen bond: insight from path-integral molecular dynamics
Walker, Brent; Li, Xin-Zheng; Michaelides, Angelos
2011-03-01
Hydrogen (H) bonds are weak, generally intermolecular bonds, that hold together much of soft matter, the condensed phases of water, network liquids, and many ferroelectric crystals. The small mass of H means H-bonds are inherently quantum mechanical; effects such as zero point motion and tunneling should be considered, although often are not. In particular, a consistent picture of quantum nuclear effects on the strength of H-bonds and consequently the structure of H-bonded systems is still absent. Here, we report ab initio path-integral molecular dynamics studies on the quantum nature of the H-bond. Systematic examination of a range of H-bonded systems shows that quantum nuclei weaken weak H-bonds but strengthen relatively strong ones. This correlation arises from a competition between anharmonic intermolecular bond bending and intramolecular bond stretching. A simple rule of thumb enables predictions to be made for H-bonded bonded materials in general with merely classical knowledge (e.g. H-bond strength or H-bond length). Our work rationalizes the contrasting influence of quantum nuclear dynamics on a wide variety of materials, including liquid water and HF, and highlights the need for flexible molecules in force-field based studies of quantum nuclear dynamics.
Wang, Qi [Department of Chemistry, Tsukuba University, 1-1-1 Tennodai, Tsukuba 305-8571 (Japan); Suzuki, Kimichi [Research Institute for Nanosystem, National Institute of Advanced Industrial Science and Technology, Chuo-2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568 (Japan); Nagashima, Umpei, E-mail: u.nagashima@aist.go.jp [Department of Chemistry, Tsukuba University, 1-1-1 Tennodai, Tsukuba 305-8571 (Japan); Research Institute for Nanosystem, National Institute of Advanced Industrial Science and Technology, Chuo-2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568 (Japan); Tachikawa, Masanori [Quantum Chemistry Division, Graduate School of Science, Yokohama-City University, Seto 22-2, Kanazawa-ku, Yokohama 236-0027 (Japan); Yan, Shiwei [College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 (China)
2013-11-29
Highlights: • PIMD simulations with PM6-DH+ potential are carried out for Cl{sup −}(H{sub 2}O){sub n} clusters. • The geometric isotope effects on the rearrangement of single and multi shell structures are presented. • The competition of intramolecular and intermolecular nuclear quantum effects on the cluster structures is shown. • The correlations between r(Cl…O) and other vibration motions are discussed. - Abstract: The geometric isotope effects on the structures of hydrated chloride ionic hydrogen bonded clusters are explored by carrying out path integral molecular dynamics simulations. First, an outer shell coordinate is selected to display the rearrangement of single and multi hydration shell cluster structures. Next, to show the competition of intramolecular and intermolecular nuclear quantum effects, the intramolecular OH{sup ∗} stretching and intermolecular ion–water wagging motions are studied for single and multi shell structures, respectively. The results indicate that the intermolecular nuclear quantum effects stabilize the ionic hydrogen bonds in single shell structures, while they are destabilized through the competition with intramolecular nuclear quantum effects in multi shell structures. In addition, the correlations between ion–water stretching motion and other cluster vibrational coordinates are discussed. The results indicate that the intermolecular nuclear quantum effects on the cluster structures are strongly related to the cooperation of the water–water hydrogen bond interactions.
Path integral Monte Carlo and density functional molecular dynamics simulations of hot, dense helium
Militzer, B.
2009-04-01
Two first-principles simulation techniques, path integral Monte Carlo (PIMC) and density functional molecular dynamics (DFT-MD), are applied to study hot, dense helium in the density-temperature range of 0.387-5.35gcm-3 and 500K-1.28×108K . One coherent equation of state is derived by combining DFT-MD data at lower temperatures with PIMC results at higher temperatures. Good agreement between both techniques is found in an intermediate-temperature range. For the highest temperatures, the PIMC results converge to the Debye-Hückel limiting law. In order to derive the entropy, a thermodynamically consistent free-energy fit is used that reproduces the internal energies and pressure derived from the first-principles simulations. The equation of state is presented in the form of a table as well as a fit and is compared with different free-energy models. Pair-correlation functions and the electronic density of states are discussed. Shock Hugoniot curves are compared with recent laser shock-wave experiments.
