International Nuclear Information System (INIS)
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation
Bistability in the Chemical Master Equation for Dual Phosphorylation Cycles
Bazzani, A; Giampieri, E; Remondini, D; Cooper, L N
2011-01-01
Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates, are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed to elucidate the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional Chemical Master Equation for a well known model of a two step phospho/dephosphorylation cycle using the quasi steady state approximation of the enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the ...
Order reduction of the chemical master equation via balanced realisation.
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Fernando López-Caamal
Full Text Available We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism, and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator.
Grima, Ramon
2011-11-01
The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion. PMID:22181475
Nguyen, Thanh Lam; Stanton, John F
2015-07-16
In the field of chemical kinetics, the solution of a two-dimensional master equation that depends explicitly on both total internal energy (E) and total angular momentum (J) is a challenging problem. In this work, a weak-E/fixed-J collisional model (i.e., weak-collisional internal energy relaxation/free-collisional angular momentum relaxation) is used along with the steady-state approach to solve the resulting (simplified) two-dimensional (E,J)-grained master equation. The corresponding solutions give thermal rate constants and product branching ratios as functions of both temperature and pressure. We also have developed a program that can be used to predict and analyze experimental chemical kinetics results. This expedient technique, when combined with highly accurate potential energy surfaces, is cable of providing results that may be meaningfully compared to experiments. The reaction of singlet oxygen with methane proceeding through vibrationally excited methanol is used as an illustrative example. PMID:25815602
Dynamics of the chemical master equation, a strip of chains of equations in d-dimensional space.
Galstyan, Vahe; Saakian, David B
2012-07-01
We investigate the multichain version of the chemical master equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a model can describe the connected diffusion processes with jumps between different types. We apply the Hamilton-Jacobi equation to solve some aspects of the model. We derive exact (in the limit of infinite number of particles) results for the dynamic of the maximum of the distribution and the variance of distribution. PMID:23005386
Computational study of p53 regulation via the chemical master equation
Vo, Huy D.; Sidje, Roger B.
2016-06-01
A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1–16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation. We report numerical results that demonstrate the extend of tackling the CME with this combination of tools.
Energy Technology Data Exchange (ETDEWEB)
Alarcón, Tomás [Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra, 08193 Bellaterra (Barcelona) (Spain); Departament de Matemàtiques, Universitat Atonòma de Barcelona, 08193 Bellaterra (Barcelona) (Spain)
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.
DEFF Research Database (Denmark)
Dyre, Jeppe
1995-01-01
energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk modelthe energy master equation...
Direct solution of the Chemical Master Equation using quantized tensor trains.
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Vladimir Kazeev
2014-03-01
Full Text Available The Chemical Master Equation (CME is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to "lift" this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species and sub-linearly in the mode size (maximum copy number, and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging hp-discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG methods from quantum chemistry. Our method automatically adapts the "basis" of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of
Direct solution of the Chemical Master Equation using quantized tensor trains.
Kazeev, Vladimir; Khammash, Mustafa; Nip, Michael; Schwab, Christoph
2014-03-01
The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to "lift" this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging hp-discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the "basis" of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage
Busch, Anna; González-García, Núria; Lendvay, György; Olzmann, Matthias
2015-07-16
The thermal decomposition of cyanonitrene, NCN, was studied behind reflected shock waves in the temperature range 1790-2960 K at pressures near 1 and 4 bar. Highly diluted mixtures of NCN3 in argon were shock-heated to produce NCN, and concentration-time profiles of C atoms as reaction product were monitored with atomic resonance absorption spectroscopy at 156.1 nm. Calibration was performed with methane pyrolysis experiments. Rate coefficients for the reaction (3)NCN + M → (3)C + N2 + M (R1) were determined from the initial slopes of the C atom concentration-time profiles. Reaction R1 was found to be in the low-pressure regime at the conditions of the experiments. The temperature dependence of the bimolecular rate coefficient can be expressed with the following Arrhenius equation: k1(bim) = (4.2 ± 2.1) × 10(14) exp[-242.3 kJ mol(-1)/(RT)] cm(3) mol(-1) s(-1). The rate coefficients were analyzed by using a master equation with specific rate coefficients from RRKM theory. The necessary molecular data and energies were calculated with quantum chemical methods up to the CCSD(T)/CBS//CCSD/cc-pVTZ level of theory. From the topography of the potential energy surface, it follows that reaction R1 proceeds via isomerization of NCN to CNN and subsequent C-N bond fission along a collinear reaction coordinate without a tight transition state. The calculations reproduce the magnitude and temperature dependence of the rate coefficient and confirm that reaction R1 is in the low-pressure regime under our experimental conditions. PMID:25853321
Martirosyan, Arax; Saakian, David B.
2012-01-01
We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the Chemical Master Equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of Van Kampen method from HJE approach using a special ansatz. Using the Van Kampen method, we...
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-04-01
The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEGs), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady-state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an overall error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of an arbitrary MEG, we also introduce an a priori method to estimate the upper bound of its truncation error. This a priori estimate can be rapidly computed from reaction rates of the network, without the need of costly trial solutions of the dCME. As examples, we show results of applying our methods to the four stochastic networks of (1) the birth and death model, (2) the single gene expression model, (3) the genetic toggle switch model, and (4) the phage lambda bistable epigenetic switch model. We demonstrate how truncation errors and steady-state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate our theories. Overall, the novel state space
Martirosyan, A; Saakian, David B
2011-08-01
We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs. PMID:21928964
International Nuclear Information System (INIS)
We propose a non-equilibrium thermodynamical description in terms of the Chemical Master Equation (CME) to characterize the dynamics of a chemical cycle chain reaction among m different species. These systems can be closed or open for energy and molecules exchange with the environment, which determines how they relax to the stationary state. Closed systems reach an equilibrium state (characterized by the detailed balance condition (D.B.)), while open systems will reach a non-equilibrium steady state (NESS). The principal difference between D.B. and NESS is due to the presence of chemical fluxes. In the D.B. condition the fluxes are absent while for the NESS case, the chemical fluxes are necessary for the state maintaining. All the biological systems are characterized by their “far from equilibrium behavior,” hence the NESS is a good candidate for a realistic description of the dynamical and thermodynamical properties of living organisms. In this work we consider a CME written in terms of a discrete Kolmogorov forward equation, which lead us to write explicitly the non-equilibrium chemical fluxes. For systems in NESS, we show that there is a non-conservative “external vector field” whose is linearly proportional to the chemical fluxes. We also demonstrate that the modulation of these external fields does not change their stationary distributions, which ensure us to study the same system and outline the differences in the system's behavior when it switches from the D.B. regime to NESS. We were interested to see how the non-equilibrium fluxes influence the relaxation process during the reaching of the stationary distribution. By performing analytical and numerical analysis, our central result is that the presence of the non-equilibrium chemical fluxes reduces the characteristic relaxation time with respect to the D.B. condition. Within a biochemical and biological perspective, this result can be related to the “plasticity property” of biological
Nakagawa, Masaki; Togashi, Yuichi
2016-01-01
Cell activities primarily depend on chemical reactions, especially those mediated by enzymes, and this has led to these activities being modeled as catalytic reaction networks. Although deterministic ordinary differential equations of concentrations (rate equations) have been widely used for modeling purposes in the field of systems biology, it has been pointed out that these catalytic reaction networks may behave in a way that is qualitatively different from such deterministic representation when the number of molecules for certain chemical species in the system is small. Apart from this, representing these phenomena by simple binary (on/off) systems that omit the quantities would also not be feasible. As recent experiments have revealed the existence of rare chemical species in cells, the importance of being able to model potential small-number phenomena is being recognized. However, most preceding studies were based on numerical simulations, and theoretical frameworks to analyze these phenomena have not been sufficiently developed. Motivated by the small-number issue, this work aimed to develop an analytical framework for the chemical master equation describing the distributional behavior of catalytic reaction networks. For simplicity, we considered networks consisting of two-body catalytic reactions. We used the probability generating function method to obtain the steady-state solutions of the chemical master equation without specifying the parameters. We obtained the time evolution equations of the first- and second-order moments of concentrations, and the steady-state analytical solution of the chemical master equation under certain conditions. These results led to the rank conservation law, the connecting state to the winner-takes-all state, and analysis of 2-molecules M-species systems. A possible interpretation of the theoretical conclusion for actual biochemical pathways is also discussed. PMID:27047384
Scott, M
2012-08-01
The time-covariance function captures the dynamics of biochemical fluctuations and contains important information about the underlying kinetic rate parameters. Intrinsic fluctuations in biochemical reaction networks are typically modelled using a master equation formalism. In general, the equation cannot be solved exactly and approximation methods are required. For small fluctuations close to equilibrium, a linearisation of the dynamics provides a very good description of the relaxation of the time-covariance function. As the number of molecules in the system decrease, deviations from the linear theory appear. Carrying out a systematic perturbation expansion of the master equation to capture these effects results in formidable algebra; however, symbolic mathematics packages considerably expedite the computation. The authors demonstrate that non-linear effects can reveal features of the underlying dynamics, such as reaction stoichiometry, not available in linearised theory. Furthermore, in models that exhibit noise-induced oscillations, non-linear corrections result in a shift in the base frequency along with the appearance of a secondary harmonic. PMID:23039692
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...
Recent developments in the Virasoro master equation
International Nuclear Information System (INIS)
The Virasoro master equation collects all possible Virasoro constructions which are quadratic in the currents of affine Lie g. The solution space of this system is immense, with generically irrational central charge, and solutions which have so far been observed are generically unitary. Other developments reviewed include the exact C-function, the superconformal master equation and partial classification of solutions by graph theory and generalized graph theories. 37 refs., 1 fig., 1 tab
From convolutionless generalized master to Pauli master equations
International Nuclear Information System (INIS)
The paper is a continuation of previous work within which it has been proved that time integrals of memory function (i.e. Markovian transfer rates from Pauli Master Equations, PME) in Time-Convolution Generalized Master Equations (TC-GME) for probabilities of finding a state of an asymmetric system interacting with a bath with a continuous spectrum are exactly zero, provided that no approximation is involved, irrespective of the usual finite-perturbation-order correspondence with the Golden Rule transition rates. In this paper, attention is paid to an alternative way of deriving the rigorous PME from the TCL-GME. Arguments are given in favor of the proposition that the long-time limit of coefficients in TCL-GME for the above probabilities, under the same assumption and presuming that this limit exists, is equal to zero. 11 refs
Master equation as a radial constraint
Hussain, Uzair; Booth, Ivan; Kunduri, Hari K.
2016-06-01
We revisit the problem of perturbations of Schwarzschild-AdS4 black holes by using a combination of the Martel-Poisson formalism for perturbations of four-dimensional spherically symmetric spacetimes [K. Martel and E. Poisson, Phys. Rev. D 71, 104003 (2005).] and the Kodama-Ishibashi formalism [H. Kodama and A. Ishibashi, Prog. Theor. Phys. 110, 701 (2003).]. We clarify the relationship between both formalisms and express the Brown-York-Balasubramanian-Krauss boundary stress-energy tensor, T¯μ ν, on a finite-r surface purely in terms of the even and odd master functions. Then, on these surfaces we find that the spacelike components of the conservation equation D¯μT¯μ ν=0 are equivalent to the wave equations for the master functions. The renormalized stress-energy tensor at the boundary r/L lim r →∞ T¯μ ν is calculated directly in terms of the master functions.
Simplified differential equations approach for Master Integrals
Papadopoulos, Costas G.
2014-01-01
A simplified differential equations approach for Master Integrals is presented. It allows to express them, straightforwardly, in terms of Goncharov Polylogarithms. As a proof-of-concept of the proposed method, results at one and two loops are presented, including the massless one-loop pentagon with up to one off-shell leg at order epsilon.
On the quantum master equation for fermions
Huang, C. F.; Huang, K. -N.
2006-01-01
A quantum master equation is obtained for identical fermions by including a relaxation term in addition to the mean-field Hamiltonian. [Huang C F and Huang K N 2004 Chinese J. Phys. ${\\bf 42}$ 221; Gebauer R and Car R 2004 Phys. Rev. B ${\\bf 70}$ 125324] It is proven in this paper that both the positivity and Pauli's exclusion principle are preserved under this equation when there exists an upper bound for the transition rate. Such an equation can be generalized to model BCS-type quasiparticl...
Directory of Open Access Journals (Sweden)
Lisa M. Bishop
2010-09-01
Full Text Available We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a individual biochemical reactions, (b nonlinear network dynamics approaching to attractors, and (c cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a and (c are stochastic while that with (b is dominantly deterministic. Both (b and (c are emergent properties of a dynamic biochemical network; We suggest that the (c is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b in a “punctuated equilibrium” manner.
Qian, Hong; Bishop, Lisa M
2010-01-01
We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner. PMID:20957107
Hasenauer, J; Wolf, V; Kazeroonian, A; Theis, F J
2014-09-01
The time-evolution of continuous-time discrete-state biochemical processes is governed by the Chemical Master Equation (CME), which describes the probability of the molecular counts of each chemical species. As the corresponding number of discrete states is, for most processes, large, a direct numerical simulation of the CME is in general infeasible. In this paper we introduce the method of conditional moments (MCM), a novel approximation method for the solution of the CME. The MCM employs a discrete stochastic description for low-copy number species and a moment-based description for medium/high-copy number species. The moments of the medium/high-copy number species are conditioned on the state of the low abundance species, which allows us to capture complex correlation structures arising, e.g., for multi-attractor and oscillatory systems. We prove that the MCM provides a generalization of previous approximations of the CME based on hybrid modeling and moment-based methods. Furthermore, it improves upon these existing methods, as we illustrate using a model for the dynamics of stochastic single-gene expression. This application example shows that due to the more general structure, the MCM allows for the approximation of multi-modal distributions. PMID:23918091
Reducing differential equations for multiloop master integrals
Lee, Roman
2014-01-01
We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter $\\epsilon$. We consider linear transformations of the functions column which are rational in the variable and in $\\epsilon$. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degen...
Master equation approach to protein folding
Cieplak, Marek; Henkel, Malte; Banavar, Jayanth R.
1998-01-01
The dynamics of two 12-monomer heteropolymers on the square lattice is studied exactly within the master equation approach. The time evolution of the occupancy of the native state is determined. At low temperatures, the median folding time follows the Arrhenius law and is governed by the longest relaxation time. For good folders, significant kinetic traps appear in the folding funnel whereas for bad folders, the traps also occur in non-native energy valleys.
Ge, Hao
2009-01-01
A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105 585-593 (2003), Annu. Rev. Phys. Chem. 58 113-142 (2007)], emerges in the signal transduction module of phosphorylation-dephosphorylation cycle (PdPC). It utilizes multiple kinetic cycles in time, in contrast to allosteric cooperativity that utilizes multiple subunits in a protein. In the present paper, we thoroughly investigate both the deterministic (microscopic) and stochastic (mesoscopic) models, and focus on the identification of the source of temporal cooperativity via comparing with allosteric cooperativity. A thermodynamic analysis confirms again the claim that the chemical equilibrium state exists if and only if the phosphorylation potential $\\triangle G=0$, in which case the amplification of sensitivity is completely abolished. Then we provide comprehensive theoretical and numerical analysis with the first-order and zero-order assumptions in phosphorylation-dephosphorylation cycle respectively. Furthermore, it is interesti...
On the quantum master equation under feedback control
Institute of Scientific and Technical Information of China (English)
QI Bo
2009-01-01
The nature of the quantum trajectories,described by stochastic master equations,may be jump-like or diffusive,depending upon different measurement processes.There are many different unravelings corresponding to different types of stochastic master equations for a given master equation.In this paper,we study the relationship between the quantum stochastic master equations and the quantum master equations in the Markovian case under feedback control.We show that the corresponding unraveling no longer exists when we further consider feedback control besides measurement.It is due to the fact that the information gained by the measurement plays an important role in the control process.The master equation governing the evolution of ensemble average cannot be restored simply by eliminating the noise term unlike the case without a control term.By establishing a fundamental limit on performance of the master equation with feedback control,we demonstrate the differences between the stochastic master equation and the master equation via theoretical proof and simulation,and show the superiority of the stochastic master equation for feedback control.
Epidemics in networks: A master equation approach
Cotacallapa, M
2016-01-01
A problem closely related to epidemiology, where a subgraph of 'infected' links is defined inside a larger network, is investigated. This subgraph is generated from the underlying network by a random variable, which decides whether a link is able to propagate a disease/information. The relaxation timescale of this random variable is examined in both annealed and quenched limits, and the effectiveness of propagation of disease/information is analyzed. The dynamics of the model is governed by a master equation and two types of underlying network are considered: one is scale-free and the other has exponential degree distribution. We have shown that the relaxation timescale of the contagion variable has a major influence on the topology of the subgraph of infected links, which determines the efficiency of spreading of disease/information over the network.
Epidemics in networks: a master equation approach
International Nuclear Information System (INIS)
A problem closely related to epidemiology, where a subgraph of ‘infected’ links is defined inside a larger network, is investigated. This subgraph is generated from the underlying network by a random variable, which decides whether a link is able to propagate a disease/information. The relaxation timescale of this random variable is examined in both annealed and quenched limits, and the effectiveness of propagation of disease/information is analyzed. The dynamics of the model is governed by a master equation and two types of underlying network are considered: one is scale-free and the other has exponential degree distribution. We have shown that the relaxation timescale of the contagion variable has a major influence on the topology of the subgraph of infected links, which determines the efficiency of spreading of disease/information over the network. (paper)
A Convergent Reaction-Diffusion Master Equation
Isaacson, Samuel A
2012-01-01
The reaction-diffusion master equation (RDME) is a lattice stochastic reaction-diffusion model that has been used to study spatially distributed cellular processes. The RDME has been shown to have the drawback of losing bimolecular reactions in the continuum limit that the lattice spacing approaches zero (in two or more dimensions). In this work we derive a new convergent RDME (CRDME) that eliminates this problem. The CRDME is obtained by finite volume discretization of a spatially-continuous stochastic reaction-diffusion model. We demonstrate the empirical numerical convergence of reaction time statistics associated with the CRDME. Although the reaction time statistics of the RDME diverge as the lattice spacing approaches zero, we show they approach those of the CRDME for sufficiently large lattice spacings or slow bimolecular reaction rates. As such, the RDME may be interpreted as an approximation to the CRDME in several asymptotic limits.
Master Equations for Correlated Quantum Channels
Giovannetti, V.; Palma, G. M.
2012-01-01
We derive the general form of a master equation describing the reduced time evolution of a sequence of subsystems “propagating” in an environment which can be described as a sequence of subenvironments. The interaction between subsystems and subenvironments is described in terms of a collision model, with the irreversible dynamics of the subenvironments between collisions explicitly taken into account. In the weak coupling regime, we show that the collisional model produces a correlated Markovian evolution for the joint density matrix of the multipartite system. The associated Lindblad superoperator contains pairwise terms describing cross correlation between the different subsystems. Such a model can describe a broad range of physical situations, ranging from quantum channels with memory to photon propagation in concatenated quantum optical systems.
The complex chemical Langevin equation
International Nuclear Information System (INIS)
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE’s main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE’s predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE’s accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the “complex CLE” predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation
Decoherence and quantum-classical master equation dynamics
Grunwald, Robbie; Kapral, Raymond
2007-03-01
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
Staying positive: going beyond Lindblad with perturbative master equations
Whitney, Robert S.
2008-05-01
The perturbative master equation (Bloch-Redfield) is used extensively to study dissipative quantum mechanics—particularly for qubits—despite the 25-year-old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom and cast its perturbative master equation (derived from a perturbative treatment of Nakajima-Zwanzig or Schoeller-Schön equations) in the form of a Lindblad master equation. We find that the equation's parameters are time dependent. This time dependence is rarely accounted for and invalidates Lindblad's dynamical semigroup analysis. We analyse one such Bloch-Redfield master equation (for a two-level system coupled to an environment with a short but non-vanishing memory time), which apparently violates positivity. We analytically show that, once the time dependence of the parameters is accounted for, positivity is preserved.
Counting master integrals. Integration by parts vs. functional equations
Energy Technology Data Exchange (ETDEWEB)
Kniehl, Bernd A.; Tarasov, Oleg V. [Hamburg Univ. (Germany). II. Inst. fuer Theoretische Physik
2016-01-15
We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.
Counting master integrals. Integration by parts vs. functional equations
International Nuclear Information System (INIS)
We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.
Counting master integrals: Integration by parts vs. functional equations
Kniehl, Bernd A
2016-01-01
We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.
Master equation for a quantum particle in a gas
Hornberger, Klaus
2006-01-01
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts non-perturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as li...
The Approach to Equilibrium: Detailed Balance and the Master Equation
Alexander, Millard H.; Hall, Gregory E.; Dagdigian, Paul J.
2011-01-01
The approach to the equilibrium (Boltzmann) distribution of populations of internal states of a molecule is governed by inelastic collisions in the gas phase and with surfaces. The set of differential equations governing the time evolution of the internal state populations is commonly called the master equation. An analytic solution to the master…
Variational master equation approach to dynamics of magnetic moments
Bogolubov, N. N.; Soldatov, A. V.
