Mean Field Forward-Backward Stochastic Differential Equations
Carmona, René; Delarue, François
2013-01-01
The purpose of this note is to provide an existence result for the solution of fully coupled Forward Backward Stochastic Differential Equations (FBSDEs) of the mean field type. These equations occur in the study of mean field games and the optimal control of dynamics of the McKean Vlasov type.
Existence of a solution to an equation arising from the theory of Mean Field Games
Gangbo, Wilfrid; Święch, Andrzej
2015-12-01
We construct a small time strong solution to a nonlocal Hamilton-Jacobi equation (1.1) introduced in [48], the so-called master equation, originating from the theory of Mean Field Games. We discover a link between metric viscosity solutions to local Hamilton-Jacobi equations studied in [2,19,20] and solutions to (1.1). As a consequence we recover the existence of solutions to the First Order Mean Field Games equations (1.2), first proved in [48], and make a more rigorous connection between the master equation (1.1) and the Mean Field Games equations (1.2).
A bilocal picture of quantum mechanics
A new, bilocal picture of quantum mechanics is developed. We show that Born’s rule supports a virtual probability for a particle to arrive, as a wave, at any two locations (but no more). We discuss two ways to implement twin detectors suitable for detecting bilocal arrivals. The bilocal picture sheds light on currents in quantum mechanics. We find there are two types of bilocal current density, whose polar form and related mean velocities are given. In the bilocal context, the definitions of both current types simplify. In the unilocal case, the two types become the usual current and a fluctuation current. Their respective mean velocity fields are the usual de Broglie–Madelung–Bohm velocity and the imaginary (osmotic) velocity. We obtain a new, probabilistic Schrödinger equation for the bilocal probability by itself, solve the example of a free particle, develop the dyadic stationary states, and find that the von Neumann equation for time-varying density of states follows directly from the new equation. We also show how to include the electromagnetic potentials in this probabilistic Schrödinger equation. (paper)
On the dynamics of mean-field equations for stochastic neural fields with delays
Touboul, Jonathan
2011-01-01
The cortex is composed of large-scale cell assemblies sharing the same individual properties and receiving the same input, in charge of certain functions, and subject to noise. Such assemblies are characterized by specific space locations and space-dependent delayed interactions. The mean-field equations for such systems were rigorously derived in a recent paper for general models, under mild assumptions on the network, using probabilistic methods. We summarize and investigate general implications of this result. We then address the dynamics of these stochastic neural field equations in the case of firing-rate neurons. This is a unique case where the very complex stochastic mean-field equations exactly reduce to a set of delayed differential or integro-differential equations on the two first moments of the solutions, this reduction being possible due to the Gaussian nature of the solutions. The obtained equations differ from more customary approaches in that it incorporates intrinsic noise levels nonlinearly ...
Bound Pairs of Fronts in a Real Ginzburg-Landau Equation Coupled to a Mean Field
Herrero, H
1995-01-01
Motivated by the observation of localized traveling-wave states (`pulses') in convection in binary liquid mixtures, the interaction of fronts is investigated in a real Ginzburg-Landau equation which is coupled to a mean field. In that system the Ginzburg-Landau equation describes the traveling-wave amplitude and the mean field corrsponds to a concentration mode which arises due to the slowness of mass diffusion. For single fronts the mean field can lead to a hysteretic transition between slow and fast fronts. Its contribution to the interaction between fronts can be attractive as well as repulsive and depends strongly on their direction of propagation. Thus, the concentration mode leads to a new localization mechanism, which does not require any dispersion in contrast to that operating in the nonlinear Schrödinger equation. Based on this mechanism alone, pairs of fronts in binary-mixture convection are expected to form {\\it stable} pulses if they travel {\\it backward}, i.e. opposite to the phase velocity. Fo...
Mean-field potential calculations of high-pressure equation of state for BeO
Zhang Qi-Li; Zhang Ping; Song Hai-Feng; Liu Hai-Feng
2008-01-01
A systematic study of the Hugoniot equation of state, phase transition, and the other thermodynamic properties including the Hugoniot temperature, the electronic and ionic heat capacities, and the Griineisen parameter for shockcompressed BeO, has been carried out by calculating the total free energy. The method of calculations combines first-principles treatment for 0 K and finite-T electronic contribution and the mean-field-potential approach for the vibrational contribution of the lattice ion to the total energy. Our calculated Hugoniot is in good agreement with the experimental data.
Chavanis, Pierre-Henri
2008-01-01
We consider a generalized class of Keller-Segel models describing the chemotaxis of biological populations (bacteria, amoebae, endothelial cells, social insects,...). We show the analogy with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. As an illustration, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). We also discuss the analogy between biological populations described by the Keller-Segel model and self-gravitating Brownian particles described by the Smoluchowski-Poisson system.