Mühlbacher, Lothar; Ankerhold, Joachim
2005-05-01
Electron transfer (ET) across molecular chains including an impurity is studied based on a recently improved real-time path-integral Monte Carlo (PIMC) approach [L. Mühlbacher, J. Ankerhold, and C. Escher, J. Chem. Phys. 121 12696 (2004)]. The reduced electronic dynamics is studied for various bridge lengths and defect site energies. By determining intersite hopping rates from PIMC simulations up to moderate times, the relaxation process in the extreme long-time limit is captured within a sequential transfer model. The total transfer rate is extracted and shown to be enhanced for certain defect site energies. Superexchange turns out to be relevant for extreme gap energies only and then gives rise to different dynamical signatures for high- and low-lying defects. Further, it is revealed that the entire bridge compound approaches a steady state on a much shorter time scale than that related to the total transfer. This allows for a simplified description of ET along donor-bridge-acceptor systems in the long-time range.
Yoshikawa, Takehiro; Sugawara, Shuichi [Department of Chemistry, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama City, Saitama 338-8570 (Japan); Takayanagi, Toshiyuki, E-mail: tako@mail.saitama-u.ac.jp [Department of Chemistry, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama City, Saitama 338-8570 (Japan); Shiga, Motoyuki [Center for Computational Science and E-systems, Japan Atomic Energy Agency, University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa City, Chiba 277-8563 (Japan); Tachikawa, Masanori [Quantum Chemistry Division, Graduate School of Nanobioscience, Yokohama-City University, Seto 22-2, Kanazawa-ku, Yokohama 236-0027 (Japan)
2012-02-06
Highlights: Black-Right-Pointing-Pointer Double proton transfer mechanisms in porphycene were studied with quantum simulations. Black-Right-Pointing-Pointer Both isotopic substitution and temperature significantly affect the transfer mechanism. Black-Right-Pointing-Pointer Nuclear quantum effects are playing important roles in the transfer mechanism. - Abstract: Path-integral molecular dynamics simulations have been performed for porphycene and its isotopic variants in order to understand the effect of isotopic substitution of inner protons on the double proton transfer mechanism. We have used an on-the-fly direct dynamics technique at the semiempirical PM6 level combined with specific reaction parameterization. Our quantum simulations show that double proton transfer of the unsubstituted porphycene at T = 300 K mainly occurs via a so-called concerted mechanism through the D{sub 2h} second-order saddle point. In addition, we found that both isotopic substitution and temperature significantly affect the double proton transfer mechanism. For example, the contribution of the stepwise mechanism increases with a temperature increase. We have also carried out hypothetical simulations with the porphycene configurations being completely planar. It has been found that out-of-plane vibrational motions significantly decrease the contribution of the concerted proton transfer mechanism.
Luis M. Sesé
2017-02-01
Full Text Available This work deals with the computation of the structure factors of quantum fluids under complex conditions involving substantial density fluctuations and/or large particle delocalization effects. The method is based on the combination of path-integral Monte Carlo (PIMC simulations and the pair Ornstein-Zernike framework (OZ2. PIMC provides the radial correlation functions (centroid, instantaneous, and thermalized-continuous total linear response, which are used as data input to the OZ2 calculations that lead to their associated structure factors. To undertake this project normal liquid 4He and supercritical 3He are selected, studying conditions in the range (T = 4.2 K; 0.01886 <ρN/Å-3 < 0.02687. Full inter-comparison between the structure factors determined via both OZ2 and direct PIMC calculations is made. In addition, comparison with experimental data, including thermodynamic properties, is made wherever possible. The results establish that, even under severe thermodynamic and/or quantum fluctuation conditions, OZ2 remains in the quantum domain as a highly reliable and cost-effective framework to determine accurate structure factors, also allowing one to understand the related isotopic shifts in fluid He.