2016-07-01
Non-equilibrium properties of a model system comprised of a subsystem of magnetic moments strongly coupled to a selected Bose field mode and weakly coupled to a heat bath made of a plurality of Bose field modes was studied on the basis of non-equilibrium master equation approach combined with the approximating Hamiltonian method. A variational master equation derived within this approach is tractable numerically and can be readily used to derive a set of ordinary differential equations for various relevant physical variables belonging to the subsystem of magnetic moments. Upon further analysis of the thus obtained variational master equation, an influence of the macroscopic filling of the selected Bose field mode at low enough temperatures on the relaxation dynamics of magnetic moments was revealed.
Master equations for photochemistry with intense infrared light. Pt. 4
Energy Technology Data Exchange (ETDEWEB)
Quack, M.
1981-04-01
A unified master equation for unimolecular reactions induced by monochromatic infrared radiation (URIMIR) is presented. Its effective rate coefficient matrix covers both case B (Pauli equation) and case C, properly including the nonlinearity of the latter. Exact quantum mechanical model solutions are compared with results from the approximate unified master equation. The exact analytical solutions of the master equation are presented for the URIMIR of some realistic molecular models. The important new properties of the transition range between case B and case C are quantitatively discussed with respect to time dependent and steady state level populations, time dependent and steady state rate coefficients and their nonlinear intensity dependence, and with respect to the influence of molecular properties. The role of case C for the interpretation of static field effects and its importance for efficient isotope separation are pointed out.
Variational master equation approach to dynamics of magnetic moments
International Nuclear Information System (INIS)
Non-equilibrium properties of a model system comprised of a subsystem of magnetic moments strongly coupled to a selected Bose field mode and weakly coupled to a heat bath made of a plurality of Bose field modes were studied on the basis of non-equilibrium master equation approach combined with the approximating Hamiltonian method. A variational master equation derived within this approach is tractable numerically and can be readily used to derive a set of ordinary differential equations for various relevant physical variables belonging to the subsystem of magnetic moments. Upon further analysis of the thus obtained variational master equation, an influence of the macroscopic filling of the selected Bose field mode at low enough temperatures on the relaxation dynamics of magnetic moments was revealed.
The pentabox Master Integrals with the Simplified Differential Equations approach
Papadopoulos, Costas G.; Tommasini, Damiano; Wever, Christopher
2016-04-01
We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many 2 → 3 amplitudes of interest at the LHC, as for instance three jet production, γ , V, H + 2 jets etc., based on the Simplified Differential Equations approach.
The Pentabox Master Integrals with the Simplified Differential Equations approach
Papadopoulos, Costas G; Wever, Christopher
2015-01-01
We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many $2\\to 3$ amplitudes of interest at the LHC, as for instance three jet production, $\\gamma, V, H +2$ jets etc., based on the Simplified Differential Equations approach.
Master equation based steady-state cluster perturbation theory
Nuss, Martin; Dorn, Gerhard; Dorda, Antonius; von der Linden, Wolfgang; Arrigoni, Enrico
2015-09-01
A simple and efficient approximation scheme to study electronic transport characteristics of strongly correlated nanodevices, molecular junctions, or heterostructures out of equilibrium is provided by steady-state cluster perturbation theory. In this work, we improve the starting point of this perturbative, nonequilibrium Green's function based method. Specifically, we employ an improved unperturbed (so-called reference) state ρ̂S, constructed as the steady state of a quantum master equation within the Born-Markov approximation. This resulting hybrid method inherits beneficial aspects of both the quantum master equation as well as the nonequilibrium Green's function technique. We benchmark this scheme on two experimentally relevant systems in the single-electron transistor regime: an electron-electron interaction based quantum diode and a triple quantum dot ring junction, which both feature negative differential conductance. The results of this method improve significantly with respect to the plain quantum master equation treatment at modest additional computational cost.
The Complexity of Relating Quantum Channels to Master Equations
Cubitt, Toby S.; Eisert, Jens; Wolf, Michael M.
2012-03-01
Completely positive, trace preserving (CPT) maps and Lindblad master equations are both widely used to describe the dynamics of open quantum systems. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski, Gorini and Sudarshan, who gave a complete characterisation of the master equations that generate completely positive semi-groups. However, the other direction has remained open: given a CPT map, is there a Lindblad master equation that generates it (and if so, can we find its form)? This is sometimes known as the Markovianity problem. Physically, it is asking how one can deduce underlying physical processes from experimental observations. We give a complexity theoretic answer to this problem: it is NP-hard. We also give an explicit algorithm that reduces the problem to integer semi-definite programming, a well-known NP problem. Together, these results imply that resolving the question of which CPT maps can be generated by master equations is tantamount to solving P = NP: any efficiently computable criterion for Markovianity would imply P = NP; whereas a proof that P = NP would imply that our algorithm already gives an efficiently computable criterion. Thus, unless P does equal NP, there cannot exist any simple criterion for determining when a CPT map has a master equation description. However, we also show that if the system dimension is fixed (relevant for current quantum process tomography experiments), then our algorithm scales efficiently in the required precision, allowing an underlying Lindblad master equation to be determined efficiently from even a single snapshot in this case. Our work also leads to similar complexity-theoretic answers to a related long-standing open problem in probability theory.
A Master Equation Approach to Quantum Chaos
Romanelli, A; Abal, G; Siri, R; Donangelo, R J
2002-01-01
We look at quantum diffusion and dynamical localization from a perspective which provides an intuitive framework to interpret known experimental and numerical results. We separate the Schr\\"{o}dinger equation into Markovian and interference terms, and show that the localized or diffusive character of the dynamics results from the competition between those terms. The procedure is illustrated through several examples.
Master Equation for Hydrogen Recombination on Grain Surfaces
Biham, O; Pirronello, V; Vidali, G; Biham, Ofer; Furman, Itay; Pirronello, Valerio; Vidali, Gianfranco
2000-01-01
Recent experimental results on the formation of molecular hydrogen on astrophysically relevant surfaces under conditions similar to those encountered in the interstellar medium provided useful quantitative information about these processes. Rate equation analysis of experiments on olivine and amorphous carbon surfaces provided the activation energy barriers for the diffusion and desorption processes relevant to hydrogen recombination on these surfaces. However, the suitability of rate equations for the simulation of hydrogen recombination on interstellar grains, where there might be very few atoms on a grain at any given time, has been questioned. To resolve this problem, we introduce a master equation that takes into account both the discrete nature of the H atoms and the fluctuations in the number of atoms on a grain. The hydrogen recombination rate on microscopic grains, as a function of grain size and temperature, is then calculated using the master equation. The results are compared to those obtained fro...
Wheeled PROPs, graph complexes and the master equation
Czech Academy of Sciences Publication Activity Database
Markl, Martin; Merkulov, S.; Shadrin, S.
2009-01-01
Roč. 213, č. 4 (2009), s. 496-535. ISSN 0022-4049 R&D Projects: GA ČR GA201/05/2117 Institutional research plan: CEZ:AV0Z10190503 Keywords : PROP * master equation Subject RIV: BA - General Mathematics Impact factor: 0.600, year: 2009
Limitations on the utility of exact master equations
International Nuclear Information System (INIS)
The low temperature solution of the exact master equation for an oscillator coupled to a linear passive heat bath is known to give rise to serious divergences. We now show that, even in the high temperature regime, problems also exist, notably the fact that the density matrix is not necessarily positive
A Contracted Path Integral Solution of the Discrete Master Equation
Helbing, Dirk
1998-01-01
A new representation of the exact time dependent solution of the discrete master equation is derived. This representation can be considered as contraction of the path integral solution of Haken. It allows the calculation of the probability distribution of the occurence time for each path and is suitable as basis of new computational solution methods.
Maxwell boundary conditions impose non-Lindblad master equation
Bamba, Motoaki
2016-01-01
From the Hamiltonian connecting the inside and outside of an Fabry-Perot cavity, which is derived from the Maxwell boundary conditions at a mirror of the cavity, a master equation of a non-Lindblad form is derived when the cavity embeds matters, although we can transform it to the Lindblad form by performing the rotating-wave approximation to that Hamiltonian. We calculate absorption spectra by these Lindblad and non-Lindblad master equations and also by the Maxwell boundary conditions in framework of the classical electrodynamics, which we consider the most reliable approach. We found that, compared to the Lindblad master equation, the absorption spectra by the non-Lindblad one agree better with those by the Maxwell boundary conditions. Although the discrepancy is highlighted only in the ultra-strong light-matter interaction regime with a relatively large broadening, the master equation of the non-Lindblad form is preferable rather than of the Lindblad one for pursuing the consistency with the classical elec...
Hierarchical quantum master equation with semiclassical Drude dissipation
Xu, Rui-Xue; Tian, Bao-Ling; Xu, Jian; Qiang SHI; Yan, YiJing
2009-01-01
We propose a nonperturbative quantum dissipation theory, in term of hierarchical quantum master equation. It may be used with a great degree of confidence to various dynamics systems in condensed phases. The theoretical development is rooted in an improved semiclassical treatment of Drude bath, beyond the conventional high temperature approximations. It leads to the new theory a simple modification but important improvement over the conventional stochastic Liouville equation theory, without e...
Post-Markovian quantum master equations from classical environment fluctuations
Budini, Adrián A.
2014-01-01
In this paper we demonstrate that two commonly used phenomenological post-Markovian quantum master equations can be derived without using any perturbative approximation. A system coupled to an environment characterized by self-classical configurational fluctuations, the latter obeying a Markovian dynamics, defines the underlying physical model. Both Shabani-Lidar equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005), 10.1103/PhysRevA.71.020101] and its associated approximated integrodifferential kernel master equation are obtained by tracing out two different bipartite Markovian Lindblad dynamics where the environment fluctuations are taken into account by an ancilla system. Furthermore, conditions under which the non-Markovian system dynamics can be unraveled in terms of an ensemble of measurement trajectories are found. In addition, a non-Markovian quantum jump approach is formulated. Contrary to recent analysis [L. Mazzola, E. M. Laine, H. P. Breuer, S. Maniscalco, and J. Piilo, Phys. Rev. A 81, 062120 (2010), 10.1103/PhysRevA.81.062120], we also demonstrate that these master equations, even with exponential memory functions, may lead to non-Markovian effects such as an environment-to-system backflow of information if the Hamiltonian system does not commutate with the dissipative dynamics.
Resummed memory kernels in generalized system-bath master equations
International Nuclear Information System (INIS)
Generalized master equations provide a concise formalism for studying reduced population dynamics. Usually, these master equations require a perturbative expansion of the memory kernels governing the dynamics; in order to prevent divergences, these expansions must be resummed. Resummation techniques of perturbation series are ubiquitous in physics, but they have not been readily studied for the time-dependent memory kernels used in generalized master equations. In this paper, we present a comparison of different resummation techniques for such memory kernels up to fourth order. We study specifically the spin-boson Hamiltonian as a model system bath Hamiltonian, treating the diabatic coupling between the two states as a perturbation. A novel derivation of the fourth-order memory kernel for the spin-boson problem is presented; then, the second- and fourth-order kernels are evaluated numerically for a variety of spin-boson parameter regimes. We find that resumming the kernels through fourth order using a Padé approximant results in divergent populations in the strong electronic coupling regime due to a singularity introduced by the nature of the resummation, and thus recommend a non-divergent exponential resummation (the “Landau-Zener resummation” of previous work). The inclusion of fourth-order effects in a Landau-Zener-resummed kernel is shown to improve both the dephasing rate and the obedience of detailed balance over simpler prescriptions like the non-interacting blip approximation, showing a relatively quick convergence on the exact answer. The results suggest that including higher-order contributions to the memory kernel of a generalized master equation and performing an appropriate resummation can provide a numerically-exact solution to system-bath dynamics for a general spectral density, opening the way to a new class of methods for treating system-bath dynamics
Master Equation for Electromagnetic Dissipation and Decoherence of Density Matrix
Haba, Z.; Kleinert, H.
2000-01-01
We set up a forward--backward path integral for a point particle in a bath of photons to derive a master equation for the density matrix which describes electromagnetic dissipation and decoherence. As an application, we recalculate the Weisskopf-Wigner formula for the natural line width of an atomic state at zero temperature and find, in addition, the temperature broadening caused by the decoherence term.
Random transition-rate matrices for the master equation
Timm, Carsten
2009-01-01
Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of independent rates of forward and backward transitions are considered. The first case leads to symmetric transition-rate matrices, whereas the second corresponds to general, asymmetric matrices. The resulting matrix ensembles are different from the standard...
Reaction rates for a generalized reaction-diffusion master equation
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
Master equation for collective spontaneous emission with quantized atomic motion
Damanet, François; Braun, Daniel; Martin, John
2015-01-01
We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including the quantization of their motion. Our equation provides a unifying picture of the effects of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, generalizing those in Ref. [1]. We find closed-form formulas for a numb...
Classes of exact solutions to the Teukolsky master equation
International Nuclear Information System (INIS)
The Teukolsky Master Equation (TME) describes perturbations of the Kerr metric in linear approximation. It admits separation of variables, thus yielding the Teukolsky Radial Equation (TRE) and the Teukolsky Angular Equation (TAE). We present here a unified description of all classes of exact solutions to these equations in terms of the confluent Heun functions and the confluent Heun polynomials. Large classes of new exact solutions are found and described together with their characteristic properties. Special attention is paid to the polynomial solutions which are singular ones and describe collimated one-way-running waves. It is shown that a proper linear combination of such singular solutions can describe bounded one-way-running waves
Umut Caglar, Mehmet; Pal, Ranadip
2010-10-01
The central dogma of molecular biology states that ``information cannot be transferred back from protein to either protein or nucleic acid.'' However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of data in the cellular level and probabilistic nature of interactions. Probabilistic models like Stochastic Master Equation (SME) or deterministic models like differential equations (DE) can be used to analyze these types of interactions. SME models based on chemical master equation (CME) can provide detailed representation of genetic regulatory system, but their use is restricted by the large data requirements and computational costs of calculations. The differential equations models on the other hand, have low calculation costs and much more adequate to generate control procedures on the system; but they are not adequate to investigate the probabilistic nature of interactions. In this work the success of the mapping between SME and DE is analyzed, and the success of a control policy generated by DE model with respect to SME model is examined. Index Terms--- Stochastic Master Equation models, Differential Equation Models, Control Policy Design, Systems biology
From convolutionless Generalized Master to finite-coupling Pauli Master Equations
International Nuclear Information System (INIS)
Time-convolutionless Generalized Master Equations (TCL-GME) for probabilities of finding a system in a general state irrespective of the state of the thermodynamic bath, are investigated. For general systems interacting with a genuine bath with a continuous spectrum described by time-independent system + bath Hamiltonians and after the thermodynamic limit for the bath, the long-time asymptotics of time-dependent coefficients can be taken as counterparts of Pauli-Master-Equation (PME) transfer rates. Here, within TCL-GME, asymptotics of these coefficients is calculated without resorting to any approximation. In the lowest order, these coefficients are known to turn into the usual Fermi Golden Rule transfer rates. Anyway, it is argued that if the exact form of these coefficients has a long-time limit, this limit is inevitably equal to zero. This makes illusory the possibility to derive standard Markovian finite-coupling Pauli, rate or balance equations as long-time asymptotics to TCL-GME. (author)
Random transition-rate matrices for the master equation.
Timm, Carsten
2009-08-01
Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of independent rates of forward and backward transitions are considered. The first case leads to symmetric transition-rate matrices, whereas the second corresponds to general asymmetric matrices. The resulting matrix ensembles are different from the standard ensembles and show different eigenvalue distributions. For example, the fraction of real eigenvalues scales anomalously with matrix dimension in the asymmetric case. PMID:19792110
Atom-wall dispersive forces from master equation formalism
Mendes, T. N. C.; Farina, C.
2007-01-01
Using the general expressions for level shifts obtained from the master equation for a small system interacting with a large one considered as a reservoir, we calculate the dispersive potentials between an atom and a wall in the dipole approximation. We analyze in detail the particular case of a two-level atom in the presence of a perfectly conducting wall. We study the van der Waals as well as the resonant interactions. All distance regimes as well as the high and low temperature regimes are...
The breakdown of the reaction-diffusion master equation with non-elementary rates
Smith, Stephen
2016-01-01
The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well-mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation, but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class then the RDME converges to the CME of the same system; while if the RDME has propensities not in this class then convergence is not guaranteed. These are revealed to be elementary and non-element...
Operator Approach to the Master Equation for the One-Step Process
Hnatič, M.; Eferina, E. G.; Korolkova, A. V.; Kulyabov, D. S.; Sevastyanov, L. A.
2016-02-01
Background. Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. The expansion of the equation in a formal Taylor series (the so called Kramers-Moyal's expansion) is used in the procedure of stochastization of one-step processes. Purpose. However, this does not eliminate the need for the study of the master equation. Method. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). Results: This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. Conclusions: We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation.
A Master Equation for Multi-Dimensional Non-Linear Field Theories
Park, Q H
1992-01-01
A master equation ( $n$ dimensional non--Abelian current conservation law with mutually commuting current components ) is introduced for multi-dimensional non-linear field theories. It is shown that the master equation provides a systematic way to understand 2-d integrable non-linear equations as well as 4-d self-dual equations and, more importantly, their generalizations to higher dimensions.
Diffusive Limits of the Master Equation in Inhomogeneous Media
Sattin, F; Salasnich, L
2015-01-01
In inhomogeneous environments several expressions for the flux of a diffusing quantity may apply--from Fick-Fourier's to Fokker-Planck's--depending upon the system studied. The integro-differential Master Equation (ME) provides a fairly generic framework for describing the dynamics of arbitrary systems driven by stochastic rules. Diffusive dynamics does arise as long-wavelength limit of the ME. However, while it is straightforward to obtain a diffusion equation with Fokker-Planck flux, its Fick-Fourier counterpart has never been worked out from the ME. In this work we show under which hypothesis the Fick's flux can actually be recovered from the ME. Analytical considerations are supported by explicit computer models.
Master equations in the microscopic theory of nuclear collective dynamics
International Nuclear Information System (INIS)
In the first half of this paper, the authors describe briefly a recent theoretical approach where the mechanism of the large-amplitude dissipative collective motions can be investigated on the basis of the microscopic theory of nuclear collective dynamics. Namely, we derive the general coupled master equations which can disclose, in the framework of the TDHF theory, not only non-linear dynamics among the collective and the single-particle modes of motion but also microscopic dynamics responsible for the dissipative processes. In the latter half, the authors investigate, without relying on any statistical hypothesis, one possible microscopic origin which leads us to the transport equation of the Fokker-Planck type so that usefullness of the general framework is demonstrated. (author)
Thermodynamics of the polaron master equation at finite bias
Energy Technology Data Exchange (ETDEWEB)
Krause, Thilo, E-mail: tkrause@physik.tu-berlin.de; Brandes, Tobias; Schaller, Gernot, E-mail: gernot.schaller@tu-berlin.de [Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin (Germany); Esposito, Massimiliano [Complex Systems and Statistical Mechanics, University of Luxembourg, L-1511 Luxembourg (Luxembourg)
2015-04-07
We study coherent transport through a double quantum dot. Its two electronic leads induce electronic matter and energy transport and a phonon reservoir contributes further energy exchanges. By treating the system-lead couplings perturbatively, whereas the coupling to vibrations is treated non-perturbatively in a polaron-transformed frame, we derive a thermodynamic consistent low-dimensional master equation. When the number of phonon modes is finite, a Markovian description is only possible when these couple symmetrically to both quantum dots. For a continuum of phonon modes however, also asymmetric couplings can be described with a Markovian master equation. We compute the electronic current and dephasing rate. The electronic current enables transport spectroscopy of the phonon frequency and displays signatures of Franck-Condon blockade. For infinite external bias but finite tunneling bandwidths, we find oscillations in the current as a function of the internal bias due to the electron-phonon coupling. Furthermore, we derive the full fluctuation theorem and show its identity to the entropy production in the system.
Master equation approach to DNA breathing in heteropolymer DNA
Ambjörnsson, Tobias; Banik, Suman K.; Lomholt, Michael A.; Metzler, Ralf
2007-02-01
After crossing an initial barrier to break the first base-pair (bp) in double-stranded DNA, the disruption of further bps is characterized by free energies up to a few kBT . Thermal motion within the DNA double strand therefore causes the opening of intermittent single-stranded denaturation zones, the DNA bubbles. The unzipping and zipping dynamics of bps at the two zipper forks of a bubble, where the single strand of the denatured zone joins the still intact double strand, can be monitored by single molecule fluorescence or NMR methods. We here establish a dynamic description of this DNA breathing in a heteropolymer DNA with given sequence in terms of a master equation that governs the time evolution of the joint probability distribution for the bubble size and position along the sequence. The transfer coefficients are based on the Poland-Scheraga free energy model. We derive the autocorrelation function for the bubble dynamics and the associated relaxation time spectrum. In particular, we show how one can obtain the probability densities of individual bubble lifetimes and of the waiting times between successive bubble events from the master equation. A comparison to results of a stochastic Gillespie simulation shows excellent agreement.
Generalized master equations for exciton dynamics in molecular systems
Schreiber, Michael; May, V.
1995-02-01
The paper demonstrates the applicability of a special type of density matrix theory for the derivation of generalized Master equations. The density matrix theory has been formulated for the description of the dissipative electron transfer dynamics in molecular complexes. The theoretical approach is based on a representation of the density matrix in appropriately taken diabatic electron-vibrational states. Dissipative effects are taken into account by a coupling of these states to further vibrational modes of the molecular complex as well as to environmental degrees of freedom. The approach is applied to a two-center system as well as to a molecular chain. Memory kernels are derived in second order with respect to the inter-center coupling. The kernels are discussed under the assumption of a quick intra-center relaxation for a part of the vibrational modes as well as for all vibrational modes. Standard expressions for the transition rates between different sites are extended to include finite life times of the vibrational levels. Results which have been obtained in the study of the so-called spin boson model can be simply reproduced. The application of the derived generalized Master equations to the investigation of the motion of Frenkel excitons in molecular chains is also presented.
Derivation of Bloch equations from the time convolution less generalized master equation
International Nuclear Information System (INIS)
The generalized Bloch equations (GBE) describing the temporal evolution of a single two-level atom interacting with a classical external field of arbitrary intensity and with a thermodynamic bath are obtained from the time convolutionless generalized master equation or equivalently from the Tokuyama-Mori identity. These GBE are then used to calculate the absorption spectrum of a single two-level atom with frequency modulated by dichotomic noise with time-dependent transition probability. (author)
Nonlinear generalized master equations and accounting for initial correlations
Los, V. F.
2009-08-01
We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can
Continuous monitoring of dynamical systems and master equations
International Nuclear Information System (INIS)
We illustrate the equivalence between the non-unitary evolution of an open quantum system governed by a Markovian master equation and a process of continuous measurements involving this system. We investigate a system of two coupled modes, only one of them interacting with external degrees of freedom, represented, in the first case, by a finite number of harmonic oscillators, and, in the second, by a sequence of atoms where each one interacts with a single mode during a limited time. Two distinct regimes appear, one of them corresponding to a Zeno-like behavior in the limit of large dissipation. -- Highlights: ► We illustrate the conjecture that non-unitary evolution can be simulated by continuous measurements. ► The relation between unitary and non-unitary couplings define distinct dynamical regimes. ► One regime with large “dissipation constant” is a Zeno-like behavior.
Master equation approach to DNA breathing in heteropolymer DNA
DEFF Research Database (Denmark)
Ambjörnsson, Tobias; Banik, Suman K; Lomholt, Michael A;
2007-01-01
After crossing an initial barrier to break the first base-pair (bp) in double-stranded DNA, the disruption of further bps is characterized by free energies up to a few k(B)T. Thermal motion within the DNA double strand therefore causes the opening of intermittent single-stranded denaturation zones......, the DNA bubbles. The unzipping and zipping dynamics of bps at the two zipper forks of a bubble, where the single strand of the denatured zone joins the still intact double strand, can be monitored by single molecule fluorescence or NMR methods. We here establish a dynamic description of this DNA breathing...... in a heteropolymer DNA with given sequence in terms of a master equation that governs the time evolution of the joint probability distribution for the bubble size and position along the sequence. The transfer coefficients are based on the Poland-Scheraga free energy model. We derive the autocorrelation function...
Master equation calculations of cluster formation in supersonic jets
International Nuclear Information System (INIS)
The kinetics of cluster formation in supersonic jets is examined by numerical integration of the master equation system. Some general characteristics of cluster kinetics could be formulated. Excellent agreement between experimental curves of p-cresol (H2O)0,1,2,3 formation as function of H2O pressure and the corresponding calculated curves were obtained assuming successive cluster formation. From the kinetic curves, and unambiguous assignment of cluster size was possible which agreed with mass-resolved REMPI measurements. The fit of the rate coefficients shows the formation of p-cresol (H2O)1 to be faster than p-cresol (H2O)2 and p-cresol (H2O)3. (orig.)
Atom-wall dispersive forces from master equation formalism
Mendes, T N C
2007-01-01
Using the general expressions for level shifts obtained from the master equation for a small system interacting with a large one considered as a reservoir, we calculate the dispersive potentials between an atom and a wall in the dipole approximation. We analyze in detail the particular case of a two-level atom in the presence of a perfectly conducting wall. We study the van der Waals as well as the resonant interactions. All distance regimes as well as the high and low temperature regimes are considered. We show that the Casimir-Polder interaction can not be considered as a direct result of the vacuum fluctuations only. Concerning the interaction between the atom and the wall at high temperature, which show that a saturation of the potential for all distances occurs. This saturated potential coincides exactly with that obtained in the London-van der Waals limit.
Solution to the Master Equation of a Free Damped Harmonic Oscillator with Linear Driving
Institute of Scientific and Technical Information of China (English)
杨洁; 逯怀新; 赵博; 赵梅生; 张永德
2003-01-01
We use the Lie algebra representation theory for superoperators to solve the master equation for a harmonic oscillator with a linear driving term in a squeezed thermal reservoir. By using the quantum displacement transformation and squeeze transformation, we show that the master equation has an su(1, 1) Lie algebra structure,with which we obtain the explicit solution to the master equation. A simple but typical example is given to illustrate our method.
Master equation solutions in the linear regime of characteristic formulation of general relativity
M., C E Cedeño
2015-01-01
From the field equations in the linear regime of the characteristic formulation of general relativity, Bishop, for a Schwarzschild's background, and M\\"adler, for a Minkowski's background, were able to show that it is possible to derive a fourth order ordinary differential equation, called master equation, for the $J$ metric variable of the Bondi-Sachs metric. Once $\\beta$, another Bondi-Sachs potential, is obtained from the field equations, and $J$ is obtained from the master equation, the other metric variables are solved integrating directly the rest of the field equations. In the past, the master equation was solved for the first multipolar terms, for both the Minkowski's and Schwarzschild's backgrounds. Also, M\\"adler recently reported a generalisation of the exact solutions to the linearised field equations when a Minkowski's background is considered, expressing the master equation family of solutions for the vacuum in terms of Bessel's functions of the first and the second kind. Here, we report new sol...
Master equation simulation analysis of immunostained Bicoid morphogen gradient
Directory of Open Access Journals (Sweden)
Reinitz John
2007-11-01
Full Text Available Abstract Background The concentration gradient of Bicoid protein which determines the developmental pathways in early Drosophila embryo is the best characterized morphogen gradient at the molecular level. Because different developmental fates can be elicited by different concentrations of Bicoid, it is important to probe the limits of this specification by analyzing intrinsic fluctuations of the Bicoid gradient arising from small molecular number. Stochastic simulations can be applied to further the understanding of the dynamics of Bicoid morphogen gradient formation at the molecular number level, and determine the source of the nucleus-to-nucleus expression variation (noise observed in the Bicoid gradient. Results We compared quantitative observations of Bicoid levels in immunostained Drosophila embryos with a spatially extended Master Equation model which represents diffusion, decay, and anterior synthesis. We show that the intrinsic noise of an autonomous reaction-diffusion gradient is Poisson distributed. We demonstrate how experimental noise can be identified in the logarithm domain from single embryo analysis, and then separated from intrinsic noise in the normalized variance domain of an ensemble statistical analysis. We show how measurement sensitivity affects our observations, and how small amounts of rescaling noise can perturb the noise strength (Fano factor observed. We demonstrate that the biological noise level in data can serve as a physical constraint for restricting the model's parameter space, and for predicting the Bicoid molecular number and variation range. An estimate based on a low variance ensemble of embryos suggests that the steady-state Bicoid molecular number in a nucleus should be larger than 300 in the middle of the embryo, and hence the gradient should extend to the posterior end of the embryo, beyond the previously assumed background limit. We exhibit the predicted molecular number gradient together with
A test for two Fokker-Planck modellings of a master equation
International Nuclear Information System (INIS)
The kind of two Fokker-Planck equation modellings of a non-equilibrium system, defined by a two-variable master equation, is studied. We compute explicitly their associated stationary potentials in the weak noise limit, and we compare them with the exact one coming from the master equation. We Find that a recently proposed Fokker-Planck equation is far better than the other coming from the commonly used Kramers-Moyal expansion. (author)
Symmetric and antisymmetric forms of the Pauli master equation.
Klimenko, A Y
2016-01-01
When applied to matter and antimatter states, the Pauli master equation (PME) may have two forms: time-symmetric, which is conventional, and time-antisymmetric, which is suggested in the present work. The symmetric and antisymmetric forms correspond to symmetric and antisymmetric extensions of thermodynamics from matter to antimatter - this is demonstrated by proving the corresponding H-theorem. The two forms are based on the thermodynamic similarity of matter and antimatter and differ only in the directions of thermodynamic time for matter and antimatter (the same in the time-symmetric case and the opposite in the time-antisymmetric case). We demonstrate that, while the symmetric form of PME predicts an equibalance between matter and antimatter, the antisymmetric form of PME favours full conversion of antimatter into matter. At this stage, it is impossible to make an experimentally justified choice in favour of the symmetric or antisymmetric versions of thermodynamics since we have no experience of thermodynamic properties of macroscopic objects made of antimatter, but experiments of this kind may become possible in the future. PMID:27440454
Closed description of arbitrariness in resolving quantum master equation
Batalin, Igor A.; Lavrov, Peter M.
2016-07-01
In the most general case of the Delta exact operator valued generators constructed of an arbitrary Fermion operator, we present a closed solution for the transformed master action in terms of the original master action in the closed form of the corresponding path integral. We show in detail how that path integral reduces to the known result in the case of being the Delta exact generators constructed of an arbitrary Fermion function.
Closed description of arbitrariness in resolving quantum master equation
Directory of Open Access Journals (Sweden)
Igor A. Batalin
2016-07-01
Full Text Available In the most general case of the Delta exact operator valued generators constructed of an arbitrary Fermion operator, we present a closed solution for the transformed master action in terms of the original master action in the closed form of the corresponding path integral. We show in detail how that path integral reduces to the known result in the case of being the Delta exact generators constructed of an arbitrary Fermion function.
Closed description of arbitrariness in resolving quantum master equation
Batalin, Igor A
2016-01-01
In the most general case of the Delta exact operator valued generators constructed of an arbitrary Fermion operator, we present a closed solution for the transformed master action in terms of the original master action in the closed form of the corresponding path integral. We show in detail how that path integral reduces to the known result in the case of being the Delta exact generators constructed of an arbitrary Fermion function.
Kinetic limits for pair-interaction driven master equations and biological swarm models
Carlen, Eric; Degond, Pierre; Wennberg, Bernt
2011-01-01
We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair interaction driven master equations. We prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well know result for the Kac model in kinetic theory. We use this result to study kinetic limits for two b...
Kinetic limits for pair-interaction driven master equations and biological swarm models
Carlen, Eric; Wennberg, Bernt
2011-01-01
We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair interaction driven master equations. We prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well know result for the Kac model in kinetic theory. We use this result to study kinetic limits for two biological swarm models. We show that propagation of chaos may be lost at large times and we exhibit an example where the invariant density is not chaotic.
Two-loop master integrals with the simplified differential equations approach
Papadopoulos, Costas; Tommasini, Damiano; Wever, Christopher
2015-01-01
We calculate the complete set of two-loop Master Integrals with two off mass-shell legs with massless internal propagators, that contribute to amplitudes of diboson $V_1V_2$ production at the LHC. This is done with the Simplified Differential Equations approach to Master Integrals, which was recently proposed by one of the authors.
Exact Solution of the Curved Dirac Equation in Polar Coordinates: Master Function Approach
Directory of Open Access Journals (Sweden)
H. Panahi
2015-01-01
Full Text Available We show that the (2+1 curved Dirac equation in polar coordinates can be transformed into Schrodinger-like differential equation for upper spinor component. We compare this equation with the Schrodinger equation derived from shape invariance property of second order differential equations of mathematical physics. This formalism enables us to determine the electrostatic potential and relativistic energy in terms of master function and corresponding weight function. We also obtain the spinor wave function in terms of orthogonal polynomials.
Number-conserving master equation theory for a dilute Bose-Einstein condensate
Schelle, Alexej; Wellens, Thomas; Delande, Dominique; Buchleitner, Andreas
2010-01-01
We describe the transition of $N$ weakly interacting atoms into a Bose-Einstein condensate within a number-conserving quantum master equation theory. Based on the separation of time scales for condensate formation and non-condensate thermalization, we derive a master equation for the condensate subsystem in the presence of the non-condensate environment under the inclusion of all two body interaction processes. We numerically monitor the condensate particle number distribution during condensa...
Modified Bloch-Redfield Master Equation for Incoherent Excitation of Multilevel Quantum Systems
Tscherbul, Timur V.; Brumer, Paul
2014-01-01
We present an efficient theoretical method for calculating the time evolution of the density matrix of a multilevel quantum system weakly interacting with incoherent light. The method combines the Bloch-Redfield theory with a partial secular approximation for one-photon coherences, resulting in a master equation that explicitly exposes the reliance on transition rates and the angles between transition dipole moments in the energy basis. The modified Bloch-Redfield master equation allows an un...
Kraus representation for maps and master equation in spin star model with layered environment
Mahdian, Mahmoud; Mehrabpour, Hadi
2013-01-01
Quantum operations are usually defined as completely positive (CP), trace preserving (TP) maps on quantum states, and can be represented by operator-sum or Kraus representations. In this paper, we calculate operator-sum representation and master equation of an exactly solvable dynamic of one-qubit open system in layered environment . On the other hand, we obtain exact Nakajima-Zwanzig (NZ) and time-convolutionless (TCL) master equation from the maps. Finally, we study a simple example to cons...
Completely positive post-Markovian master equation via a measurement approach
International Nuclear Information System (INIS)
A post-Markovian quantum master equation is derived, which includes bath memory effects via a phenomenologically introduced memory kernel k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t) is presented, in addition to a prescription for the experimental determination of k(t). The formalism is illustrated with examples
A classical Master equation approach to modeling an artificial protein motor
International Nuclear Information System (INIS)
Inspired by biomolecular motors, as well as by theoretical concepts for chemically driven nanomotors, there is significant interest in constructing artificial molecular motors. One driving force is the opportunity to create well-controlled model systems that are simple enough to be modeled in detail. A remaining challenge is the fact that such models need to take into account processes on many different time scales. Here we describe use of a classical Master equation approach, integrated with input from Langevin and molecular dynamics modeling, to stochastically model an existing artificial molecular motor concept, the Tumbleweed, across many time scales. This enables us to study how interdependencies between motor processes, such as center-of-mass diffusion and track binding/unbinding, affect motor performance. Results from our model help guide the experimental realization of the proposed motor, and potentially lead to insights that apply to a wider class of molecular motors.
Generalized Quantum Master Equations In and Out of Equilibrium: When Can One Win?
Kelly, Aaron; Wang, Lu; Markland, Thomas E
2016-01-01
Generalized quantum master equations (GQMEs) are an important tool in modeling chemical and physical processes. The central quantity in these approaches is the memory kernel, which encodes the effect of the projected dynamical degrees of freedom on the observable of interest. For a large number of problems it has been shown that exact and approximate methods can be made dramatically more efficient, and in the latter case more accurate, by proceeding via the GQME formalism. However, there are many situations where utilizing the GQME approach seems to offer no advantage over a direct evaluation of the property of interest. The development of a more detailed understanding of the conditions under which these methods will offer benefits would thus greatly enhance their utility. Here, we derive exact expressions for the memory kernel obtained from projection operators for systems both in and out of equilibrium, and show the conditions under which these expressions will be guaranteed to return an identical result to...
Alfonso, L.
2015-01-01
In cloud modeling studies, the time evolution of droplet size distributions due to collision–coalescence events is usually modeled with the Smoluchowski coagulation equation, also known as the kinetic collection equation (KCE). However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equat...
Gelin, Maxim F
2014-12-01
We consider a classical point particle bilinearly coupled to a harmonic bath. Assuming that the evolution of the particle is monitored on a timescale which is longer than the characteristic bath correlation time, we derive the Markovian master equation for the probability density of the particle. The relaxation operator of this master equation is evaluated analytically, without invoking the perturbation theory and the approximation of weak system-bath coupling. When the bath correlation time tends to zero, the Fokker-Planck equation is recovered. For a finite bath correlation time, the relaxation operator contains contributions of all orders in the system-bath coupling. PMID:25481131
Dynamics of open quantum spin systems: An assessment of the quantum master equation approach
Zhao, P; Miyashita, S; Jin, F; Michielsen, K
2016-01-01
Data of the numerical solution of the time-dependent Schr\\"odinger equation of a system containing one spin-1/2 particle interacting with a bath of up to 32 spin-1/2 particles is used to construct a Markovian quantum master equation describing the dynamics of the system spin. The procedure of obtaining this quantum master equation, which takes the form of a Bloch equation with time-independent coefficients, accounts for all non-Markovian effects in as much the general structure of the quantum master equation allows. Our simulation results show that, with a few rather exotic exceptions, the Bloch-like equation with time-independent coefficients provides a simple and accurate description of the dynamics of a spin-1/2 particle in contact with a thermal bath. A calculation of the coefficients that appear in the Redfield master equation in the Markovian limit shows that this equation yields a rather poor description of the original data.
Fleming, C H; Hu, B L
2010-01-01
We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.
Quantal Brownian Motion from RPA dynamics: The master and Fokker-Planck equations
International Nuclear Information System (INIS)
From the purely quantal RPA description of the damped harmonic oscillator and of the corresponding Brownian Motion within the full space (phonon subspace plus reservoir), a master equation (as well as a Fokker-Planck equation) for the reduced density matrix (for the reduced Wigner function, respectively) within the phonon subspace is extracted. The RPA master equation agrees with the master equation derived by the time-dependent perturbative approaches which utilize Tamm-Dancoff Hilbert spaces and invoke the rotating wave approximation. Since the RPA yields a full, as well as a contracted description, it can account for both the kinetic and the unperturbed oscillator momenta. The RPA description of the quantal Brownian Motion contrasts with the descriptions provided by the time perturbative approaches whether they invoke or not the rotating wave approximation. The RPA description also contrasts with the phenomenological phase space quantization. (orig.)
Theory of Electron Transport in Small Semiconductor Devices Using the Pauli Master Equation
M. V. Fischetti
1998-01-01
It is argued that the Pauli master equation can be used to simulate electron transport in very small electronic devices under steady-state conditions. Written in a basis of suitable wavefunctions and with the appropriate open boundary conditions, this equation removes some of the approximations which render the Boltzmann equation unsatisfactory at small length-scales. The main problems consist in describing the interaction of the system with the reservoirs and in assessing the ...
An extended master-equation approach applied to aggregation in freeway traffic
Institute of Scientific and Technical Information of China (English)
Li Jun-Wei; Lin Bo-Liang; Huang Yong-Chang
2008-01-01
We restudy the master-equation approach applied to aggregation in a one-dimensional freeway,where the decay transition probabilities for the jump processes are reconstructed based on a car-following model. According to the reconstructed transition probabilities,the clustering behaviours and the stochastic properties of the master equation in a one-lane freeway traffic model are investigated in detail.The numerical results show that the size of the clusters initially below the critical size of the unstable cluster and initially above that of the unstable cluster all enter the same stable state,which also accords with the nucleation theory and is known from the result in earlier work.Moreover,we have obtained more reasonable parameters of the master equation based on some results of cellular automata models.
Master Equation Approach to Current-Voltage Characteristics of Solar Cells
Oh, Sangchul; Zhang, Yiteng; Alharbi, Fahhad; Kais, Sabre
2015-03-01
The current-voltage characteristics of solar cells is obtained using quantum master equations for electrons, holes, and excitons, in which generation, recombination, and transport processes are taken into account. As a first example, we simulate a photocell with a molecular aggregate donor to investigate whether a delocalized quantum state could enhance the efficiency. As a second example, we calculate the current-voltage characteristics of conventional p-n junction solar cells and perovskite solar cells using the master equation. The connection between the drift-diffusion model and the master equation method is established. The short-circuit current and the open-circuit voltage are calculated numerically as a function of the intensity of the sunlight and material properties such as energy gaps, diffusion constants, etc.
Vibrational energy flow in the villin headpiece subdomain: Master equation simulations
International Nuclear Information System (INIS)
We examine vibrational energy flow in dehydrated and hydrated villin headpiece subdomain HP36 by master equation simulations. Transition rates used in the simulations are obtained from communication maps calculated for HP36. In addition to energy flow along the main chain, we identify pathways for energy transport in HP36 via hydrogen bonding between residues quite far in sequence space. The results of the master equation simulations compare well with all-atom non-equilibrium simulations to about 1 ps following initial excitation of the protein, and quite well at long times, though for some residues we observe deviations between the master equation and all-atom simulations at intermediate times from about 1–10 ps. Those deviations are less noticeable for hydrated than dehydrated HP36 due to energy flow into the water
Complete positivity of a spin-1/2 master equation with memory
International Nuclear Information System (INIS)
We study a non-Markovian spin-1/2 master equation with exponential memory. We derive the conditions under which the dynamical map describing the reduced system dynamics is completely positive, i.e., the nonunitary evolution of the system is compatible with a description in terms of a closed total spin-reservoir system. Our results show that for a zero-T reservoir, the dynamical map of the model here considered is never completely positive. For moderate- and high-T reservoirs, on the contrary, positivity is a necessary and sufficient condition for complete positivity. We also consider the Shabani-Lidar master equation recently introduced [A. Shabani and D.A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and we demonstrate that such a master equation is always completely positive
Vibrational energy flow in the villin headpiece subdomain: Master equation simulations
Energy Technology Data Exchange (ETDEWEB)
Leitner, David M., E-mail: dml@unr.edu, E-mail: stock@physik.uni-freiburg.de [Department of Chemistry and Chemical Physics Program, University of Nevada, Reno, Nevada 89557 (United States); Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, Freiburg (Germany); Buchenberg, Sebastian; Brettel, Paul [Biomolecular Dynamics, Institute of Physics, University of Freiburg, Freiburg (Germany); Stock, Gerhard, E-mail: dml@unr.edu, E-mail: stock@physik.uni-freiburg.de [Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, Freiburg (Germany); Biomolecular Dynamics, Institute of Physics, University of Freiburg, Freiburg (Germany)
2015-02-21
We examine vibrational energy flow in dehydrated and hydrated villin headpiece subdomain HP36 by master equation simulations. Transition rates used in the simulations are obtained from communication maps calculated for HP36. In addition to energy flow along the main chain, we identify pathways for energy transport in HP36 via hydrogen bonding between residues quite far in sequence space. The results of the master equation simulations compare well with all-atom non-equilibrium simulations to about 1 ps following initial excitation of the protein, and quite well at long times, though for some residues we observe deviations between the master equation and all-atom simulations at intermediate times from about 1–10 ps. Those deviations are less noticeable for hydrated than dehydrated HP36 due to energy flow into the water.