Numerical path integral solution to strong Coulomb correlation in one dimensional Hooke's atom
Ruokosenmäki, Ilkka; Kylänpää, Ilkka; Rantala, Tapio T
2015-01-01
We present a new approach based on real time domain Feynman path integrals (RTPI) for electronic structure calculations and quantum dynamics, which includes correlations between particles exactly but within the numerical accuracy. We demonstrate that incoherent propagation by keeping the wave function real is a novel method for finding and simulation of the ground state, similar to Diffusion Monte Carlo (DMC) method, but introducing new useful tools lacking in DMC. We use 1D Hooke's atom, a two-electron system with very strong correlation, as our test case, which we solve with incoherent RTPI (iRTPI) and compare against DMC. This system provides an excellent test case due to exact solutions for some confinements and because in 1D the Coulomb singularity is stronger than in two or three dimensional space. The use of Monte Carlo grid is shown to be efficient for which we determine useful numerical parameters. Furthermore, we discuss another novel approach achieved by combining the strengths of iRTPI and DMC. We...
Equilibrium fractionation of H and O isotopes in water from path integral molecular dynamics
Pinilla, Carlos; Blanchard, Marc; Balan, Etienne; Ferlat, Guillaume; Vuilleumier, Rodolphe; Mauri, Francesco
2014-06-01
The equilibrium fractionation factor between two phases is of importance for the understanding of many planetary and environmental processes. Although thermodynamic equilibrium can be achieved between minerals at high temperature, many natural processes involve reactions between liquids or aqueous solutions and solids. For crystals, the fractionation factor α can be theoretically determined using a statistical thermodynamic approach based on the vibrational properties of the phases. These calculations are mostly performed in the harmonic approximation, using empirical or ab-initio force fields. In the case of aperiodic and dynamic systems such as liquids or solutions, similar calculations can be done using finite-size molecular clusters or snapshots obtained from molecular dynamics (MD) runs. It is however difficult to assess the effect of these approximate models on the isotopic fractionation properties. In this work we present a systematic study of the calculation of the D/H and 18O/16O equilibrium fractionation factors in water for the liquid/vapour and ice/vapour phases using several levels of theory within the simulations. Namely, we use a thermodynamic integration approach based on Path Integral MD calculations (PIMD) and an empirical potential model of water. Compared with standard MD, PIMD takes into account quantum effects in the thermodynamic modeling of systems and the exact fractionation factor for a given potential can be obtained. We compare these exact results with those of modeling strategies usually used, which involve the mapping of the quantum system on its harmonic counterpart. The results show the importance of including configurational disorder for the estimation of isotope fractionation in liquid phases. In addition, the convergence of the fractionation factor as a function of parameters such as the size of the simulated system and multiple isotope substitution is analyzed, showing that isotope fractionation is essentially a local effect in
Path integration and cognitive mapping in a continuous attractor neural network model.