Modified Bloch-Redfield Master Equation for Incoherent Excitation of Multilevel Quantum Systems
Tscherbul, Timur V
2014-01-01
We present an efficient theoretical method for calculating the time evolution of the density matrix of a multilevel quantum system weakly interacting with incoherent light. The method combines the Bloch-Redfield theory with a partial secular approximation for one-photon coherences, resulting in a master equation that explicitly exposes the reliance on transition rates and the angles between transition dipole moments in the energy basis. The modified Bloch-Redfield master equation allows an unambiguous distinction between the regimes of quantum coherent vs. incoherent energy transfer under incoherent light illumination. The fully incoherent regime is characterized by orthogonal transition dipole moments in the energy basis, leading to a dynamical evolution governed by a coherence-free Pauli-type master equation. The coherent regime requires non-orthogonal transition dipole moments in the energy basis, and leads to the generation of noise-induced quantum coherences and population-to-coherence couplings. As a fi...
A multi-site variational master equation approach to dissipative energy transfer
International Nuclear Information System (INIS)
Unitary transformations can allow one to study open quantum systems in situations for which standard, weak-coupling type approximations are not valid. We develop here an extension of the variational (polaron) transformation approach to open system dynamics, which applies to arbitrarily large exciton transport networks with local environments. After deriving a time-local master equation in the transformed frame, we go on to compare the population dynamics predicted using our technique with other established master equations. The variational frame dynamics are found to agree with both weak coupling and full polaron master equations in their respective regions of validity. In parameter regimes considered difficult for these methods, the dynamics predicted by our technique are found to interpolate between the two. The variational method thus gives insight, across a broad range of parameters, into the competition between coherent and incoherent processes in determining the dynamical behaviour of energy transfer networks. (paper)
Number-resolved master equation approach to quantum measurement and quantum transport
Li, Xin-Qi
2016-08-01
In addition to the well-known Landauer-Büttiker scattering theory and the nonequilibrium Green's function technique for mesoscopic transports, an alternative (and very useful) scheme is quantum master equation approach. In this article, we review the particle-number ( n)-resolved master equation ( n-ME) approach and its systematic applications in quantum measurement and quantum transport problems. The n-ME contains rich dynamical information, allowing efficient study of topics such as shot noise and full counting statistics analysis. Moreover, we also review a newly developed master equation approach (and its n-resolved version) under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.
Chiral Bosons as solutions of the BV master equation 2D chiral gauge theories
Braga, N. R. F.; Montani, H.
1994-01-01
We construct the chiral Wess-Zumino term as a solution for the Batalin-Vilkovisky master equation for anomalous two-dimensional gauge theories, working in an extended field-antifield space, where the gauge group elements are introduced as additional degrees of freedom. We analyze the Abelian and the non-Abelian cases, calculating in both cases the BRST generator in order to show the physical equivalence between this chiral solution for the master equation and the usual (non-chiral) one.
Unified Einstein-Virasoro Master Equation in the General Non-Linear Sigma Model
de Boer, J
2009-01-01
The Virasoro master equation (VME) describes the general affine-Virasoro construction $T=L^abJ_aJ_b+iD^a \\dif J_a$ in the operator algebra of the WZW model, where $L^ab$ is the inverse inertia tensor and $D^a $ is the improvement vector. In this paper, we generalize this construction to find the general (one-loop) Virasoro construction in the operator algebra of the general non-linear sigma model. The result is a unified Einstein-Virasoro master equation which couples the spacetime spin-two f...
Unified Einstein-Virasoro Master Equation in the General Non-Linear Sigma Model
de Boer, J; Halpern, M. B.
1996-01-01
The Virasoro master equation (VME) describes the general affine-Virasoro construction $T=L^{ab}J_aJ_b+iD^a \\dif J_a$ in the operator algebra of the WZW model, where $L^{ab}$ is the inverse inertia tensor and $D^a $ is the improvement vector. In this paper, we generalize this construction to find the general (one-loop) Virasoro construction in the operator algebra of the general non-linear sigma model. The result is a unified Einstein-Virasoro master equation which couples the spacetime spin-t...
Directory of Open Access Journals (Sweden)
L. Alfonso
2015-04-01
Full Text Available In cloud modeling studies, the time evolution of droplet size distributions due to collision-coalescence events is usually modeled with the kinetic collection equation (KCE or Smoluchowski coagulation equation. However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitude of correlations are significant, and must be taken into account when modeling the evolution of a finite volume coalescing system.
Alfonso, L.
2015-11-01
In cloud modeling studies, the time evolution of droplet size distributions due to collision-coalescence events is usually modeled with the Smoluchowski coagulation equation, also known as the kinetic collection equation (KCE). However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain types of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was seen by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitudes of correlations are significant and must be taken into account when modeling the evolution of a finite volume coalescing system.
The Master Ward Identity and generalized Schwinger-Dyson Equation in classical field theory
International Nuclear Information System (INIS)
In the framework of perturbative quantum field theory a new, universal renormalization condition (called Master Ward Identity) was recently proposed by one of us (M.D.) in a joint paper with F.-M. Boas. The main aim of the present paper is to get a better understanding of the Master Ward Identity by analyzing its meaning in classical field theory. It turns out that it is the most general identity for classical local fields which follows from the field equations. It is equivalent to a generalization of the Schwinger-Dyson Equation and is closely related to the Quantum Action Principle of Lowenstein and Lam. The validity of the Master Ward Identity makes possible a local construction of quantum gauge theories. (orig.)
DEFF Research Database (Denmark)
Iles-Smith, Jake; Dijkstra, Arend G.; Lambert, Neill; Nazir, Ahsan
2016-01-01
We explore excitonic energy transfer dynamics in a molecular dimer system coupled to both structured and unstructured oscillator environments. By extending the reaction coordinate master equation technique developed by Iles-Smith et al. [Phys. Rev. A 90, 032114 (2014)], we go beyond the commonly ...
Quantum dot as a spin-current diode: A master-equation approach
DEFF Research Database (Denmark)
Souza, F.M.; Egues, J.C.; Jauho, Antti-Pekka
2007-01-01
We report a study of spin-dependent transport in a system composed of a quantum dot coupled to a normal metal lead and a ferromagnetic lead NM-QD-FM. We use the master equation approach to calculate the spin-resolved currents in the presence of an external bias and an intradot Coulomb interaction...
Solution of the Master Equation for Bak-Sneppen Model of Biological Evolution in Finite Ecosystem
Pis'mak, Yu. M.
1996-01-01
The master equations describing processes of biological evolution in the framework of the random neighbor Bak-Sneppen model are studied. For the eqosystem of $N$ species they are solved exactly and asymptotical behavior of this solution for large $N$ is analyzed.
Fuchsia and master integrals for splitting functions from differential equations in QCD
Gituliar, O
2016-01-01
We report on the recent progress in reducing differential equations for Feynman master integrals to canonical form with the help of a method proposed by Roman Lee. For the first time, we present Fuchsia --- our open-source implementation of the Lee algorithm written in Python using mathematical routines of a free computer algebra system SageMath. We demonstrate Fuchsia by reducing differential equations for NLO contributions to splitting functions in QCD, which contain both loops and legs integrals.
Vol, E. D.
2014-01-01
We propose a new representation for several quantum master equations in so-called quasithermodynamic form. This representation (when it exists) let one to write down dynamical equations both for diagonal and non-diagonal elements of density matrix of the quantum system of interest in unified form by means of nonequilibrium potential ("entropy") that is a certain quadratic function depending on all variables describing the state. We prove that above representation exists for the general Pauli ...
Generalized quantum master equations in and out of equilibrium: When can one win?
Kelly, Aaron; Montoya-Castillo, Andrés; Wang, Lu; Markland, Thomas E.
2016-05-01
Generalized quantum master equations (GQMEs) are an important tool in modeling chemical and physical processes. For a large number of problems, it has been shown that exact and approximate quantum dynamics methods can be made dramatically more efficient, and in the latter case more accurate, by proceeding via the GQME formalism. However, there are many situations where utilizing the GQME approach with an approximate method has been observed to return the same dynamics as using that method directly. Here, for systems both in and out of equilibrium, we provide a more detailed understanding of the conditions under which using an approximate method can yield benefits when combined with the GQME formalism. In particular, we demonstrate the necessary manipulations, which are satisfied by exact quantum dynamics, that are required to recast the memory kernel in a form that can be analytically shown to yield the same result as a direct application of the dynamics regardless of the approximation used. By considering the connections between these forms of the kernel, we derive the conditions that approximate methods must satisfy if they are to offer different results when used in conjunction with the GQME formalism. These analytical results thus provide new insights as to when proceeding via the GQME approach can be used to improve the accuracy of simulations.
Rodrigues solution of the Dirac equation for fields obtained from the master function formalism
International Nuclear Information System (INIS)
We show that the radial Dirac equation with constant electrostatic potential and for a large class of the field potentials which are obtained from the master function formalism can be solved by the Rodrigues representation of the orthogonal polynomials. We also show that the Schrödinger-like differential equation obtained from the Dirac equation satisfies the supersymmetry and shape invariant conditions in non-relativistic quantum mechanics. The relativistic energy spectrum for a given potential function is calculated from its corresponding non-relativistic energy spectrum. (paper)
Driven stochastic processes with metastable states: Fokker-Planck versus master equations
International Nuclear Information System (INIS)
Processes occurring either in living systems on the cellular or molecular level or in artificial nanoscopic systems are most often strongly influenced by ambient thermal noise. On the other hand, intrinsic nonlinearities of the dynamical laws of these systems typically lead to multistability. The influences of the combined effects of noise and nonlinear intrinsic dynamics can often be described by a Fokker-Planck equation. As a consequence of the interaction of nonlinearities and noise the multistability is changed into metastability , i.e., on sufficiently long time scales transitions occur between the locally stable states of the noiseless dynamics. In a coarse grained description these transitions can be characterized by rates which enter a master equation, the states of which correspond to the metastable states of the system. We will demonstrate that such a description continues to hold even if the parameters of the system change with time either due to some external forcing of the system, for example by periodic or ramp-like forcing. As a prominent effect occurring in the presence of periodic forcing we mention stochastic resonance. Steadily increasing forces are often applied to molecules in order to determine their transition states and activation energies. In such forced systems the rates that determine the master equation change in time. In this talk we will discuss methods to calculate these time dependent rates. These methods also allow one to reconstruct the long time dynamics of the continuous Fokker-Planck process from the master equation. An analytic approximations for the rates will be discussed in the limit of slow driving. We will exemplify this method by a periodically driven bistable Brownian oscillator and demonstrate the use of the master equation in order to determine the point process of subsequent transition times. (author)
On the accuracy of the Padé-resummed master equation approach to dissipative quantum dynamics.
Chen, Hsing-Ta; Berkelbach, Timothy C; Reichman, David R
2016-04-21
Well-defined criteria are proposed for assessing the accuracy of quantum master equations whose memory functions are approximated by Padé resummation of the first two moments in the electronic coupling. These criteria partition the parameter space into distinct levels of expected accuracy, ranging from quantitatively accurate regimes to regions of parameter space where the approach is not expected to be applicable. Extensive comparison of Padé-resummed master equations with numerically exact results in the context of the spin-boson model demonstrates that the proposed criteria correctly demarcate the regions of parameter space where the Padé approximation is reliable. The applicability analysis we present is not confined to the specifics of the Hamiltonian under consideration and should provide guidelines for other classes of resummation techniques. PMID:27389208
Multi-qubit joint measurements in circuit QED: stochastic master equation analysis
Energy Technology Data Exchange (ETDEWEB)
Criger, Ben; Ciani, Alessandro [RWTH, JARA Institut fuer Quanteninformation, Aachen (Germany); DiVincenzo, David P. [RWTH, JARA Institut fuer Quanteninformation, Aachen (Germany); Forschungszentrum Juelich, Juelich (Germany)
2016-12-15
We derive a family of stochastic master equations describing homodyne measurement of multi-qubit diagonal observables in circuit quantum electrodynamics. In the regime where qubit decay can be neglected, our approach replaces the polaron-like transformation of previous work, which required a lengthy calculation for the physically interesting case of three qubits and two resonator modes. The technique introduced here makes this calculation straightforward and manifestly correct. Using this technique, we are able to show that registers larger than one qubit evolve under a non-Markovian master equation. We perform numerical simulations of the three-qubit, two-mode case from previous work, obtaining an average post-measurement state fidelity of ∝94%, limited by measurement-induced decoherence and dephasing. (orig.)
Xiang-Guo, Meng; Ji-Suo, Wang; Hong-Yi, Fan; Cheng-Wei, Xia
2016-04-01
We solve the fermionic master equation for a thermal bath to obtain its explicit Kraus operator solutions via the fermionic state approach. The normalization condition of the Kraus operators is proved. The matrix representation for these solutions is obtained, which is incongruous with the result in the book completed by Nielsen and Chuang [Quantum Computation and Quantum Information, Cambridge University Press, 2000]. As especial cases, we also present the Kraus operator solutions to master equations for describing the amplitude-decay model and the diffusion process at finite temperature. Project supported by the National Natural Science Foundation of China (Grant No. 11347026), the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2013AM012 and ZR2012AM004), and the Research Fund for the Doctoral Program and Scientific Research Project of Liaocheng University, Shandong Province, China.
Simulating open quantum systems by applying SU(4) to quantum master equations
Xu, Minghui; Tieri, D. A.; Holland, M J
2013-01-01
We show that open quantum systems of two-level atoms symmetrically coupled to a single-mode photon field can be efficiently simulated by applying a SU(4) group theory to quantum master equations. This is important since many foundational examples in quantum optics fall into this class. We demonstrate the method by finding exact solutions for many-atom open quantum systems such as lasing and steady state superradiance.
International Nuclear Information System (INIS)
In this thesis the influence of continuously repeting measurement by the environment on a quantum mechanical system is studied. The interaction with the environment which is in principle present for each physical object can have an important influence on the considered system as the ''watchdog'' effect has clearly demonstrated. These situations are however quite special. Of much more importance is the effect of the environment on macroscopical objects for which only regarding this interaction a derivation of master equations can be founded. (HSI)
Master Equation for the Motion of a Polarizable Particle in a Multimode Cavity
Nimmrichter, Stefan; Hammerer, Klemens; Asenbaum, Peter; Ritsch, Helmut; Arndt, Markus
2010-01-01
We derive a master equation for the motion of a polarizable particle weakly interacting with one or several strongly pumped cavity modes. We focus here on massive particles with complex internal structure such as large molecules and clusters, for which we assume a linear scalar polarizability mediating the particle-light interaction. The predicted friction and diffusion coefficients are in good agreement with former semiclassical calculations for atoms and small molecules in weakly pumped cav...
Tscherbul, Timur V.; Brumer, Paul
2015-03-01
We present an efficient theoretical method for calculating the time evolution of the density matrix of a multilevel quantum system weakly interacting with incoherent light. The method combines the Bloch-Redfield theory with a partial secular approximation for one-photon coherences, resulting in a master equation that explicitly exposes the reliance on transition rates and the angles between transition dipole moments in the energy basis. The partial secular Bloch-Redfield master equation allows an unambiguous distinction between the regimes of quantum coherent vs. incoherent energy transfer under incoherent light illumination. The fully incoherent regime is characterized by orthogonal transition dipole moments in the energy basis, leading to a dynamical evolution governed by a coherence-free Pauli-type master equation. The coherent regime requires non-orthogonal transition dipole moments in the energy basis and leads to the generation of noise-induced quantum coherences and population-to-coherence couplings. As a first application, we consider the dynamics of excited state coherences arising under incoherent light excitation from a single ground state and observe population-to-coherence transfer and the formation of non-equilibrium quasisteady states in the regime of small excited state splitting. Analytical expressions derived earlier for the V-type system [T. V. Tscherbul and P. Brumer, Phys. Rev. Lett. 113, 113601 (2014)] are found to provide a nearly quantitative description of multilevel excited-state populations and coherences in both the small- and large-molecule limits.
Energy Technology Data Exchange (ETDEWEB)
Tscherbul, Timur V., E-mail: ttscherb@chem.utoronto.ca; Brumer, Paul [Chemical Physics Theory Group, Department of Chemistry, and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario M5S 3H6 (Canada)
2015-03-14
We present an efficient theoretical method for calculating the time evolution of the density matrix of a multilevel quantum system weakly interacting with incoherent light. The method combines the Bloch-Redfield theory with a partial secular approximation for one-photon coherences, resulting in a master equation that explicitly exposes the reliance on transition rates and the angles between transition dipole moments in the energy basis. The partial secular Bloch-Redfield master equation allows an unambiguous distinction between the regimes of quantum coherent vs. incoherent energy transfer under incoherent light illumination. The fully incoherent regime is characterized by orthogonal transition dipole moments in the energy basis, leading to a dynamical evolution governed by a coherence-free Pauli-type master equation. The coherent regime requires non-orthogonal transition dipole moments in the energy basis and leads to the generation of noise-induced quantum coherences and population-to-coherence couplings. As a first application, we consider the dynamics of excited state coherences arising under incoherent light excitation from a single ground state and observe population-to-coherence transfer and the formation of non-equilibrium quasisteady states in the regime of small excited state splitting. Analytical expressions derived earlier for the V-type system [T. V. Tscherbul and P. Brumer, Phys. Rev. Lett. 113, 113601 (2014)] are found to provide a nearly quantitative description of multilevel excited-state populations and coherences in both the small- and large-molecule limits.
Solve the Master Equation by Python-An Introduction to the Python Computing Environment
Fan, Wei; Xu, Yan; Chen, Bing; Ye, Qianqian
2011-01-01
A brief introduction to the Python computing environment is given. By solving the master equation encountered in quantum transport, we give an example of how to solve the ODE problems in Python. The ODE solvers used are the ZVODE routine in Scipy and the bsimp solver in GSL. For the former, the equation can be in its complex-valued form, while for the latter, it has to be rewritten to a real-valued form. The focus is on the detailed workflow of the implementation process, rather than on the s...
Variance estimates for transport in stochastic media by means of the master equation
International Nuclear Information System (INIS)
The master equation has been used to examine properties of transport in stochastic media. It has been shown previously that not only may the Levermore-Pomraning (LP) model be derived from the master equation for a description of ensemble-averaged transport quantities, but also that equations describing higher-order statistical moments may be obtained. We examine in greater detail the equations governing the second moments of the distribution of the angular fluxes, from which variances may be computed. We introduce a simple closure for these equations, as well as several models for estimating the variances of derived transport quantities. We revisit previous benchmarks for transport in stochastic media in order to examine the error of these new variance models. We find, not surprisingly, that the errors in these variance estimates are at least as large as the corresponding estimates of the average, and sometimes much larger. We also identify patterns in these variance estimates that may help guide the construction of more accurate models. (authors)
Birth and death master equation for the evolution of complex networks
Alvarez-Martínez, R.; Cocho, G.; Rodríguez, R. F.; Martínez-Mekler, G.
2014-05-01
Master equations for the evolution of complex networks with positive (birth) and negative (death) transition probabilities per unit time are analyzed. Explicit equations for the time evolution of the total number of nodes and for the relative node frequencies are given. It is shown that, in the continuous limit, the master equation reduces to a Fokker-Planck equation (FPE). The basic dynamical function for its stationary solution is the ratio between its drift and diffusion coefficients. When this ratio is approximated by partial fractions (Padé's approximants), a hierarchy of stationary solutions of the FPE is obtained analytically, which are expressed as an exponential times the product of powers of monomials and binomials. It is also shown that if the difference between birth and death transition probabilities goes asymptotically to zero, the exponential factor in the solution is absent. Fits to real complex network probability distribution functions are shown. Comparison with rank-ordered data shows that, in general, the value of this exponential factor is close to unity, evidencing crossovers among power-law scale invariant regimes which might be associated to an underlying criticality and are related to a generalization of the beta distribution. The time dependent solution is also obtained analytically in terms of hyper-geometric functions. It is also shown that the FPE has similarity solutions. The limitations of the approach here presented are also discussed.
International Nuclear Information System (INIS)
In the framework of the dissipative dynamics of coupled qubits interacting with independent reservoirs, a comparison between non-Markovian master equation techniques and an exact solution is presented here. We study various regimes in order to find the limits of validity of the Nakajima-Zwanzig and the time-convolutionless master equations in the description of the entanglement dynamics. A comparison between the performances of the concurrence and the negativity as entanglement measures for the system under study is also presented.
Vaccaro, S R
2016-01-01
The Na+ current in nerve and muscle membranes may be described in terms of the activation variable m(t) and the inactivation variable h(t), which are dependent on the transitions of S4 sensors in each of the ion channel domains DI to DIV. The time-dependence of the Na+ current and the rate equations satisfied by m(t) and h(t) may be derived from the solution to a master equation which describes the coupling between two activation sensors regulating the Na+ channel conductance and a two stage inactivation process. The voltage dependence of the rate functions for inactivation and recovery from inactivation are consistent with the empirically determined Hodgkin-Huxley expressions, and exhibit saturation for both depolarized and hyperpolarized clamp potentials.
Iles-Smith, Jake; Lambert, Neill; Nazir, Ahsan
2015-01-01
We explore excitonic energy transfer dynamics in a molecular dimer system coupled to both structured and unstructured oscillator environments. By extending the reaction coordinate master equation technique developed in [J. Iles-Smith, N. Lambert, and A. Nazir, Phys. Rev. A 90, 032114 (2014)], we go beyond the commonly used Born-Markov approximations to incorporate system-environment correlations and the resultant non-Markovian dynamical effects. We obtain energy transfer dynamics for both underdamped and overdamped oscillator environments that are in perfect agreement with the numerical hierarchical equations of motion over a wide range of parameters. Furthermore, we show that the Zusman equations, which may be obtained in a semiclassical limit of the reaction coordinate model, are often incapable of describing the correct dynamical behaviour. This demonstrates the necessity of properly accounting for quantum correlations generated between the system and its environment when the Born-Markov approximations no ...
Hybrid Numerical Solution of the Chemical Master Equation
Henzinger, Thomas A.; Mateescu, Maria; Mikeev, Linar; Wolf, Verena
2010-01-01
We present a numerical approximation technique for the analysis of continuous-time Markov chains that describe networks of biochemical reactions and play an important role in the stochastic modeling of biological systems. Our approach is based on the construction of a stochastic hybrid model in which certain discrete random variables of the original Markov chain are approximated by continuous deterministic variables. We compute the solution of the stochastic hybrid model using a numeri...
Pdf - Transport equations for chemically reacting flows
Kollmann, W.
1989-01-01
The closure problem for the transport equations for pdf and the characteristic functions of turbulent, chemically reacting flows is addressed. The properties of the linear and closed equations for the characteristic functional for Eulerian and Lagrangian variables are established, and the closure problem for the finite-dimensional case is discussed for pdf and characteristic functions. It is shown that the closure for the scalar dissipation term in the pdf equation developed by Dopazo (1979) and Kollmann et al. (1982) results in a single integral, in contrast to the pdf, where double integration is required. Some recent results using pdf methods obtained for turbulent flows with combustion, including effects of chemical nonequilibrium, are discussed.
Master Equation for the Motion of a Polarizable Particle in a Multimode Cavity
Nimmrichter, Stefan; Asenbaum, Peter; Ritsch, Helmut; Arndt, Markus
2010-01-01
We derive a master equation for the motion of a polarizable particle weakly interacting with one or several strongly pumped cavity modes. We focus here on massive particles with complex internal structure such as large molecules and clusters, for which we assume a linear scalar polarizability mediating the particle-light interaction. The predicted friction and diffusion coefficients are in good agreement with former semiclassical calculations for atoms and small molecules in weakly pumped cavities, while the current rigorous quantum treatment and numerical assessment sheds a light on the feasibility of experiments that aim at optically manipulating beams of massive molecules with multimode cavities.
Stimulated Raman adiabatic passage in an open quantum system: Master equation approach
International Nuclear Information System (INIS)
A master equation approach to the study of environmental effects in the adiabatic population transfer in three-state systems is presented. A systematic comparison with the non-Hermitian Hamiltonian approach [Vitanov and Stenholm, Phys. Rev. A 56, 1463 (1997)] shows that, in the weak-coupling limit, the two treatments lead to essentially the same results. In contrast, in the strong-damping limit the predictions are quite different: In particular, the counterintuitive sequences in the STIRAP scheme turn out to be much more efficient than expected before. This point is explained in terms of quantum Zeno dynamics.
Intravaia, F; Messina, Andrea
2003-01-01
An original method to exactly solve the non-Markovian Master Equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak coupling limit is reported. By using a superoperatorial approach we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.
Lane, Thomas
2012-01-01
Motivated by claims about the nature of the observed timescales in protein systems said to fold downhill, we have studied the finite, linear master equation which is a model of the downhill process. By solving for the system eigenvalues, we prove the often stated claim that in situations where there is no free energy barrier, a transition between single and multi-exponential kinetics occurs at sufficient bias (towards the native state). Consequences for protein folding, especially the downhill folding scenario, are briefly discussed.
Splitting of the rate matrix as a definition of time reversal in master equation systems
International Nuclear Information System (INIS)
Motivated by recent progress in nonequilibrium fluctuation relations, we present a generalized time reversal for stochastic master equation systems with discrete states, which is defined as a splitting of the rate matrix into irreversible and reversible parts. An immediate advantage of this definition is that a variety of fluctuation relations can be attributed to different matrix splittings. Additionally, we find that the accustomed total entropy production formula and conditions of the detailed balance must be modified appropriately to account for the reversible rate part, which was previously ignored. (paper)
Fermionic Stochastic Schr\\"{o}dinger Equation and Master Equation: An Open System Model
Zhao, Xinyu; Shi, Wufu; Wu, Lian-Ao; Yu, Ting
2012-01-01
This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath, to the models in which the system operators commute with the fermion bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\\"{o}dinger equation (SSE), called fermionic quantum state diffusion (...
General features and master equations for structurization in complex dusty plasmas
International Nuclear Information System (INIS)
Dust structurization is considered to be typical for complex plasmas. Homogeneous dusty plasmas are shown to be universally unstable. The dusty plasma structurization instability is similar to the gravitational instability and can results in creation of different compact dust structures. A general approach for investigation of the nonlinear stage of structurization in dusty plasmas is proposed and master equations for the description of self-organized structures are formulated in the general form that can be used for any nonlinear model of dust screening. New effects due to the scattering of ions on the nonlinearly screened grains are calculated: nonlinear ion dust drag force and nonlinear ion diffusion. The physics of confinement of dust and plasma components in the equilibria of compact dust structures is presented and is supported by numerical calculations of master equations. The necessary conditions for the existence of equilibrium structures are found for an arbitrary nonlinearity in dust screening. Features of compact dust structures observed in recent experiments agree with the numerically calculated ones. Some proposals for future experiments in spherical chamber are given.
Dou, Wenjie; Subotnik, Joseph E
2016-01-14
A broadened classical master equation (BCME) is proposed for modeling nonadiabatic dynamics for molecules near metal surfaces over a wide range of parameter values and with arbitrary initial conditions. Compared with a standard classical master equation-which is valid in the limit of weak molecule-metal couplings-this BCME should be valid for both weak and strong molecule-metal couplings. (The BCME can be mapped to a Fokker-Planck equation that captures level broadening correctly.) Finally, our BCME can be solved with a simple surface hopping algorithm; numerical tests of equilibrium and dynamical observables look very promising. PMID:26772563
The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators, and master equations
Kaufmann, Ralph M.; Ward, Benjamin C.; Zúñiga, J. Javier
2015-10-01
Using five basic principles, we treat Gerstenhaber/Lie brackets, Batalin-Vilkovisky (BV) operators, and master equations appearing in mathematical and physical contexts in a unified way. The different contexts for this are given by the different types of (Feynman) graphs that underlie the particular situation. Two of the maxims we bring forth are (1) that extending to the non-connected graphs gives a commutative multiplication forming a part of the BV structure and (2) that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs and is responsible for the BV operator being a differential. Our treatment results in uniform, general theorems. These allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature. The more general point of view also allows us to disentangle the necessary from the circumstantial.
Rapidity window dependences of higher order cumulants and diffusion master equation
International Nuclear Information System (INIS)
We study the rapidity window dependences of higher order cumulants of conserved charges observed in relativistic heavy ion collisions. The time evolution and the rapidity window dependence of the non-Gaussian fluctuations are described by the diffusion master equation. Analytic formulas for the time evolution of cumulants in a rapidity window are obtained for arbitrary initial conditions. We discuss that the rapidity window dependences of the non-Gaussian cumulants have characteristic structures reflecting the non-equilibrium property of fluctuations, which can be observed in relativistic heavy ion collisions with the present detectors. It is argued that various information on the thermal and transport properties of the hot medium can be revealed experimentally by the study of the rapidity window dependences, especially by the combined use, of the higher order cumulants. Formulas of higher order cumulants for a probability distribution composed of sub-probabilities, which are useful for various studies of non-Gaussian cumulants, are also presented
Critical assessment of two-qubit post-Markovian master equations
Campbell, S; Mazzola, L; Gullo, N Lo; Vacchini, B; Busch, Th; Paternostro, M
2012-01-01
A post-Markovian master equation has been recently proposed as a tool to describe the evolution of a system coupled to a memory-keeping environment [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101 (R) (2005)]. For a single qubit affected by appropriately chosen environmental conditions, the corresponding dynamics is always legitimate and physical. Here we extend such situation to the case of two qubits, only one of which experiences the environmental effects. We show how, despite the innocence of such an extension, the introduction of the second qubit should be done cum grano salis to avoid consequences such as the breaking of the positivity of the associated dynamical map. This hints at the necessity of using care when adopting phenomenologically derived models for evolutions occurring outside the Markovian framework.
Paillotin, G; Swenberg, C E; Breton, J; Geacintov, N E
1979-03-01
A Pauli master equation is formulated and solved to describe the fluorescence quantum yield, phi, and the fluorescence temporal decay curves. F(t), obtained in picosecond laser excitation experiments of photosynthetic systems. It is assumed that the lowering of phi with increasing pulse intensity is due to bimolecular singlet exciton annihilation processes which compete with the monomolecular exciton decay processes; Poisson statistics are taken into account. Calculated curves of phi as a function of the number of photon hits per domain are compared with experimental data, and it is concluded that these domains contain at least two to four connected photosynthetic units (depending on the temperature), where each photosynthetic unit is assumed to contain approximately 300 pigment molecules. It is shown that under conditions of high excitation intensities, the fluorescence decays approximately according to the (time)1/2 law. PMID:262402
Entropy and Entanglement in Master Equation of Effective Hamiltonian for Jaynes-Cummings Model
International Nuclear Information System (INIS)
In this paper, we analytically solve the master equation for Jaynes-Cummings model in the dispersive regime including phase damping and the field is assumed to be initially in a superposition of coherent states. Using an established entanglement measure based on the negativity of the eigenvalues of the partially transposed density matrix we find a very strong sensitivity of the maximally generated entanglement to the amount of phase damping. Qualitatively this behavior is also reflected in alternative entanglement measures, but the quantitative agreement between different measures depends on the chosen noise model. The phase decoherence for this model results in monotonic increase in the total entropy while the atomic sub-entropy keeps its periodic behaviour without any effect. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
Energy Technology Data Exchange (ETDEWEB)
Iles-Smith, Jake, E-mail: Jakeilessmith@gmail.com [Controlled Quantum Dynamics Theory, Imperial College London, London SW7 2PG (United Kingdom); Photon Science Institute and School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL (United Kingdom); Department of Photonics Engineering, DTU Fotonik, Ørsteds Plads, 2800 Kongens Lyngby (Denmark); Dijkstra, Arend G. [Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg (Germany); Lambert, Neill [CEMS, RIKEN, Saitama 351-0198 (Japan); Nazir, Ahsan, E-mail: ahsan.nazir@manchester.ac.uk [Photon Science Institute and School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL (United Kingdom)
2016-01-28
We explore excitonic energy transfer dynamics in a molecular dimer system coupled to both structured and unstructured oscillator environments. By extending the reaction coordinate master equation technique developed by Iles-Smith et al. [Phys. Rev. A 90, 032114 (2014)], we go beyond the commonly used Born-Markov approximations to incorporate system-environment correlations and the resultant non-Markovian dynamical effects. We obtain energy transfer dynamics for both underdamped and overdamped oscillator environments that are in perfect agreement with the numerical hierarchical equations of motion over a wide range of parameters. Furthermore, we show that the Zusman equations, which may be obtained in a semiclassical limit of the reaction coordinate model, are often incapable of describing the correct dynamical behaviour. This demonstrates the necessity of properly accounting for quantum correlations generated between the system and its environment when the Born-Markov approximations no longer hold. Finally, we apply the reaction coordinate formalism to the case of a structured environment comprising of both underdamped (i.e., sharply peaked) and overdamped (broad) components simultaneously. We find that though an enhancement of the dimer energy transfer rate can be obtained when compared to an unstructured environment, its magnitude is rather sensitive to both the dimer-peak resonance conditions and the relative strengths of the underdamped and overdamped contributions.
International Nuclear Information System (INIS)
We explore excitonic energy transfer dynamics in a molecular dimer system coupled to both structured and unstructured oscillator environments. By extending the reaction coordinate master equation technique developed by Iles-Smith et al. [Phys. Rev. A 90, 032114 (2014)], we go beyond the commonly used Born-Markov approximations to incorporate system-environment correlations and the resultant non-Markovian dynamical effects. We obtain energy transfer dynamics for both underdamped and overdamped oscillator environments that are in perfect agreement with the numerical hierarchical equations of motion over a wide range of parameters. Furthermore, we show that the Zusman equations, which may be obtained in a semiclassical limit of the reaction coordinate model, are often incapable of describing the correct dynamical behaviour. This demonstrates the necessity of properly accounting for quantum correlations generated between the system and its environment when the Born-Markov approximations no longer hold. Finally, we apply the reaction coordinate formalism to the case of a structured environment comprising of both underdamped (i.e., sharply peaked) and overdamped (broad) components simultaneously. We find that though an enhancement of the dimer energy transfer rate can be obtained when compared to an unstructured environment, its magnitude is rather sensitive to both the dimer-peak resonance conditions and the relative strengths of the underdamped and overdamped contributions
Nonequilibrium quantum dynamics in the condensed phase via the generalized quantum master equation
Zhang, Ming-Liang; Ka, Being J.; Geva, Eitan
2006-07-01
The Nakajima-Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system's dynamics, and the inhomogeneous term accounts for initial system-bath correlations. In this paper, we propose a new approach for calculating the memory kernel and inhomogeneous term for arbitrary initial state and system-bath coupling. The memory kernel and inhomogeneous term are obtained by numerically solving a single inhomogeneous Volterra equation of the second kind for each. The new approach can accommodate a very wide range of projection operators, and requires projection-free two-time correlation functions as input. An application to the case of a two-state system with diagonal coupling to an arbitrary bath is described in detail. Finally, the utility and self-consistency of the formalism are demonstrated by an explicit calculation on a spin-boson model.
Master equation approach to charge injection and transport in organic insulators.
Freire, José A; Voss, Grasiela
2005-03-22
We develop a master equation model of a disordered organic insulator sandwiched between metallic electrodes by treating as rate processes both the injection and the internal transport. We show how the master equation model allows for the inclusion of crucial correlation effects in the charge transport, particularly of the Pauli exclusion principle and of space-charge effects, besides, being dependent on just the microscopic form of the transfer rate between the localized electronic states, it allows for the investigation of different microscopic scenarios in the organic, such as polaronic hopping, correlated energy levels, interaction with image charge, etc. The model allows for a separate analysis of the injection and the recombination currents. We find that the disorder, besides increasing the injection current, eliminates the possibility of observation of a Fowler-Nordheim injection current at zero temperature, and that it does not alter the Schottky barrier size of the zero-field thermionic injection current from the value based on the energy difference between the electrode Fermi level and the highest occupied molecular orbital/lowest unoccupied molecular orbital levels in the organic, but it makes the Arrhenius temperature dependence appear at larger temperatures. We investigate how the I(V) characteristics of a device is affected by the presence of correlations in the site energy distribution and by the form of the internal hopping rate, specifically the Miller-Abrahams rate and the Marcus or small-polaron rate. We show that the disorder does not modify significantly the ebeta square root E field dependence of the net current due to the Schottky barrier lowering caused by the attraction between the charge and its image in the electrode. PMID:15836407
Ghaderi, Nima
2016-03-01
Expressions for a K-adiabatic master equation for a bimolecular recombination rate constant krec are derived for a bimolecular reaction forming a complex with a single well or complexes with multiple well, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product. The K-active master equation is also considered. The exact analytic solutions, i.e., the K-adiabatic and K-active steady-state population distribution function of reactive complexes, g(EJK) and g(EJ), respectively, are derived for the K-adiabatic and K-active master equation cases using properties of inhomogeneous integral equations (Fredholm type). The solutions accommodate arbitrary intermolecular energy transfer models, e.g., the single exponential, double exponential, Gaussian, step-ladder, and near-singularity models. At the high pressure limit, the krec for both the K-adiabatic and K-active master equations reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high pressure limit expressions). Ozone and its formation from O + O2 are known to exhibit an adiabatic K. The ratio of the K-adiabatic to the K-active recombination rate constants for ozone formation at the high pressure limit is calculated to be ˜0.9 at 300 K. Results on the temperature and pressure dependence of the recombination rate constants and populations of O3 will be presented elsewhere.
A master equation formalism for macroscopic modeling of asynchronous irregular activity states.
El Boustani, Sami; Destexhe, Alain
2009-01-01
Many efforts have been devoted to modeling asynchronous irregular (AI) activity states, which resemble the complex activity states seen in the cerebral cortex of awake animals. Most of models have considered balanced networks of excitatory and inhibitory spiking neurons in which AI states are sustained through recurrent sparse connectivity, with or without external input. In this letter we propose a mesoscopic description of such AI states. Using master equation formalism, we derive a second-order mean-field set of ordinary differential equations describing the temporal evolution of randomly connected balanced networks. This formalism takes into account finite size effects and is applicable to any neuron model as long as its transfer function can be characterized. We compare the predictions of this approach with numerical simulations for different network configurations and parameter spaces. Considering the randomly connected network as a unit, this approach could be used to build large-scale networks of such connected units, with an aim to model activity states constrained by macroscopic measurements, such as voltage-sensitive dye imaging. PMID:19210171
On the structure of the master equation for a two-level system coupled to a thermal bath
International Nuclear Information System (INIS)
We derive a master equation from the exact stochastic Liouville–von-Neumann (SLN) equation (Stockburger and Grabert 2002 Phys. Rev. Lett. 88 170407). The latter depends on two correlated noises and describes exactly the dynamics of an oscillator (which can be either harmonic or present an anharmonicity) coupled to an environment at thermal equilibrium. The newly derived master equation is obtained by performing analytically the average over different noise trajectories. It is found to have a complex hierarchical structure that might be helpful to explain the convergence problems occurring when performing numerically the stochastic average of trajectories given by the SLN equation (Koch et al 2008 Phys. Rev. Lett. 100 230402, Koch 2010 PhD thesis Fakultät Mathematik und Naturwissenschaften der Technischen Universitat Dresden). (paper)
Indian Academy of Sciences (India)
Marko Žnidarič
2011-11-01
We discuss recent ﬁndings about properties of quantum nonequilibrium steady states. In particular we focus on transport properties. It is shown that the time-dependent density matrix renormalization method can be used successfully to ﬁnd a stationary solution of Lindblad master equation. Furthermore, for a speciﬁc model an exact solution is presented.
A semiclassical generalized quantum master equation for an arbitrary system-bath coupling
Shi, Qiang; Geva, Eitan
2004-06-01
The Nakajima-Zwanzig generalized quantum master equation (GQME) provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a, possibly anharmonic, quantum bath. In this equation, a memory kernel superoperator accounts for the influence of the bath on the dynamics of the system. In a previous paper [Q. Shi and E. Geva, J. Chem. Phys. 119, 12045 (2003)] we proposed a new approach to calculating the memory kernel, in the case of arbitrary system-bath coupling. Within this approach, the memory kernel is obtained by solving a set of two integral equations, which requires a new type of two-time system-dependent bath correlation functions as input. In the present paper, we consider the application of the linearized semiclassical (LSC) approximation for calculating those correlation functions, and subsequently the memory kernel. The new approach is tested on a benchmark spin-boson model. Application of the LSC approximation for calculating the relatively short-lived memory kernel, followed by a numerically exact solution of the GQME, is found to provide an accurate description of the relaxation dynamics. The success of the proposed LSC-GQME methodology is contrasted with the failure of both the direct application of the LSC approximation and the weak coupling treatment to provide an accurate description of the dynamics, for the same model, except at very short times. The feasibility of the new methodology to anharmonic systems is also demonstrated in the case of a two level system coupled to a chain of Lennard-Jones atoms.
Reduced master equation analysis of multiple-tunnel junction single-electron memory device
Jalil, M. B. A.; Wagner, M.