Samsonovich, A; McNaughton, B L
1997-08-01
A minimal synaptic architecture is proposed for how the brain might perform path integration by computing the next internal representation of self-location from the current representation and from the perceived velocity of motion. In the model, a place-cell assembly called a "chart" contains a two-dimensional attractor set called an "attractor map" that can be used to represent coordinates in any arbitrary environment, once associative binding has occurred between chart locations and sensory inputs. In hippocampus, there are different spatial relations among place fields in different environments and behavioral contexts. Thus, the same units may participate in many charts, and it is shown that the number of uncorrelated charts that can be encoded in the same recurrent network is potentially quite large. According to this theory, the firing of a given place cell is primarily a cooperative effect of the activity of its neighbors on the currently active chart. Therefore, it is not particularly useful to think of place cells as encoding any particular external object or event. Because of its recurrent connections, hippocampal field CA3 is proposed as a possible location for this "multichart" architecture; however, other implementations in anatomy would not invalidate the main concepts. The model is implemented numerically both as a network of integrate-and-fire units and as a "macroscopic" (with respect to the space of states) description of the system, based on a continuous approximation defined by a system of stochastic differential equations. It provides an explanation for a number of hitherto perplexing observations on hippocampal place fields, including doubling, vanishing, reshaping in distorted environments, acquiring directionality in a two-goal shuttling task, rapid formation in a novel environment, and slow rotation after disorientation. The model makes several new predictions about the expected properties of hippocampal place cells and other cells of the
Jang, Seogjoo; Voth, Gregory A
2017-05-07
Despite the fact that quantum mechanical principles do not allow the establishment of an exact quantum analogue of the classical transition state theory (TST), the development of a quantum TST (QTST) with a proper dynamical justification, while recovering the TST in the classical limit, has been a long standing theoretical challenge in chemical physics. One of the most recent efforts of this kind was put forth by Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)], which can be specified for any cyclically invariant dividing surface defined in the space of the imaginary time path integral. The present work revisits the issue of the non-uniqueness of QTST and provides a detailed theoretical analysis of HA-QTST for a general class of such path integral dividing surfaces. While we confirm that HA-QTST reproduces the result based on the ring polymer molecular dynamics (RPMD) rate theory for dividing surfaces containing only a quadratic form of low frequency Fourier modes, we find that it produces different results for those containing higher frequency imaginary time paths which accommodate greater quantum fluctuations. This result confirms the assessment made in our previous work [Jang and Voth, J. Chem. Phys. 144, 084110 (2016)] that HA-QTST does not provide a derivation of RPMD-TST in general and points to a new ambiguity of HA-QTST with respect to its justification for general cyclically invariant dividing surfaces defined in the space of imaginary time path integrals. Our analysis also offers new insights into similar path integral based QTST approaches.
Interest rates in quantum finance: the Wilson expansion and Hamiltonian.
Baaquie, Belal E
2009-10-01
Interest rate instruments form a major component of the capital markets. The Libor market model (LMM) is the finance industry standard interest rate model for both Libor and Euribor, which are the most important interest rates. The quantum finance formulation of the Libor market model is given in this paper and leads to a key generalization: all the Libors, for different future times, are imperfectly correlated. A key difference between a forward interest rate model and the LMM lies in the fact that the LMM is calibrated directly from the observed market interest rates. The short distance Wilson expansion [Phys. Rev. 179, 1499 (1969)] of a Gaussian quantum field is shown to provide the generalization of Ito calculus; in particular, the Wilson expansion of the Gaussian quantum field A(t,x) driving the Libors yields a derivation of the Libor drift term that incorporates imperfect correlations of the different Libors. The logarithm of Libor phi(t,x) is defined and provides an efficient and compact representation of the quantum field theory of the Libor market model. The Lagrangian and Feynman path integrals of the Libor market model of interest rates are obtained, as well as a derivation given by its Hamiltonian. The Hamiltonian formulation of the martingale condition provides an exact solution for the nonlinear drift of the Libor market model. The quantum finance formulation of the LMM is shown to reduce to the industry standard Bruce-Gatarek-Musiela-Jamshidian model when the forward interest rates are taken to be exactly correlated.