2000-07-01
We employ a master equation (ME) approach in the charge transport analysis across a uniform multiple-tunnel junction (MTJ) memory trap, using a much-reduced state list derived from circuit symmetry, and previous assumptions by Jensen and Martinis. This enables all significant single tunneling and higher-order cotunneling sequences to be accounted for, while avoiding the computational cost of the full ME method. The reduced ME method is conceptually simpler and yields greater accuracy, compared with previous approximations based on tunneling probabilities. For an MTJ trap with zero stray capacitance C0, the results obtained are found to agree very closely with the full ME results up to a temperature of T≈3T0/10, where T0=e2/kBC, whereas previous methods break down at T≈T0/10. Furthermore, unlike the earlier methods, the reduced ME approach can be applied to the realistic but less symmetric case of a trap with finite C0, and remains valid up to the trap's maximum operating temperature of T≈T0/100. Finally, our reduced ME results are in close agreement with available experimental data at T
Wu, Fuke; Tian, Tianhai; Rawlings, James B; Yin, George
2016-05-01
The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766-1793 (1996); ibid. 56, 1794-1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence. PMID:27155630
Neutron fluctuations in accelerator driven and power reactors via backward master equations
International Nuclear Information System (INIS)
The transport of neutrons in a reactor is a random process, and thus the number of neutrons in a reactor is a random variable. Fluctuations in the number of neutrons in a reactor can be divided into two categories, namely zero noise and power reactor noise. As the name indicates, they dominate (i.e. are observable) at different power levels. The reasons for their occurrences and utilization are also different. In addition, they are described via different mathematical tools, namely master equations and the Langevin equation, respectively. Zero noise carries information about some nuclear properties such as reactor reactivity. Hence methods such as Feynman- and Rossi-alpha methods have been established to determine the subcritical reactivity of a subcritical system. Such methods received a renewed interest recently with the advent of the so-called accelerator driven systems (ADS). Such systems, intended to be used either for energy production or transuranium transmutation, will use a subcritical core with a strong spallation source. A spallation source has statistical properties that are different from those of the traditionally used radioactive sources which were also assumed in the derivation of the Feynman- and Rossi-alpha formulae. Therefore it is necessary to re-derive the Feynman- and Rossi-alpha formulae. Such formulae for ADS have been derived recently but in simpler neutronic models. One subject of this thesis is the extension of such formulae to a more general case in which six groups of delayed neutron precursors are taken into account, and the full joint statistics of the prompt and all delayed groups is included. The involved complexity problems are solved with a combination of effective analytical techniques and symbolic algebra codes. Power reactor noise carries information about parametric perturbation of the system. Langevin technique has been used to extract such information. In such a treatment, zero noise has been neglected. This is a pragmatic
International Nuclear Information System (INIS)
The generalized quantum master equation provides a powerful tool to describe the dynamics in quantum impurity models driven away from equilibrium. Two complementary approaches, one based on Nakajima–Zwanzig–Mori time-convolution (TC) and the other on the Tokuyama–Mori time-convolutionless (TCL) formulations provide a starting point to describe the time-evolution of the reduced density matrix. A key in both approaches is to obtain the so called “memory kernel” or “generator,” going beyond second or fourth order perturbation techniques. While numerically converged techniques are available for the TC memory kernel, the canonical approach to obtain the TCL generator is based on inverting a super-operator in the full Hilbert space, which is difficult to perform and thus, nearly all applications of the TCL approach rely on a perturbative scheme of some sort. Here, the TCL generator is expressed using a reduced system propagator which can be obtained from system observables alone and requires the calculation of super-operators and their inverse in the reduced Hilbert space rather than the full one. This makes the formulation amenable to quantum impurity solvers or to diagrammatic techniques, such as the nonequilibrium Green’s function. We implement the TCL approach for the resonant level model driven away from equilibrium and compare the time scales for the decay of the generator with that of the memory kernel in the TC approach. Furthermore, the effects of temperature, source-drain bias, and gate potential on the TCL/TC generators are discussed
Application of quantum master equation for long-term prognosis of asset-prices
Khrennikova, Polina
2016-05-01
This study combines the disciplines of behavioral finance and an extension of econophysics, namely the concepts and mathematical structure of quantum physics. We apply the formalism of quantum theory to model the dynamics of some correlated financial assets, where the proposed model can be potentially applied for developing a long-term prognosis of asset price formation. At the informational level, the asset price states interact with each other by the means of a "financial bath". The latter is composed of agents' expectations about the future developments of asset prices on the finance market, as well as financially important information from mass-media, society, and politicians. One of the essential behavioral factors leading to the quantum-like dynamics of asset prices is the irrationality of agents' expectations operating on the finance market. These expectations lead to a deeper type of uncertainty concerning the future price dynamics of the assets, than given by a classical probability theory, e.g., in the framework of the classical financial mathematics, which is based on the theory of stochastic processes. The quantum dimension of the uncertainty in price dynamics is expressed in the form of the price-states superposition and entanglement between the prices of the different financial assets. In our model, the resolution of this deep quantum uncertainty is mathematically captured with the aid of the quantum master equation (its quantum Markov approximation). We illustrate our model of preparation of a future asset price prognosis by a numerical simulation, involving two correlated assets. Their returns interact more intensively, than understood by a classical statistical correlation. The model predictions can be extended to more complex models to obtain price configuration for multiple assets and portfolios.
Energy Technology Data Exchange (ETDEWEB)
Kidon, Lyran [School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978 (Israel); The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978 (Israel); Wilner, Eli Y. [School of Physics and Astronomy, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978 (Israel); Rabani, Eran [The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978 (Israel); Department of Chemistry, University of California and Lawrence Berkeley National Laboratory, Berkeley California 94720-1460 (United States)
2015-12-21
The generalized quantum master equation provides a powerful tool to describe the dynamics in quantum impurity models driven away from equilibrium. Two complementary approaches, one based on Nakajima–Zwanzig–Mori time-convolution (TC) and the other on the Tokuyama–Mori time-convolutionless (TCL) formulations provide a starting point to describe the time-evolution of the reduced density matrix. A key in both approaches is to obtain the so called “memory kernel” or “generator,” going beyond second or fourth order perturbation techniques. While numerically converged techniques are available for the TC memory kernel, the canonical approach to obtain the TCL generator is based on inverting a super-operator in the full Hilbert space, which is difficult to perform and thus, nearly all applications of the TCL approach rely on a perturbative scheme of some sort. Here, the TCL generator is expressed using a reduced system propagator which can be obtained from system observables alone and requires the calculation of super-operators and their inverse in the reduced Hilbert space rather than the full one. This makes the formulation amenable to quantum impurity solvers or to diagrammatic techniques, such as the nonequilibrium Green’s function. We implement the TCL approach for the resonant level model driven away from equilibrium and compare the time scales for the decay of the generator with that of the memory kernel in the TC approach. Furthermore, the effects of temperature, source-drain bias, and gate potential on the TCL/TC generators are discussed.
Jang, Seogjoo; Hoyer, Stephan; Fleming, Graham; Whaley, K Birgitta
2014-10-31
A generalized master equation (GME) governing quantum evolution of modular exciton density (MED) is derived for large scale light harvesting systems composed of weakly interacting modules of multiple chromophores. The GME-MED offers a practical framework to incorporate real time coherent quantum dynamics calculations of small length scales into dynamics over large length scales, and also provides a non-Markovian generalization and rigorous derivation of the Pauli master equation employing multichromophoric Förster resonance energy transfer rates. A test of the GME-MED for four sites of the Fenna-Matthews-Olson complex demonstrates how coherent dynamics of excitonic populations over coupled chromophores can be accurately described by transitions between subgroups (modules) of delocalized excitons. Application of the GME-MED to the exciton dynamics between a pair of light harvesting complexes in purple bacteria demonstrates its promise as a computationally efficient tool to investigate large scale exciton dynamics in complex environments. PMID:25396397
Energy Technology Data Exchange (ETDEWEB)
Boyd, Iain D., E-mail: iainboyd@umich.edu [Department of Aerospace Engineering, University of Michigan, Ann Arbor, Michigan 48109 (United States); Josyula, Eswar [U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433 (United States)
2016-01-15
The direct simulation Monte Carlo (DSMC) method is the primary numerical technique for analysis of rarefied gas flows. While recent progress in computational chemistry is beginning to provide vibrationally resolved transition and reaction cross sections that can be employed in DSMC calculations, the particle nature of the standard DSMC method makes it difficult to use this information in a statistically significant way. The current study introduces a new technique that makes it possible to resolve all of the vibrational energy levels by using a master equation approach along with temperature-dependent transition rates. The new method is compared to the standard DSMC technique for several heat bath and shock wave conditions and demonstrates the ability to resolve the full vibrational manifold at the expected overall rates of relaxation. The ability of the new master equation approach to the DSMC method for resolving, in particular, the high-energy states addresses a well-known, longstanding deficiency of the standard DSMC method.
Smith, Charles E.
2016-01-01
There is increasing interest concerning the details about how quantum systems interact with their surroundings. A number of methodologies have been used to describe these interactions, including Master Equations (ME) based on a system-plus-reservoir (S + R) approach, and more recently, Steepest Entropy Ascent Quantum Thermodynamics (SEAQT) which asserts that entropy is a fundamental physical property and that isolated quantum systems that are not at stable equilibrium may spontaneously relax ...
International Nuclear Information System (INIS)
In this article, we show how Ehrenfest mean field theory can be made both a more accurate and efficient method to treat nonadiabatic quantum dynamics by combining it with the generalized quantum master equation framework. The resulting mean field generalized quantum master equation (MF-GQME) approach is a non-perturbative and non-Markovian theory to treat open quantum systems without any restrictions on the form of the Hamiltonian that it can be applied to. By studying relaxation dynamics in a wide range of dynamical regimes, typical of charge and energy transfer, we show that MF-GQME provides a much higher accuracy than a direct application of mean field theory. In addition, these increases in accuracy are accompanied by computational speed-ups of between one and two orders of magnitude that become larger as the system becomes more nonadiabatic. This combination of quantum-classical theory and master equation techniques thus makes it possible to obtain the accuracy of much more computationally expensive approaches at a cost lower than even mean field dynamics, providing the ability to treat the quantum dynamics of atomistic condensed phase systems for long times
Lee, Keumsook; Park, Jong Soo; Jung, Woo-Sung; Choi, M Y; 10.1088/1751-8113/44/11/115007
2011-01-01
The master equation approach is proposed to describe the evolution of passengers in a subway system. With the transition rate constructed from simple geographical consideration, the evolution equation for the distribution of subway passengers is found to bear skew distributions including log-normal, Weibull, and power-law distributions. This approach is then applied to the Metropolitan Seoul Subway system: Analysis of the trip data of all passengers in a day reveals that the data in most cases fit well to the log-normal distributions. Implications of the results are also discussed.
International Nuclear Information System (INIS)
The master equation approach is proposed to describe the evolution of passengers in a subway system. With the transition rate constructed from simple geographical consideration, the evolution equation for the distribution of subway passengers is found to bear skew distributions including log-normal, Weibull, and power-law distributions. This approach is then applied to the Metropolitan Seoul Subway system: analysis of the trip data of all passengers in a day reveals that the data in most cases fit well to the log-normal distributions. Implications of the results are also discussed.
Institute of Scientific and Technical Information of China (English)
MU Wei-Hua; OU-YANG Zhong-Can; Li Xiao-Qing
2011-01-01
The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful. However, what are the sufficient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reflect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems.By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker-Planck equation of the chemical reaction systems. The obtained Fokker-Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies.
International Nuclear Information System (INIS)
The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful. However, what are the sufficient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reflect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems. By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker-Planck equation of the chemical reaction systems. The obtained Fokker-Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies. (general)
Huestis, D. L.
2008-12-01
gases will have more than one relaxation time constant. Thus we write the chemical kinetics master equation as (1) (d/dt) Nm = Σnpq [ - kmn→pq Nm Nn + kpq→mn Np Nq ] which has the well-known time-dependent solution given by (2) Nm(t) = Σn Cmn exp(-λn t) where the λn values are the decay frequencies and the Cmn coefficients depend on how the gas was initially excited. What we have contributed is the frequency-dependent sound absorption solution to Equation (1): (3) cvint(ω) = Σnk Wn / (1 + i ω/λn) where cvint(ω) is the complex heat capacity (per molecule), ω is the circular sound frequency, 2πf, the λn are the calculated decay frequencies [as in Equation (2)] and k Wn is the real effective heat capacity for decay mode n. As pointed out by Landau and Teller [Phys. Z. Sowjet. 10, 34-43 (1936)], for a simple case when the decay modes correspond to vibrational modes, Wn is the ordinary heat capacity of the vibrational mode. In the more complicated case involving one or more reversible energy-transfer steps, e.g., water and oxygen, the vibrational modes and the decay modes do not correspond to each other, and we need to use the rate coefficients in Equation (1) to calculate both λn and Wn.
International Nuclear Information System (INIS)
We study a certain family of Schroedinger operators whose eigenfunctions φ(x, λ) satisfy a differential equation in the spectral parameter λ of the form B(λ, dλ)φ=Θ(x)φ. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class of bispectral potentials. This extends and complements a result of Duistermaat and Gruenbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy. (orig.)
Parameter Estimates in Differential Equation Models for Chemical Kinetics
Winkel, Brian
2011-01-01
We discuss the need for devoting time in differential equations courses to modelling and the completion of the modelling process with efforts to estimate the parameters in the models using data. We estimate the parameters present in several differential equation models of chemical reactions of order n, where n = 0, 1, 2, and apply more general…
Directory of Open Access Journals (Sweden)
Charles E. Smith
2016-05-01
Full Text Available There is increasing interest concerning the details about how quantum systems interact with their surroundings. A number of methodologies have been used to describe these interactions, including Master Equations (ME based on a system-plus-reservoir (S + R approach, and more recently, Steepest Entropy Ascent Quantum Thermodynamics (SEAQT which asserts that entropy is a fundamental physical property and that isolated quantum systems that are not at stable equilibrium may spontaneously relax without environmental influences. In this paper, the ME, SEAQT approaches, and a simple linear difference equation (DE model are compared with each other and experimental data in order to study the behavior of a single trapped ion as it interacts with one or more external heat reservoirs. The comparisons of the models present opportunities for additional study to verify the validity and limitations of these approaches.
Jump-diffusion unravelling of a non-Markovian generalized Lindblad master equation
International Nuclear Information System (INIS)
The ''correlated-projection technique'' has been successfully applied to derive a large class of highly non-Markovian dynamics, the so called non-Markovian generalized Lindblad-type equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unraveling can be interpreted in terms of measurements continuous in time but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.
Brasil, Carlos Alexandre
2011-01-01
The most general form for the generator of quantum dynamical semigroups is the one proposed by Lindblad, which can be used in several approaches involving quantum mechanics for open systems, from analysis of noise and dissipation to fundamental aspects of the quantum theory of measurement. When dealing with a system interacting with its environment, the trace of the environmental degrees of freedom using the traditional approach of exponentiation of the Hamiltonian terms, originates prohibitive and difficult calculations. This paper presents an alternative analytic method to derive, through superoperator algebra and Nakajima-Zwanzig thermodynamic projectors, a compact and fairly simple master equation describing the reduced system dynamics. As a simple example of the present approach, we analyze a two-level system in contact with an environment, which allows us to observe the decoherence intensification by the interaction.
Nogawa, Tomoaki
2012-10-18
We examine the effectiveness of assuming an equal probability for states far from equilibrium. For this aim, we propose a method to construct a master equation for extensive variables describing nonstationary nonequilibrium dynamics. The key point of the method is the assumption that transient states are equivalent to the equilibrium state that has the same extensive variables, i.e., an equal probability holds for microscopic states in nonequilibrium. We demonstrate an application of this method to the critical relaxation of the two-dimensional Potts model by Monte Carlo simulations. While the one-variable description, which is adequate for equilibrium, yields relaxation dynamics that are very fast, the redundant two-variable description well reproduces the true dynamics quantitatively. These results suggest that some class of the nonequilibrium state can be described with a small extension of degrees of freedom, which may lead to an alternative way to understand nonequilibrium phenomena. © 2012 American Physical Society.
Global asymptotic stability for a class of nonlinear chemical equations
Anderson, David F.
2007-01-01
We consider a class of nonlinear differential equations that arises in the study of chemical reaction systems that are known to be locally asymptotically stable and prove that they are in fact globally asymptotically stable. More specifically, we will consider chemical reaction systems that are weakly reversible, have a deficiency of zero, and are equipped with mass action kinetics. We show that if for each $c \\in \\R_{> 0}^m$ the intersection of the stoichiometric compatibility class $c + S$ ...
Chemical Equilibrium and Polynomial Equations: Beware of Roots.
Smith, William R.; Missen, Ronald W.
1989-01-01
Describes two easily applied mathematical theorems, Budan's rule and Rolle's theorem, that in addition to Descartes's rule of signs and intermediate-value theorem, are useful in chemical equilibrium. Provides examples that illustrate the use of all four theorems. Discusses limitations of the polynomial equation representation of chemical…
International Nuclear Information System (INIS)
It is known that many irreversible processes have a high degree of symmetry. Thus, spontaneous emission, thermal collisions, and the collisions of atoms or molecules of a gas with container walls lead to completely isotropic relaxation. The assumption of isotropy is also a good approximation in the presence of an external magnetic field if the energy of the Zeeman level splitting of the system is appreciably less than kT. Relaxation in the presence of a weak fluctuating perturbation, which is typical of liquids and gases, is also isotropic. In the presence of a strong magnetic field the environment is, as a rule, axisymmetric. In this connection, semigroups with different degrees of symmetry are widely used in the phenomenological description of the irreversible evolution of open quantum systems. In this paper, quantum Markov master equations of spin systems are classified (up to conjugation) with respect to the continuous symmetry groups of the environment. The Bloch equations are derived from the general theory of completely positive quantum dynamical semigroups
Singh, Navinder
2011-01-01
A direct numerical algorithm for solving the time-nonlocal non-Markovian master equation in the second Born approximation is introduced and the range of utility of this approximation, and of the Markov approximation, is analyzed for the traditional dimer system that models excitation energy transfer in photosynthesis. Specifically, the coupled integro-differential equations for the reduced density matrix are solved by an efficient auxiliary function method in both the energy and site representations. In addition to giving exact results to this order, the approach allows us to computationally assess the range of the reorganization energy and decay rates of the phonon auto-correlation function for which the Markovian Redfield theory and the second order approximation is valid. For example, the use of Redfield theory for $\\lambda> 10 \\textrm{cm}^{-1}$ in systems like Fenna-Mathews-Olson (FMO) type systems is shown to be in error. In addition, analytic inequalities are obtained for the regime of validity of the M...
Karimi, F.; Davoody, A. H.; Knezevic, I.
2016-05-01
We introduce a method for calculating the dielectric function of nanostructures with an arbitrary band dispersion and Bloch wave functions. The linear response of a dissipative electronic system to an external electromagnetic field is calculated by a self-consistent-field approach within a Markovian master-equation formalism (SCF-MMEF) coupled with full-wave electromagnetic equations. The SCF-MMEF accurately accounts for several concurrent scattering mechanisms. The method captures interband electron-hole-pair generation, as well as the interband and intraband electron scattering with phonons and impurities. We employ the SCF-MMEF to calculate the dielectric function, complex conductivity, and loss function for supported graphene. From the loss-function maximum, we obtain plasmon dispersion and propagation length for different substrate types [nonpolar diamondlike carbon (DLC) and polar SiO2 and hBN], impurity densities, carrier densities, and temperatures. Plasmons on the two polar substrates are suppressed below the highest surface phonon energy, while the spectrum is broad on the nonpolar DLC. Plasmon propagation lengths are comparable on polar and nonpolar substrates and are on the order of tens of nanometers, considerably shorter than previously reported. They improve with fewer impurities, at lower temperatures, and at higher carrier densities.
Caglar, Mehmet Umut; Pal, Ranadip
2011-03-01
Central dogma of molecular biology states that ``information cannot be transferred back from protein to either protein or nucleic acid''. However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of cell level data and probabilistic - nonlinear nature of interactions. Several models widely used to analyze and simulate these types of nonlinear interactions. Stochastic Master Equation (SME) models give probabilistic nature of the interactions in a detailed manner, with a high calculation cost. On the other hand Probabilistic Boolean Network (PBN) models give a coarse scale picture of the stochastic processes, with a less calculation cost. Differential Equation (DE) models give the time evolution of mean values of processes in a highly cost effective way. The understanding of the relations between the predictions of these models is important to understand the reliability of the simulations of genetic regulatory networks. In this work the success of the mapping between SME, PBN and DE models is analyzed and the accuracy and affectivity of the control policies generated by using PBN and DE models is compared.
Development of a chlorine chemistry module for the Master Chemical Mechanism
Xue, L. K.; Saunders, S. M.; Wang, T.; Gao, R.; Wang, X. F.; Zhang, Q. Z.; Wang, W. X.