Böhm, Michael C.; Schulte, Joachim; Utrera, Luis
Feynman path-integral quantum Monte Carlo (QMC) simulations and an analytic many-body approach are used to study the ground state properties of one-dimensional (1D) chains in the theoretical framework of model Hamiltonians of the Hubbard type. The QMC algorithm is employed to derive position-space quantities, while band structure properties are evaluated by combining QMC data with expressions derived in momentum (k) space. Bridging link between both representations is the quasi-chemical approximation (QCA). Electronic charge fluctuations and the fluctuations of the magnetic local moments are studied as a function of the on-site density and correlation strength, which is given by the ratio between two-electron interaction and kinetic hopping. Caused by the non-analytic behaviour of the chemical potential μ = ∂E/∂ (with E denoting the electronic energy), strict 1D systems with an on-site density of 1·0 do not exhibit the properties of a conductor for any non-zero interaction beyond the mean-field approximation. The QMC simulations lead to straightforward access to the probabilities Pi(n) of finding n = 0, 1, 2 electrons at the ith lattice site. The Pi(n) elements allow to calculate the enhancement factors on the electron spin susceptibility χ, effective electronic mass m* and Knight shift κ. m* is enhanced by a bandwidth renormalization factor D-10, κ by an element ηK mapping the additional localization of the correlated electrons in the presence of an external magnetic field B and χ by the product D-10 ηK. Available experimental data are discussed in the light of the present theoretical findings.
New approaches to generalized Hamiltonian realization of autonomous nonlinear systems
王玉振; 李春文; 程代展
2003-01-01
The Hamiltonian function method plays an important role in stability analysis and stabilization. The key point in applying the method is to express the system under consideration as the form of dissipative Hamiltonian systems, which yields the problem of generalized Hamiltonian realization. This paper deals with the generalized Hamiltonian realization of autonomous nonlinear systems. First, this paper investigates the relation between traditional Hamiltonian realizations and first integrals, proposes a new method of generalized Hamiltonian realization called the orthogonal decomposition method, and gives the dissipative realization form of passive systems. This paper has proved that an arbitrary system has an orthogonal decomposition realization and an arbitrary asymptotically stable system has a strict dissipative realization. Then this paper studies the feedback dissipative realization problem and proposes a control-switching method for the realization. Finally,this paper proposes several sufficient conditions for feedback dissipative realization.
THE HAMILTONIAN SYSTEMS OF THE LCZ HIERARCHY BY NONLINEARIZATION
Li Lu
2000-01-01
In this paper, we first search for the Hamiltonian structure of LCZ hierarchy by use of a trace identity. Then we determine a higher-order constraint condition between the potentials and the eigenfunctions of the LCZ spectral problem, and under this constraint condition, the Lax pairs of LCZ hierarchy are all nonlinearized into the finite-dimensional integrable Hamiltonian systems in Liouville sense.
FEEDBACK REALIZATION OF HAMILTONIAN SYSTEMS
CHENG Daizhan; XI Zairong
2002-01-01
This paper investigates the relationship between state feedback and Hamiltonian realizatiou. First, it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian realization. Necessary and sufficient conditions are obtained for it to have a Hamiltonian realization with natural outpnt. Then some conditions for an affine nonlinear system to have a Hamiltonian realization arc given.For generalized outputs, the conditions of the feedback, keeping Hamiltonian, are discussed. Finally, the admissible feedback controls for generalized Hamiltonian systems are considered.
FEEDBACK REALIZATION OF HAMILTONIAN SYSTEMS
CHENGDaizhan; XIZairong
2002-01-01
This paper investigates the relationship between state feedback and Hamiltonican realization.Firest,it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian realization.Necessary and sufficient conditions are obtained for it to have a Hamiltonian realization with natural output.Then some conditions for an affine nonlinear system to have a Hamiltonian realization are given.some conditions for an affine nonlinear system to have a Hamiltonian realization are given.For generalized outputs,the conditions of the feedback,keeping Hamiltonian,are discussed.Finally,the admissible feedback controls for generalized Hamiltonian systems are considered.
Singh, Upendra N.; Refaat, Tamer F.; Petros, Mulugeta
2017-01-01
The societal benefits of understanding climate change through identification of global carbon dioxide sources and sinks led to the desired NASA's active sensing of carbon dioxide emissions over nights, days, and seasons (ASCENDS) space-based missions of global carbon dioxide measurements. For more than 15 years, NASA Langley Research Center (LaRC) have developed several carbon dioxide active remote sensors using the differential absorption lidar (DIAL) technique operating at the two-micron wavelength. Currently, an airborne two-micron triple-pulse integrated path differential absorption (IPDA) lidar is under development. This IPDA lidar measures carbon dioxide as well as water vapor, the dominant interfering molecule on carbon dioxide remote sensing. Advancement of this triple-pulse IPDA lidar development is presented.
Mouhat, Félix; Sorella, Sandro; Vuilleumier, Rodolphe; Saitta, Antonino Marco; Casula, Michele
2017-06-13
We introduce a novel approach for a fully quantum description of coupled electron-ion systems from first principles. It combines the variational quantum Monte Carlo solution of the electronic part with the path integral formalism for the quantum nuclear dynamics. On the one hand, the path integral molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational quantum Monte Carlo can provide Born-Oppenheimer potential energy surfaces with a precision comparable to the most-advanced post-Hartree-Fock approaches, and with a favorable scaling with the system size. In order to cope with the intrinsic noise due to the stochastic nature of quantum Monte Carlo methods, we generalize the path integral molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of quantum Monte Carlo nuclear forces. The variational parameters of the quantum Monte Carlo wave function are evolved during the nuclear dynamics, such that the Born-Oppenheimer potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well-controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated, even in the presence of the Langevin thermostat, thanks to the mapping onto an Ornstein-Uhlenbeck process. This framework turns out to be also very efficient in the case of noiseless (deterministic) ionic forces. The new implementation is validated on the Zundel ion (H5O2(+)) by direct comparison with standard path integral Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond
Jolicard, Georges; Viennot, David; Killingbeck, John P
2016-01-01
A global solution of the Schr\\"odinger equation, obtained recently within the wave operator formalism for explicitly time-dependent Hamiltonians [J. Phys. A: Math. Theor. 48, 225205 (2015)], is generalized to take into account the case of multidimensional active spaces. An iterative algorithm is derived to obtain the Fourier series of the evolution operator issuing from a given multidimensional active subspace and then the effective Hamiltonian corresponding to the model space is computed and analysed as a measure of the cyclic character of the dynamics. Studies of the laser controlled dynamics of diatomic models clearly show that a multidimensional active space is required if the wavefunction escapes too far from the initial subspace. A suitable choice of the multidimensional active space, including the initial and target states, increases the cyclic character and avoids divergences occuring when one-dimensional active spaces are used. The method is also proven to be efficient in describing dissipative proce...
Lagrangian tetragons and instabilities in Hamiltonian dynamics
Entov, Michael; Polterovich, Leonid
2017-01-01
We present a new existence mechanism, based on symplectic topology, for orbits of Hamiltonian flows connecting a pair of disjoint subsets in the phase space. The method involves function theory on symplectic manifolds combined with rigidity of Lagrangian submanifolds. Applications include superconductivity channels in nearly integrable systems and dynamics near a perturbed unstable equilibrium.
Different strategies for spatial updating in yaw and pitch path integration
Caspar Mathias Goeke
2013-02-01
Full Text Available Research in spatial navigation revealed the existence of discrete strategies defined by the use of distinct reference frames during virtual path integration. The present study investigated the distribution of these navigation strategies as a function of gender, video gaming experience, and self-estimates of spatial navigation abilities in a population of 300 subjects. Participants watched videos of virtual passages through a star-field with one turn in either the horizontal (yaw or the vertical (pitch axis. At the end of a passage they selected one out of four homing arrows to indicate the initial starting location. To solve the task, participants could employ two discrete strategies, navigating within either an egocentric or an allocentric reference frame. The majority of valid subjects (232/260 consistently used the same strategy in more than 75% of all trials. With that approach 33.1% of all participants were classified as Turners (using an egocentric reference frame on both axes and 46.5% as Nonturners (using an allocentric reference frame on both axes. 9.2% of all participants consistently used an egocentric reference frame in the yaw plane but an allocentric reference frame in the pitch plane (Switcher. Investigating the influence of gender on navigation strategies revealed that females predominantly used the Nonturner strategy while males used both the Turner and the Nonturner strategy with comparable probabilities. Other than expected, video gaming experience did not influence strategy use. Based on a strong quantitative basis with the sample size about an order of magnitude larger than in typical psychophysical studies these results demonstrate that most people reliably use one out of three possible navigation strategies (Turners, Nonturners, Switchers for spatial updating and provides a sound estimate of how those strategies are distributed within the general population.