2015-10-01
The chlorine atom (Cl·) has a high potential to perturb atmospheric photochemistry by oxidizing volatile organic compounds (VOCs), but the exact role it plays in the polluted troposphere remains unclear. The Master Chemical Mechanism (MCM) is a near-explicit mechanism that has been widely applied in the atmospheric chemistry research. While it addresses comprehensively the chemistry initiated by the OH, O3 and NO3 radicals, its representation of the Cl· chemistry is incomplete as it only considers the reactions for alkanes. In this paper, we develop a more comprehensive Cl· chemistry module that can be directly incorporated within the MCM framework. A suite of 205 chemical reactions describes the Cl·-initiated degradation of alkenes, aromatics, alkynes, aldehydes, ketones, alcohols, and some organic acids and nitrates, along with the inorganic chemistry involving Cl· and its precursors. To demonstrate the potential influence of the new chemistry module, it was incorporated into a MCM box model to evaluate the impacts of nitryl chloride (ClNO2), a product of nocturnal halogen activation by nitrogen oxides (NOX), on the following day's atmospheric photochemistry. With constraints of recent observations collected at a coastal site in Hong Kong, southern China, the modeling analyses suggest that the Cl· produced from ClNO2 photolysis may substantially enhance the atmospheric oxidative capacity, VOC oxidation and O3 formation, particularly in the early morning period. The results demonstrate the critical need for photochemical models to include more detailed chlorine chemistry in order to better understand the atmospheric photochemistry in polluted environments subject to intense emissions of NOX, VOCs and chlorine-containing constituents.
Boltzmann Equation Solver Adapted to Emergent Chemical Non-equilibrium
Birrell, Jeremiah
2014-01-01
We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature $T(t)$ and phase space occupation factor $\\Upsilon(t)$. In this first paper we address (effectively) massless fermions and derive dynamical equations for $T(t)$ and $\\Upsilon(t)$ such that the zeroth order term of the basis alone captures the number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component ($e^\\pm$-annihilation).
Luo, JunYan; Jin, Jinshuang; Wang, Shi-Kuan; Hu, Jing; Huang, Yixiao; He, Xiao-Ling
2016-03-01
We present a generic unraveling scheme for a detailed-balance-preserved quantum master equation applicable for stochastic point processes in mesoscopic transport. It enables us to investigate continuous measurement of a qubit on the level of single quantum trajectories, where essential correlations between the inherent dynamics of the qubit and detector current fluctuations are revealed. Based on this unraveling scheme, feedback control of the charge qubit is implemented to achieve a desired pure state in the presence of the detailed-balance condition. With sufficient feedback strength, coherent oscillations of the measured qubit can be maintained for arbitrary qubit-detector coupling. Competition between the loss and restoration of coherence entailed, respectively, by measurement back action and feedback control is reflected in the noise power spectrum of the detector's output. It is demonstrated unambiguously that the signal-to-noise ratio is significantly enhanced with increasing feedback strength and could even exceed the well-known Korotkov-Averin bound in quantum measurement. The proposed unraveling and feedback scheme offers a transparent and straightforward approach to effectively sustaining ideal coherent oscillations of a charge qubit in the field of quantum computation.
International Nuclear Information System (INIS)
We construct a number(n)-resolved master equation (ME) approach under self-consistent Born approximation (SCBA) for noise spectrum calculation. The formulation is essentially non-Markovian and incorporates properly the interlay of the multi-tunneling processes and many-body correlations. We apply this approach to the challenging nonequilibrium Kondo system and predict a profound nonequilibrium Kondo signature in the shot noise spectrum. The proposed n-SCBA-ME scheme goes completely beyond the scope of the Born-Markovian master equation approach, in the sense of being applicable to the shot noise of transport under small bias voltage, in non-Markovian regime, and with strong Coulomb correlations as favorably demonstrated in the nonequilibrium Kondo system. (condensed matter: electronic structure, electrical, magnetic, and optical properties)
Kritsotakis, M.; Kominis, I. K.
2014-01-01
Radical-ion-pair reactions, central in photosynthesis and the avian magnetic compass mechanism, have recently shown to be a paradigm system for applying quantum information science in a biochemical setting. The fundamental quantum master equation describing radical-ion-pair reactions is still under debate. We here use quantum retrodiction to produce a rigorous refinement of the theory put forward in Phys. Rev. E {\\bf 83}, 056118 (2011). We also provide a rigorous analysis of the measure of si...
Reineker, P.; Kühne, R.
1980-03-01
Starting from the stochastic Liouville equation of the full Haken-Strobl model, describing the coupled coherent and incoherent motion of excitons in molecular crystals, the Nakajima-Zwanzig generalized master equation (GME) for the probability of finding an exciton at a specific lattice site is derived by an exact straightforward evaluation of its memory function. Various recently derived generalized master equations describing the excition motion are obtained as limiting cases and the Born approximation is discussed. It is shown that, even in the case of nearest-neighbor interaction in the stochastic Liouville equation, in the GME generalized time-dependent transition rates evolve between non-nearest neighbors and that their time behavior shows damped oscillations. Applying the Born approximation to the GME, the range of the generalized transition rates reduces to that of the interaction in the stochastic Liouville equation. Furthermore in this approximation the transition rates show a purely exponential decay with increasing time. Taking into account the interaction with an arbitrary number of neighbors, the mean square displacement of the exciton motion is calculated exactly from the GME. Finally the GME is solved exactly in the general case and several limiting expressions are discussed.
Brett, Tobias; Galla, Tobias
2013-01-01
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation a further reduction leads to the linear-noise approximation. We apply the formali...
Shi, Qiang; Geva, Eitan
2003-12-01
The Nakajima-Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system's dynamics. The standard approach is based on using a perturbative treatment of the system-bath coupling for calculating this kernel, and is therefore restricted to systems weakly coupled to the bath. In this paper, we propose a new approach for calculating the memory kernel for an arbitrary system-bath coupling. The memory kernel is obtained by solving a set of two coupled integral equations that relate it to a new type of two-time system-dependent bath correlation functions. The feasibility of the method is demonstrated in the case of an asymetrical two-level system linearly coupled to a harmonic bath.
A New Pseudoinverse Matrix Method For Balancing Chemical Equations And Their Stability
Energy Technology Data Exchange (ETDEWEB)
Risteski, Ice B. [2 Milepost Place, Ontario (Canada)
2008-06-15
In this work is given a new pseudoniverse matrix method for balancing chemical equations. Here offered method is founded on virtue of the solution of a Diophantine matrix equation by using of a Moore-Penrose pseudoinverse matrix. The method has been tested on several typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method, which works successfully without any limitations, also has the capability to determine the feasibility of a new chemical reaction, and if it is feasible, then it will balance the equation. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are introduced necessary and sufficient criteria for stability of chemical equations over stability of their extended matrices.
Manson, Ross; Roy-Choudhury, Kaushik; Hughes, Stephen
2016-04-01
We introduce an intuitive and semianalytical polaron master equation approach to model pulse-driven population inversion and emitted single photons from a quantum dot exciton. The master equation theory allows one to identify important phonon-induced scattering rates analytically and fully includes the role of the time-dependent pump field. As an application of the theory, we first study a quantum dot driven by a time-varying laser pulse on and off resonance, showing the population inversion caused by acoustic phonon emission in direct agreement with recent experiments of Quilter et al. [Phys. Rev. Lett. 114, 137401 (2015), 10.1103/PhysRevLett.114.137401]. We then model quantum dots in weakly coupled cavities and show the difference in population response between exciton-driven and cavity-driven systems. Finally, we assess the nonresonant phonon-assisted loading scheme with a quantum dot resonantly coupled to a cavity as a deterministic single-photon source. We also compare and contrast the important single photon figures of merit with direct Rabi oscillation of the population using a resonant π pulse, and show that the resonant scheme is much more efficient.
Håkansson, Pär; Westlund, Per-Olof
2005-01-01
This paper discusses the process of energy migration transfer within reorientating chromophores using the stochastic master equation (SME) and the stochastic Liouville equation (SLE) of motion. We have found that the SME over-estimates the rate of the energy migration compared to the SLE solution for a case of weakly interacting chromophores. This discrepancy between SME and SLE is caused by a memory effect occurring when fluctuations in the dipole-dipole Hamiltonian ( H( t)) are on the same timescale as the intrinsic fast transverse relaxation rate characterized by (1/ T2). Thus the timescale critical for energy-transfer experiments is T2≈10 -13 s. An extended SME is constructed, accounting for the memory effect of the dipole-dipole Hamiltonian dynamics. The influence of memory on the interpretation of experiments is discussed.
Directory of Open Access Journals (Sweden)
Andrei Khrennikov
2016-07-01
Full Text Available We present a new conceptual approach for modeling of fluid flows in random porous media based on explicit exploration of the treelike geometry of complex capillary networks. Such patterns can be represented mathematically as ultrametric spaces and the dynamics of fluids by ultrametric diffusion. The images of p-adic fields, extracted from the real multiscale rock samples and from some reference images, are depicted. In this model the porous background is treated as the environment contributing to the coefficients of evolutionary equations. For the simplest trees, these equations are essentially less complicated than those with fractional differential operators which are commonly applied in geological studies looking for some fractional analogs to conventional Euclidean space but with anomalous scaling and diffusion properties. It is possible to solve the former equation analytically and, in particular, to find stationary solutions. The main aim of this paper is to attract the attention of researchers working on modeling of geological processes to the novel utrametric approach and to show some examples from the petroleum reservoir static and dynamic characterization, able to integrate the p-adic approach with multifractals, thermodynamics and scaling. We also present a non-mathematician friendly review of trees and ultrametric spaces and pseudo-differential operators on such spaces.
US Agency for International Development — OPS Master is a management tool and database for integrated financial planning and portfolio management in USAID Missions. Using OPS Master, the three principal...
Uses and abuses of the Langevin equation for chemical reactions in condensed phases
International Nuclear Information System (INIS)
The Langevin and Fokker-Planck equations are useful in the description of many classical and quantum mechanical systems. However, these equations are justifiable from molecular considerations under very restricted conditions. These conditions include weak coupling. Brownian motion, and systems with special Hamiltonians. The application of these equations to chemical reactions in condensed phases is fraught with peril, particularly for fluid systems. The authors examine the molecular derivations of these equations and describe the conditions under which they are justifiable. It is, of course, possible that the equations are useful under other conditions
Prosen, Tomaz
2009-01-01
We generalize the method of third quantization to a unified exact treatment of Redfield and Lindblad master equations for open quadratic systems of n fermions in terms of diagonalization of 4n x 4n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how to compute all physically relevant quantities, such as non-equilibrium expectation values of local observables, various entropies or information measures, or time evolution and properties of relaxation. We also discuss how to exactly treat explicitly time dependent problems. The general formalism is then applied to study a thermally driven open XY spin 1/2 chain. We find that recently proposed non-equilibrium quantum phase transition in the open XY chain survives the thermal driving within the Redfield model. In particular, the phase of long-range magnetic correlations can be characterized by hypersensitivity of the non-equilibrium-steady state to external (bath or bulk) parameters. Studying the heat tran...
The equation of state of QCD at finite chemical potential
Gupta, Sourendu; Majumdar, Pushan
2014-01-01
We obtain the baryon number density, n, and the excess contribution to the pressure, Delta P, at finite chemical potential, mu_B, and temperature, T, by resumming the Taylor series expansion in a lattice computation with lattice spacing of 1/(4T) and two flavours of quarks at three different quark masses. The method proceeds by giving a critical mu_B and limits on the critical exponent, and permits reliable estimations of the errors in resummed quantities. We find that n and Delta P are insensitive to the quark mass. We also report the bulk isothermal compressibility, kappa, over a range of T and mu_B.
Energy Technology Data Exchange (ETDEWEB)
Brett, Tobias, E-mail: tobias.brett@postgrad.manchester.ac.uk; Galla, Tobias, E-mail: tobias.galla@manchester.ac.uk [Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL (United Kingdom)
2014-03-28
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.
International Nuclear Information System (INIS)
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period
Härtle, R.; Cohen, G.; Reichman, D. R.; Millis, A. J.
2015-08-01
We give a detailed comparison of the hierarchical quantum master equation (HQME) method to a continuous-time quantum Monte Carlo (CT-QMC) approach, assessing the usability of these numerically exact schemes as impurity solvers in practical nonequilibrium calculations. We review the main characteristics of the methods and discuss the scaling of the associated numerical effort. We substantiate our discussion with explicit numerical results for the nonequilibrium transport properties of a single-site Anderson impurity. The numerical effort of the HQME scheme scales linearly with the simulation time but increases (at worst exponentially) with decreasing temperature. In contrast, CT-QMC is less restricted by temperature at short times, but in general the cost of going to longer times is also exponential. After establishing the numerical exactness of the HQME scheme, we use it to elucidate the influence of different ways to induce transport through the impurity on the initial dynamics, discuss the phenomenon of coherent current oscillations, known as current ringing, and explain the nonmonotonic temperature dependence of the steady-state magnetization as a result of competing broadening effects. We also elucidate the pronounced nonlinear magnetization dynamics, which appears on intermediate time scales in the presence of an asymmetric coupling to the electrodes.
Chemical oceanography of the Indian Ocean, north of the equator
Gupta, R. Sen; Naqvi, S. W. A.
Chemical oceanographic studies in the North Indian Ocean have revealed several interesting and unique features. These are caused by the diverse conditions prevailing in the area which include immense river runoff in the northeast (Bay of Bengal) and a large excess of evaporation over precipitation and runoff in the northwest (Arabian Sea, Persian Gulf and Red Sea), resulting in the formation of several low- and high-salinity water masses. The occurrence of coastal upwelling seasonally makes the region highly fertile, and the existence of Asian landmass, forming the northern boundary, prevents quick renewal of subsurface layers. Consequently, dissolved oxygen gets severely depleted below the thermocline and reducing conditions prevail at intermediate depths (ca. 150-1200m) resulting in the reduction of nitrate (denitrification). The North Indian Ocean may contribute up to 10% of the global marine denitrification. The "denitrified" nitrogen, when combined with the rate of photosynthetic production reaching below the euphotic zone, gives the average residence time of water between 75 and 1200m as 43-51 years. The inorganic nutrient concentrations in the subsurface layers are very high in close proximity of the euphotic zone. The two-layered circulation leads to an active recycling of nutrients. The presence of organic fractions of nitrogen and phosphorus in significant concentrations in the deep water suggest that oxidation of organic matter is incomplete even great depths. The relationships between the apparent oxygen utilization (AOU) and nutrients and the stoichiometric composition of organic matter, deduced from the oxidative ratios and by analysis of plankton, are not very different from other oceanic areas. Higher nutrients and lower oxygen concentrations occur in the bottom layer as compared to the overlying water column in deep waters of the Bay of Bengal and Arabian Sea, suggesting that considerable quantities of organic matter reach the deep-sea floor
A New Mathematical Formulation of the Governing Equations for the Chemical Compositional Simulation
Bekbauov, Bakhbergen E; Berdyshev, Abdumauvlen
2015-01-01
It is the purpose of this work to develop new approach for chemical compositional reservoir simulation, which may be regarded as a sequential method. The development process can be roughly divided into the following two stages: (1) development of a new mathematical formulation for the sequential chemical compositional reservoir simulation, (2) implementation of a sequential solution approach for chemical compositional reservoir simulation based on the formulation described in this paper. This paper addresses the first stage of the development process by presenting a new mathematical formulation of the chemical compositional reservoir flow equations for the sequential simulation. The newly developed mathematical formulation is extended from the model formulation used in existing chemical compositional simulators. During the model development process, it was discovered that the currently used chemical compositional model estimates the adsorption effect on the transport of a component reasonably well but it viol...
Institute of Scientific and Technical Information of China (English)
王双进; 李凌云; 张建
2011-01-01
An amendment to the original Master equation was put forward, and a mechanism for node increases was introduced. The modified Master equation was of discreteness, which was more accurate, more efficient to calculate the evolution law of degree distribution of real complex network. The analysis formula of degree distribution of model BA and its calculation by the modified Master equation were discussed. From this. the logarithmic figure of degree distribution of model BA was gotlen. Then a compassion was made between the discrete Master equation and mean - field theory, and logarithmic figure was gotten, that of degree distribution of model BA wich two kinds calculative theory in the same coordinate system.%Master方程是计算无标度网络度分布演化规律的一种常用方法.提出了对原始的Master方程进行修正,加入了节点增长机制,修正后的Master方程具有离散性,能够更精确、更有效的计算真实复杂网络的度分布演化规律.用修正的Master方程分析BA模型度分布的解析武并计算,由此得到BA模型度分布对数图.把离散性的Master方程与连续性的平均场理论进行对比分析,并在同一坐标系下分别做出用两种理论计算的BA模型度分布的对数图.
Institute of Scientific and Technical Information of China (English)
王双进; 李凌云; 李佳
2011-01-01
An amendment to the original Master equation and add a mechanism for node increases is be put for ward. The modified Master equation is of discreteness, which is more accurate, more efficient to calculate the evo lution law of degree distribution of real complex network. The analysis formula of degree distribution of model BA and its calculation by the modified Master equation is discussed. From this we get the logarithmic figure of degree distribution of model BA. Then we make a compassion between the discrete Master equation and mean-field theory, and get logarithmic figure of degree distribution of model BA with two kinds calculative theory in the same coor dinate system.%提出了对原始的Master方程进行修正,加入了节点增长机制,修正后的Master方程具有离散性,能够更精确、更有效地计算真实复杂网络的度分布演化规律.用修正的Master方程分析BA模型度分布的解析式并计算,由此得出BA模型度分布对数图.把离散性的Master方程与连续性的平均场理论进行对比分析,并在同一坐标系下分别作出用2种理论计算的BA模型度分布的对数图.
DEFF Research Database (Denmark)
Maribo-Mogensen, Bjørn
This thesis extends the Cubic Plus Association (CPA) equation of state (EoS) to handle mixtures containing ions from fully dissociated salts. The CPA EoS has during the past 18 years been applied to thermodynamic modeling of a wide range of industrially important chemicals, mainly in relation to ...
Institute of Scientific and Technical Information of China (English)
LUO Zhen-dong; ZHOU Yan-jie; ZHU Jiang
2007-01-01
The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical modes by the following governing nonlinear partial differential equations containing velocity vector,temperature field,pressure field,and gas mass field.The mixed finite element(MFE)method is employed to study the system of equations for the vapor deposition chemical reaction processes.The semidiscrete and fully discrete MFE formulations are derived.And the existence and convergence(error estimate)of the semidiscrete and fully discrete MFE solutions are deposition chemical reaction processes,the numerical solutions of the velocity vector,the temperature field,the pressure field,and the gas mass field can be found out simultaneonsly.Thus,these researches are not only of important theoretical means,but also of extremely extensive applied vistas.
Model reduction for stochastic chemical systems with abundant species
Smith, Stephen; Cianci, Claudia; Grima, Ramon
2015-12-01
Biochemical processes typically involve many chemical species, some in abundance and some in low molecule numbers. We first identify the rate constant limits under which the concentrations of a given set of species will tend to infinity (the abundant species) while the concentrations of all other species remains constant (the non-abundant species). Subsequently, we prove that, in this limit, the fluctuations in the molecule numbers of non-abundant species are accurately described by a hybrid stochastic description consisting of a chemical master equation coupled to deterministic rate equations. This is a reduced description when compared to the conventional chemical master equation which describes the fluctuations in both abundant and non-abundant species. We show that the reduced master equation can be solved exactly for a number of biochemical networks involving gene expression and enzyme catalysis, whose conventional chemical master equation description is analytically impenetrable. We use the linear noise approximation to obtain approximate expressions for the difference between the variance of fluctuations in the non-abundant species as predicted by the hybrid approach and by the conventional chemical master equation. Furthermore, we show that surprisingly, irrespective of any separation in the mean molecule numbers of various species, the conventional and hybrid master equations exactly agree for a class of chemical systems.
Model reduction for stochastic chemical systems with abundant species
Energy Technology Data Exchange (ETDEWEB)
Smith, Stephen; Cianci, Claudia; Grima, Ramon [School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh EH93JR, Scotland (United Kingdom)
2015-12-07
Biochemical processes typically involve many chemical species, some in abundance and some in low molecule numbers. We first identify the rate constant limits under which the concentrations of a given set of species will tend to infinity (the abundant species) while the concentrations of all other species remains constant (the non-abundant species). Subsequently, we prove that, in this limit, the fluctuations in the molecule numbers of non-abundant species are accurately described by a hybrid stochastic description consisting of a chemical master equation coupled to deterministic rate equations. This is a reduced description when compared to the conventional chemical master equation which describes the fluctuations in both abundant and non-abundant species. We show that the reduced master equation can be solved exactly for a number of biochemical networks involving gene expression and enzyme catalysis, whose conventional chemical master equation description is analytically impenetrable. We use the linear noise approximation to obtain approximate expressions for the difference between the variance of fluctuations in the non-abundant species as predicted by the hybrid approach and by the conventional chemical master equation. Furthermore, we show that surprisingly, irrespective of any separation in the mean molecule numbers of various species, the conventional and hybrid master equations exactly agree for a class of chemical systems.