Hayes, Robin L; Paddison, Stephen J; Tuckerman, Mark E
2009-12-31
The mono-, di-, and tetrahydrates of trifluoromethanesulfonic acid, which contain characteristic H(3)O(+), H(5)O(2)(+), and H(9)O(4)(+) structures, provide model systems for understanding proton transport in materials with high perfluorosulfonic acid density such as perfluorosulfonic acid membranes commonly employed in hydrogen fuel cells. Ab initio molecular dynamics simulations indicate that protons in these solids are predisposed to transfer to the water most strongly bound to sulfonate groups via a Grotthuss-type mechanism, but quickly return to the most solvated defect structure either due to the lack of a nearby species to stabilize the new defect or a preference for the proton to be maximally hydrated. Path integral molecular dynamics of the mono- and dihydrate reveal significant quantum effects that facilitate proton transfer to the "presolvated" water or SO(3)(-) in the first solvation shell and increase the Zundel character of all the defects. These trends are quantified in free energy profiles for each bonding environment. Hydrogen bonding criteria for HOH-OH(2) and HOH-O(3)S are extracted from the two-dimensional potential of mean force. The quantum radial distribution function, radius of gyration, and root-mean-square displacement position correlation function show that the protonic charge is distributed over two or more water molecules. Metastable structural defects with one excess proton shared between two sulfonate groups and another Zundel or Eigen type cation defect are found for the mono- and dihydrate but not for the tetrahydrate crystal. Results for the tetrahydrate native crystal exhibit minor differences at 210 and 250 K. IR spectra are calculated for all native and stable defect structures. Graph theory techniques are used to characterize the chain lengths and ring sizes in the hydrogen bond network. Low conductivities when limited water is present may be attributable to trapping of protons between SO(3)(-) groups and the increased
Statistical mechanics and field theory. [Path integrals, lattices, pseudofree vertex model
Samuel, S.A.
1979-05-01
Field theory methods are applied to statistical mechanics. Statistical systems are related to fermionic-like field theories through a path integral representation. Considered are the Ising model, the free-fermion model, and close-packed dimer problems on various lattices. Graphical calculational techniques are developed. They are powerful and yield a simple procedure to compute the vacuum expectation value of an arbitrary product of Ising spin variables. From a field theorist's point of view, this is the simplest most logical derivation of the Ising model partition function and correlation functions. This work promises to open a new area of physics research when the methods are used to approximate unsolved problems. By the above methods a new model named the 128 pseudo-free vertex model is solved. Statistical mechanics intuition is applied to field theories. It is shown that certain relativistic field theories are equivalent to classical interacting gases. Using this analogy many results are obtained, particularly for the Sine-Gordon field theory. Quark confinement is considered. Although not a proof of confinement, a logical, esthetic, and simple picture is presented of how confinement works. A key ingredient is the insight gained by using an analog statistical system consisting of a gas of macromolecules. This analogy allows the computation of Wilson loops in the presence of topological vortices and when symmetry breakdown occurs in the topological quantum number. Topological symmetry breakdown calculations are placed on approximately the same level of rigor as instanton calculations. The picture of confinement that emerges is similar to the dual Meissner type advocated by Mandelstam. Before topological symmetry breakdown, QCD has monopoles bound linearly together by three topological strings. Topological symmetry breakdown corresponds to a new phase where these monopoles are liberated. It is these liberated monopoles that confine quarks. 64 references.