On Validity of Linear Phenomenological Nonequilibrium Thermodynamics Equations in Chemical Kinetics
International Nuclear Information System (INIS)
The chemical equilibrium state is treated as a fundamental ''reference frame'' in description of chemical reaction. In a definition of reactive absolute activities for components in chemical reaction the difference of chemical potential and its value in the equilibrium is used. The chemical reaction rate is shown to be proportional to the force Xnew defined as the difference of reactive absolute activities of reactants and products. The force Xnew is shown to be equivalent to the force following from chemical kinetics equations and compared with the reduced affinity X as well as with the force of Ross and Mazur XRM = 1 - exp(-X). The force Xnew coincides with X and XRM near to the chemical equilibrium state. A range of the molar fraction of product, in which a difference between the forces Xnew and X is relatively small, is larger than it would be for the forces XRM and X. It means that for some chemical reactions the formalism of linear nonequilibrium thermodynamics can be used in wider ranges than usually expected. Particular analysis is presented for simple reactions. (author)
Gałdzicki, Z; Miekisz, S
1984-04-01
The role of viscosity in coupling between chemical reaction (complex formation) and diffusion in membranes has been investigated. The Fick law was replaced by the momentum balance equation with the viscous term. The irreversible thermodynamics admits coupling of the chemical reaction rate with the gradient of velocity. The proposed model has shown the contrary effect of viscosity and confirmed the experimental results. The chemical reaction rate increases only above the limit value of viscosity. The parameter Q (degree of complex formation) was introduced to investigate coupling. Q equals to the ratio of the chemical contribution into the flux of the complex to the total flux of the substance transported. For different values of the parameters of the model the dependence of Q upon position inside the membrane has been numerically calculated. The assumptions of the model limit it to a specific case and they only roughly model the biological situation. PMID:6537360
Hot QCD equation of state and quark-gluon plasma-- finite quark chemical potential
Chandra, Vinod
2008-01-01
We explore the relevance of a hot QCD equation of state of $O[g^6\\ln(1/g)]$, which has been obtained\\cite{avrn} for non-vanishing quark-chemical potentials to heavy ion collisions. Employing a method proposed in a recent paper \\cite{chandra1}, we use the EOS to determine a host of thermodynamic quantities, the energy density, specific heat, entropy dnesity, and the temperature dependence of screening lengths, with the behaviour of QGP at RHIC and LHC in mind. We also investigate the sensitivity of these observables to the quark chemical potential.
Mélykúti, Bence
2010-01-01
The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions there is another, simple formulation of the CLE with only m1 + m2 Wiener processes, whereas the standard approach uses 2 m1 + m2. We demonstrate that there are considerable computational savings when using this latter formulation. Such transformations of the CLE do not cause a loss of accuracy and are therefore distinct from model reduction techniques. We illustrate our findings by considering alternative formulations of the CLE for a human ether a-go-go related gene ion channel model and the Goldbeter-Koshland switch. © 2010 American Institute of Physics.
Fan, Hong-Yi; Hu, Li-Yun
2009-04-01
By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation langleη|, which can arrange master equations of density operators ρ(t) in quantum statistics as state-vector evolution equations due to the elegant properties of langleη|. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of the decay constant κ we find that the matrix element of ρ(t) at time t in langleη| representation is proportional to that of the initial ρ0 in the decayed entangled state langleηe-κt| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ρ = ∫(d2η/π)langleη|ρrangleD(η), which is different from all the previous known representations.
Directory of Open Access Journals (Sweden)
S. M. Saunders
2003-01-01
Full Text Available Kinetic and mechanistic data relevant to the tropospheric degradation of volatile organic compounds (VOC, and the production of secondary pollutants, have previously been used to define a protocol which underpinned the construction of a near-explicit Master Chemical Mechanism. In this paper, an update to the previous protocol is presented, which has been used to define degradation schemes for 107 non-aromatic VOC as part of version 3 of the Master Chemical Mechanism (MCM v3. The treatment of 18 aromatic VOC is described in a companion paper. The protocol is divided into a series of subsections describing initiation reactions, the reactions of the radical intermediates and the further degradation of first and subsequent generation products. Emphasis is placed on updating the previous information, and outlining the methodology which is specifically applicable to VOC not considered previously (e.g. a- and b-pinene. The present protocol aims to take into consideration work available in the open literature up to the beginning of 2001, and some other studies known by the authors which were under review at the time. Application of MCM v3 in appropriate box models indicates that the representation of isoprene degradation provides a good description of the speciated distribution of oxygenated organic products observed in reported field studies where isoprene was the dominant emitted hydrocarbon, and that the a-pinene degradation chemistry provides a good description of the time dependence of key gas phase species in a-pinene/NOX photo-oxidation experiments carried out in the European Photoreactor (EUPHORE. Photochemical Ozone Creation Potentials (POCP have been calculated for the 106 non-aromatic non-methane VOC in MCM v3 for idealised conditions appropriate to north-west Europe, using a photochemical trajectory model. The POCP values provide a measure of the relative ozone forming abilities of the VOC. Where applicable, the values are compared with
Li, Daniel
2014-01-01
This easy-to-understand tutorial provides you with several engaging projects that show you how to utilize Grunt with various web technologies, teaching you how to master build automation and testing with Grunt in your applications.If you are a JavaScript developer who is looking to streamline their workflow with build-automation, then this book will give you a kick start in fully understanding the importance of the described web technologies and automate their processes using Grunt.
International Nuclear Information System (INIS)
Numerical simulation method was examined for chemical reactions of actinide elements U, Pu, Np, and Tc etc. in an aqueous nitric acid solution. It is known that the numerical calculation for the Purex process with chemical reactions and liquid flow becomes stiff, because time constant for the chemical reactions is two to three order of magnitude smaller due to the very fast reactions than that of mass transfer or of reaching distribution equilibrium. Recently in order to increase a time step Δt the partial equilibrium (P.E.) model, in which some very fast reactions are treated by the equilibrium law whereas other reactions are by the rate law, has been proposed. In the present study concentration change of the solutes in an aqueous solution with 30 chemical reactions, of which 4 are expressed by equilibrium equations, has been calculated. Description of the P.E. model and the comparison of the results and cpu time between the kinetic and the P.E. models are given. (author)
Directory of Open Access Journals (Sweden)
S. M. Saunders
2002-11-01
Full Text Available Kinetic and mechanistic data relevant to the tropospheric degradation of volatile organic compounds (VOC, and the production of secondary pollutants, have previously been used to define a protocol which underpinned the construction of a near-explicit Master Chemical Mechanism. In this paper, an update to the previous protocol is presented, which has been used to define degradation schemes for 107 non-aromatic VOC as part of version 3 of the Master Chemical Mechanism (MCM v3. The treatment of 18 aromatic VOC is described in a companion paper. The protocol is divided into a series of subsections describing initiation reactions, the reactions of the radical intermediates and the further degradation of first and subsequent generation products. Emphasis is placed on updating the previous information, and outlining the methodology which is specifically applicable to VOC not considered previously (e.g. a- and b-pinene. The present protocol aims to take into consideration work available in the open literature up to the beginning of 2001, and some other studies known by the authors which were under review at the time. Application of MCM v3 in appropriate box models indicates that the representation of isoprene degradation provides a good description of the speciated distribution of oxygenated organic products observed in reported field studies where isoprene was the dominant emitted hydrocarbon, and that the a-pinene degradation chemistry provides a good description of the time dependence of key gas phase species in a-pinene/NO_{X} photo-oxidation experiments carried out in the European Photoreactor (EUPHORE. Photochemical Ozone Creation Potentials (POCP have been calculated for the 106 non-aromatic non-methane VOC in MCM v3 for idealised conditions appropriate to north-west Europe, using a photochemical trajectory model. The POCP
Molecular finite-size effects in stochastic models of equilibrium chemical systems
Cianci, Claudia; Smith, Stephen; Grima, Ramon
2016-01-01
The reaction-diffusion master equation (RDME) is a standard modelling approach for understanding stochastic and spatial chemical kinetics. An inherent assumption is that molecules are point-like. Here, we introduce the excluded volume reaction-diffusion master equation (vRDME) which takes into account volume exclusion effects on stochastic kinetics due to a finite molecular radius. We obtain an exact closed form solution of the RDME and of the vRDME for a general chemical system in equilibriu...
Directory of Open Access Journals (Sweden)
M. E. Jenkin
2002-11-01
Full Text Available Kinetic and mechanistic data relevant to the tropospheric degradation of aromatic volatile organic compounds (VOC have been used to define a mechanism development protocol, which has been used to construct degradation schemes for 18 aromatic VOC as part of version 3 of the Master Chemical Mechanism (MCM v3. This is complementary to the treatment of 107 non-aromatic VOC, presented in a companion paper. The protocol is divided into a series of subsections describing initiation reactions, the degradation chemistry to first generation products via a number of competitive routes, and the further degradation of first and subsequent generation products. Emphasis is placed on describing where the treatment differs from that applied to the non-aromatic VOC. The protocol is based on work available in the open literature up to the beginning of 2001, and some other studies known by the authors which were under review at the time. Photochemical Ozone Creation Potentials (POCP have been calculated for the 18 aromatic VOC in MCM v3 for idealised conditions appropriate to north-west Europe, using a photochemical trajectory model. The POCP values provide a measure of the relative ozone forming abilities of the VOC. These show distinct differences from POCP values calculated previously for the aromatics, using earlier versions of the MCM, and reasons for these differences are discussed.
Directory of Open Access Journals (Sweden)
M. E. Jenkin
2003-01-01
Full Text Available Kinetic and mechanistic data relevant to the tropospheric degradation of aromatic volatile organic compounds (VOC have been used to define a mechanism development protocol, which has been used to construct degradation schemes for 18 aromatic VOC as part of version 3 of the Master Chemical Mechanism (MCM v3. This is complementary to the treatment of 107 non-aromatic VOC, presented in a companion paper. The protocol is divided into a series of subsections describing initiation reactions, the degradation chemistry to first generation products via a number of competitive routes, and the further degradation of first and subsequent generation products. Emphasis is placed on describing where the treatment differs from that applied to the non-aromatic VOC. The protocol is based on work available in the open literature up to the beginning of 2001, and some other studies known by the authors which were under review at the time. Photochemical Ozone Creation Potentials (POCP have been calculated for the 18 aromatic VOC in MCM v3 for idealised conditions appropriate to north-west Europe, using a photochemical trajectory model. The POCP values provide a measure of the relative ozone forming abilities of the VOC. These show distinct differences from POCP values calculated previously for the aromatics, using earlier versions of the MCM, and reasons for these differences are discussed.
Validity conditions for moment closure approximations in stochastic chemical kinetics
International Nuclear Information System (INIS)
Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here, we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show that for bistable and oscillatory chemical systems with deterministic initial conditions, the solution of the MA equations can be interpreted as a valid approximation to the true moments of the chemical master equation, only when the steady-state mean molecule numbers obtained from the chemical master equation fall within a certain finite range. The same validity criterion for monostable systems implies that the steady-state mean molecule numbers obtained from the chemical master equation must be above a certain threshold. For mean molecule numbers outside of this range of validity, the MA equations lead to either qualitatively wrong oscillatory dynamics or to unphysical predictions such as negative variances in the molecule numbers or multiple steady-state moments of the stationary distribution as the initial conditions are varied. Our results clarify the range of validity of the MA approach and show that pitfalls in the interpretation of the results can only be overcome through the systematic comparison of the solutions of the MA equations of a certain order with those of higher orders
Noise-induced multistability in chemical systems: Discrete versus continuum modeling
Czech Academy of Sciences Publication Activity Database
Duncan, A.; Liao, S.; Vejchodský, Tomáš; Erban, R.; Grima, R.
2015-01-01
Roč. 91, č. 4 (2015), s. 042111. ISSN 1539-3755 EU Projects: European Commission(XE) 328008 - STOCHDETBIOMODEL Institutional support: RVO:67985840 Keywords : chemical master equation * chemical Fokker-Planck equation * multimodality Subject RIV: BA - General Mathematics Impact factor: 2.288, year: 2014 http://journals.aps.org/pre/abstract/10.1103/PhysRevE.91.042111
Ab initio studies of equations of state and chemical reactions of reactive structural materials
Zaharieva, Roussislava
subject of studies of the shock or thermally induced chemical reactions of the two solids comprising these reactive materials, from first principles, is a relatively new field of study. The published literature on ab initio techniques or quantum mechanics based approaches consists of the ab initio or ab initio-molecular dynamics studies in related fields that contain a solid and a gas. One such study in the literature involves a gas and a solid. This is an investigation of the adsorption of gasses such as carbon monoxide (CO) on Tungsten. The motivation for these studies is to synthesize alternate or synthetic fuel technology by Fischer-Tropsch process. In this thesis these studies are first to establish the procedure for solid-solid reaction and then to extend that to consider the effects of mechanical strain and temperature on the binding energy and chemisorptions of CO on tungsten. Then in this thesis, similar studies are also conducted on the effect of mechanical strain and temperature on the binding energies of Titanium and hydrogen. The motivations are again to understand the method and extend the method to such solid-solid reactions. A second motivation is to seek strained conditions that favor hydrogen storage and strain conditions that release hydrogen easily when needed. Following the establishment of ab initio and ab initio studies of chemical reactions between a solid and a gas, the next step of research is to study thermally induced chemical reaction between two solids (Ni+Al). Thus, specific new studies of the thesis are as follows: (1) Ab initio Studies of Binding energies associated with chemisorption of (a) CO on W surfaces (111, and 100) at elevated temperatures and strains and (b) adsorption of hydrogen in titanium base. (2) Equations of state of mixtures of reactive material structures from ab initio methods. (3) Ab initio studies of the reaction initiation, transition states and reaction products of intermetallic mixtures of (Ni+Al) at elevated
Energy Technology Data Exchange (ETDEWEB)
Honey, D.A.
1989-12-01
The collisional Boltzmann equation was solved numerically to obtain excitation rates for use in a CO{sub 2} laser design program. The program was written in Microsoft QuickBasic for use on the IBM Personal Computer or equivalent. Program validation involved comparisons of computed transport coefficients with experimental data and previous theoretical work. Four different numerical algorithms were evaluated in terms of accuracy and efficiency. L-U decomposition was identified as the preferred approach. The calculated transport coefficients were found to agree with empirical data within one to five percent. The program was integrated into a CO{sub 2} laser design program. Studies were then performed to evaluate the effects on predicted laser output power and energy density as parameters affecting electron kinetics were changed. Plotting routines were written for both programs.
Malkin, Tamsin L; Heard, Dwayne E; Hood, Christina; Stocker, Jenny; Carruthers, David; MacKenzie, Ian A; Doherty, Ruth M; Vieno, Massimo; Lee, James; Kleffmann, Jörg; Laufs, Sebastian; Whalley, Lisa K
2016-07-18
Air pollution is the environmental factor with the greatest impact on human health in Europe. Understanding the key processes driving air quality across the relevant spatial scales, especially during pollution exceedances and episodes, is essential to provide effective predictions for both policymakers and the public. It is particularly important for policy regulators to understand the drivers of local air quality that can be regulated by national policies versus the contribution from regional pollution transported from mainland Europe or elsewhere. One of the main objectives of the Coupled Urban and Regional processes: Effects on AIR quality (CUREAIR) project is to determine local and regional contributions to ozone events. A detailed zero-dimensional (0-D) box model run with the Master Chemical Mechanism (MCMv3.2) is used as the benchmark model against which the less explicit chemistry mechanisms of the Generic Reaction Set (GRS) and the Common Representative Intermediates (CRIv2-R5) schemes are evaluated. GRS and CRI are used by the Atmospheric Dispersion Modelling System (ADMS-Urban) and the regional chemistry transport model EMEP4UK, respectively. The MCM model uses a near-explicit chemical scheme for the oxidation of volatile organic compounds (VOCs) and is constrained to observations of VOCs, NOx, CO, HONO (nitrous acid), photolysis frequencies and meteorological parameters measured during the ClearfLo (Clean Air for London) campaign. The sensitivity of the less explicit chemistry schemes to different model inputs has been investigated: Constraining GRS to the total VOC observed during ClearfLo as opposed to VOC derived from ADMS-Urban dispersion calculations, including emissions and background concentrations, led to a significant increase (674% during winter) in modelled ozone. The inclusion of HONO chemistry in this mechanism, particularly during wintertime when other radical sources are limited, led to substantial increases in the ozone levels predicted
Department of Veterans Affairs — As of June 28, 2010, the Master Veteran Index (MVI) database based on the enhanced Master Patient Index (MPI) is the authoritative identity service within the VA,...
Sattsangi, Prem D.
2014-01-01
A laboratory method for teaching inorganic qualitative analysis and chemical equations is described. The experiment has been designed to focus attention on cations and anions that react to form products. This leads to a logical approach to understand and write chemical equations. The procedure uses 3 mL plastic micropipettes to store and deliver…
Family name distributions: Master equation approach
Baek, Seung Ki; Kiet, Hoang Anh Tuan; Kim, Beom Jun
2008-01-01
Although cumulative family name distributions in many countries exhibit power-law forms, there also exist counterexamples. The origin of different family name distributions across countries is discussed analytically in the framework of a population dynamics model. Combined with empirical observations made, it is suggested that those differences in distributions are closely related with the rate of appearance of new family names.
Master equation as a radial constraint
Hussain, Uzair; Kunduri, Hari K
2015-01-01
We revisit the problem of perturbations of Schwarzschild-AdS$_4$ black holes by using a combination of the Martel-Poisson formalism for perturbations of four-dimensional spherically symmetric spacetimes and the Kodama-Ishibashi formalism. We clarify the relationship between both formalisms and express the Brown-York-Balasubramanian-Krauss boundary stress-energy tensor, $\\bar{T}_{\\mu\
Are safe results obtained when SAFT equations are applied to ordinary chemicals?
DEFF Research Database (Denmark)
Privat, Romain; Conte, Elisa; Jaubert, Jean-Noël; Gani, Rafiqul
2012-01-01
In a previous work, some irregular behaviours of the PC-SAFT EoS – and more generally of SAFT-type EoS – were pointed out for pure components at low temperatures. In particular, it was shown that for pure fluids at a fixed temperature and pressure, these equations of state were likely to exhibit ...
Thorne, Lawrence R
2011-01-01
I propose a novel approach to balancing equations that is applicable to all chemical-reaction equations; it is readily accessible to students via scientific calculators and basic computer spreadsheets that have a matrix-inversion application. The new approach utilizes the familiar matrix-inversion operation in an unfamiliar and innovative way; its purpose is not to identify undetermined coefficients as usual, but, instead, to compute a matrix null space (or matrix kernel). The null space then provides the coefficients that balance the equation. Indeed, the null space contains everything there is to know about balancing any chemical-reaction equation!
Validity conditions for moment closure approximations in stochastic chemical kinetics
Schnoerr, David; Sanguinetti, Guido; Grima, Ramon
2014-01-01
Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here, we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show tha...
Mastering mathematics geometry & measures
Various
2014-01-01
Deliver outstanding lessons that build fluency, problem-solving and mathematical reasoning skills to enable sustained progress at Key Stage 3, in preparation for GCSE. Mastering Mathematics provides flexible online and print teaching and learning resources. The service focuses on strands within the curriculum to improve progression throughout Secondary Mathematics . Mastering Mathematics Student Books and Whiteboard eTextbooks are organised into progression strands in line with Mastering Mathematics Teaching and Learning Resources:. - Enable students to identify appropriate remediation or exte
Mastering mathematics statistics & probability
Various
2014-01-01
Mastering Mathematics provides flexible online and print teaching and learning resources. The service focuses on strands within the curriculum to improve progression throughout Secondary Mathematics. Mastering Mathematics Student Books and eBooks are organised into progression strands in line with Mastering Mathematics Teaching and Learning Resources:. - Enable students to identify appropriate remediation or extension steps they need in order to progress, through easy to follow progression charts. - Clear explanations of the tools needed for the chapter followed by questions that develop fluen
Various
2014-01-01
Mastering Mathematics provides flexible online and print teaching and learning resources. The service focuses on strands within the curriculum to improve progression throughout Secondary Mathematics. Mastering Mathematics Student Books and Whiteboard eTextbooks are organised into progression strands in line with Mastering Mathematics Teaching and Learning Resources:. - Enable students to identify appropriate remediation or extension steps they need in order to progress, through easy to follow progression charts. - Clear explanations of the tools needed for the chapter followed by questions tha
Masters Colors -meikkisarjan lanseeraus
Muhonen, Veera; Renlund, Siri
2013-01-01
Toiminnallisen opinnäytetyön tarkoituksena oli suunnitella ja toteuttaa Masters Colors –meikkisarjan lanseeraustoimenpiteet. Opinnäytetyö toteutettiin yhteistyössä hoitolakosmetiikan maahantuontiyritys Benecom Oy:n kanssa. Yrityksen päätoimisena maahantuontisarjana toimii Guinot-hoitolakosmetiikkasarja, jonka lisäksi Benecom Oy tuo maahan Guinot-konsernin Masters Colors –meikkisarjaa sekä Cosmecology –kosmetiikkaa. Masters Colors on kehitetty laajentamaan Guinot-kauneushoitoloiden palveluvali...
MASTER TELEVISION ANTENNA SYSTEM.
Rhode Island State Dept. of Education, Providence.
SPECIFICATIONS FOR THE FURNISHING AND INSTALLATION OF TELEVISION MASTER ANTENNA SYSTEMS FOR SECONDARY AND ELEMENTARY SCHOOLS ARE GIVEN. CONTRACTOR REQUIREMENTS, EQUIPMENT, PERFORMANCE STANDARDS, AND FUNCTIONS ARE DESCRIBED. (MS)
Institute of Scientific and Technical Information of China (English)
YUQIAN
2004-01-01
Celestial burial is worshipped in Tibet as the highest pursuit of life. Of three elements indispensable for celestial burial-celestial rock (also known as altar), cinereous vultures, and masters of celestial burial, celestial burial masters are the most mysteriously important.