Lagin, L.J. [Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 (United States)], E-mail: lagin1@llnl.gov; Bettenhausen, R.C.; Bowers, G.A.; Carey, R.W.; Edwards, O.D.; Estes, C.M.; Demaret, R.D.; Ferguson, S.W.; Fisher, J.M.; Ho, J.C.; Ludwigsen, A.P.; Mathisen, D.G.; Marshall, C.D.; Matone, J.T.; McGuigan, D.L.; Sanchez, R.J.; Stout, E.A.; Tekle, E.A.; Townsend, S.L.; Van Arsdall, P.J. [Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 (United States)] (and others)
2008-04-15
final optics, target positioners and diagnostics. Additional capabilities to support fusion ignition shots in a National Ignition Campaign (NIC) beginning in 2010 will include a cryogenic target system, target diagnostics, and integrated experimental shot data analysis with tools for data visualization and archiving. This talk discusses the current status of the control system implementation and discusses the plan to complete the control system on the path to ignition.
S. Ishii
2013-05-01
Full Text Available The National Institute of Information and Communications Technology (NICT has made a great deal of effort to develop a coherent 2 μm differential absorption and wind lidar (Co2DiaWiL for measuring CO2 and wind speed. First, coherent Integrated Path Differential Absorption (IPDA lidar experiments were conducted using the Co2DiaWiL and a foothill target (tree and ground surface located about 7.12 km south of NICT on 11, 27, and 28 December 2010. The detection sensitivity of a 2 μm IPDA lidar was examined in detail using the CO2 concentration measured by the foothill reflection. The precisions of CO2 measurements for the foothill target and 900, 4500 and 27 000 shot pairs were 6.5, 2.8, and 1.2%, respectively. The results indicated that a coherent IPDA lidar with a laser operating at a high pulse repetition frequency of a few tens of KHz is necessary for XCO2 (column-averaged dry air mixing ratio of CO2 measurement with a precision of 1–2 ppm in order to observe temporal and spatial variations in the CO2. Statistical comparisons indicated that, although a small amount of in situ data and the fact that they were not co-located with the foothill target made comparison difficult, the CO2 volume mixing ratio obtained by the Co2DiaWiL measurements for the foothill target and atmospheric returns was about −5 ppm lower than the 5 min running averages of the in situ sensor. Not only actual difference of sensing volume or the natural variability of CO2 but also the fluctuations of temperature could cause this difference. The statistical results indicated that there were no biases between the foothill target and atmospheric return measurements. The 2 μm coherent IPDA lidar can detect the CO2 volume mixing ratio change of 3% in the 5 min signal integration. In order to detect the position of the foothill target, to measure a range with a high SNR (signal-to-noise ratio, and to reduce uncertainty due to the presence of aerosols and clouds, it is
Remarks on hamiltonian digraphs
Gutin, Gregory; Yeo, Anders
2001-01-01
This note is motivated by A.Kemnitz and B.Greger, Congr. Numer. 130 (1998)127-131. We show that the main result of the paper by Kemnitz and Greger is an easy consequence of the characterization of hamiltonian out-locally semicomplete digraphs by Bang-Jensen, Huang, and Prisner, J. Combin. Theory...... of Fan's su#cient condition [5] for an undirected graph to be hamiltonian. In this note we give another, more striking, example of this kind, which disproves a conjecture from [6]. We also show that the main result of [6] 1 is an easy consequence of the characterization of hamiltonian out......-tournaments by Bang-Jensen, Huang and Prisner [4]. For further information and references on hamiltonian digraphs, see e.g. the chapter on hamiltonicity in [1] as well as recent survey papers [2, 8]. We use the standard terminology and notation on digraphs as described in [1]. A digraph D has vertex set V (D) and arc...
Microscopic plasma Hamiltonian
Peng, Y.-K. M.
1974-01-01
A Hamiltonian for the microscopic plasma model is derived from the Low Lagrangian after the dual roles of the generalized variables are taken into account. The resulting Hamilton equations are shown to agree with the Euler-Lagrange equations of the Low Lagrangian.