Mean-field treatment of the many-body Fokker-Planck equation
We review some properties of the stationary states of the Fokker-Planck equation for N interacting particles within a mean-field approximation, which yields a non-linear integrodifferential equation for the particle density. Analytical results show that for attractive long-range potentials the steady state is always a precipitate containing one or several clusters of small size. For arbitrary potential, linear stability analysis allows the statement of the conditions under which the uniform equilibrium state is unstable against small perturbations and, via the Einstein relation, definition of a critical temperature Tc separating the two phases, uniform and precipitate. The corresponding phase diagram turns out to be strongly dependent on the pair-potential. In addition, numerical calculations reveal that the transition is hysteretic. We finally discuss the dynamics of relaxation for the uniform state suddenly cooled below Tc. (author)
Relativistic mean-field theory and the high-density nuclear equation of state
Müller, H; Mueller, Horst; Serot, Brian D
1996-01-01
The properties of high-density nuclear and neutron matter are studied using a relativistic mean-field approximation to the nuclear matter energy functional. Based on ideas of effective field theory, nonlinear interactions between the fields are introduced to parametrize the density dependence of the energy functional. Various types of nonlinearities involving scalar-isoscalar (\\sigma), vector-isoscalar (\\omega), and vector-isovector (\\rho) fields are studied. After calibrating the model parameters at equilibrium nuclear matter density, the model and parameter dependence of the resulting equation of state is examined in the neutron-rich and high-density regime. It is possible to build different models that reproduce the same observed properties at normal nuclear densities, but which yield maximum neutron star masses that differ by more than one solar mass. Implications for the existence of kaon condensates or quark cores in neutron stars are discussed.
Relativistic Mean-Field Theory and the High-Density Nuclear Equation of State
Mueller, Horst; Serot, Brian D.
1996-01-01
The properties of high-density nuclear and neutron matter are studied using a relativistic mean-field approximation to the nuclear matter energy functional. Based on ideas of effective field theory, nonlinear interactions between the fields are introduced to parametrize the density dependence of the energy functional. Various types of nonlinearities involving scalar-isoscalar ($\\sigma$), vector-isoscalar ($\\omega$), and vector-isovector ($\\rho$) fields are studied. After calibrating the model...
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
Touboul, Jonathan
2012-08-01
In this manuscript we analyze the collective behavior of mean-field limits of large-scale, spatially extended stochastic neuronal networks with delays. Rigorously, the asymptotic regime of such systems is characterized by a very intricate stochastic delayed integro-differential McKean-Vlasov equation that remain impenetrable, leaving the stochastic collective dynamics of such networks poorly understood. In order to study these macroscopic dynamics, we analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics and sigmoidal interactions. In that case, we prove that the solution of the mean-field equation is Gaussian, hence characterized by its two first moments, and that these two quantities satisfy a set of coupled delayed integro-differential equations. These equations are similar to usual neural field equations, and incorporate noise levels as a parameter, allowing analysis of noise-induced transitions. We identify through bifurcation analysis several qualitative transitions due to noise in the mean-field limit. In particular, stabilization of spatially homogeneous solutions, synchronized oscillations, bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow further exploring the role of noise in the nervous system.
Jiang, Wei-Zhou; Li, Bao-An; Chen, Lie-Wen
2007-01-01
Using in-medium hadron properties according to the Brown-Rho scaling due to the chiral symmetry restoration at high densities and considering naturalness of the coupling constants, we have newly constructed several relativistic mean-field Lagrangians with chiral limits. The model parameters are adjusted such that the symmetric part of the resulting equation of state at supra-normal densities is consistent with that required by the collective flow data from high energy heavy-ion reactions, whi...
Investigation Of Mean-Field Equations Of ANNNI Model For The Phase Transition In UNi2Si2
The phase transition from the uncompensated antiferromagnetic state to a simple antiferromagnet, experimentally observed in UNi2Si2 is a subject of intensive discussion in recent few years. We present a study of initial mean-field equations of Axial-Next-Next-Neighbour Ising (ANNNI) model applied to the transition. The temperature evolution of a magnetic phase diagram within ANNNI model is presented. Results are discussed in comparison with other previous investigations of this topic. (Authors)
Bilocal field theory of elementary particles
The basic idea of this article is that in a bilocal field theory the Lorentz group can be replaced by the larger group SO(4;4) which contains SO(3;1) as a subgroup. On this basis a system of 24 coupled differential equations is introduced. In the special case of the pion the connection with the local field theory is discussed. It turns out that the application of the local field theory is limited by the reduced Compton wave length lambda/2π of the pion. (orig.)
Perturbative construction of the periodic orbits of the mean-field equations in nuclear physics
We have presented a perturbative construction method of the periodic orbits of the time-dependent Hartree-Fock equations (TDHF). The solutions are found in the form of a power series in the amplitude of the collective motion. We have performed calculations using third order expansions to determine the splitting of two-phonon states of the low-lying octupole vibration in oxygen-16 and calcium-40. We have also investigated giant quadrupole vibrations. We had to generalize our method in order to account for the resonant coupling between the two-phonon state and one-phonon states in the continuum. This was done by introducing admixture of the resonant mode in the first order expression of the periodic orbit. Our results demonstrate that the method of quantization of periodic orbits of TDHF equation is a powerful tool to investigate the energy spectra of many-body systems. We have used our method to build the classical periodic orbits of a Skyrmion. The method, used up to second order, has been applied to the Roper resonance described in terms of monopole vibrations. To first order the method is equivalent to linear response theory and we find that the response function displays a well developed peak. We have also presented a powerful method which uses the knowledge of periodic orbits to construct a collective Bohr-type Hamiltonian. We have applied it to the case of monopole vibrations of the Skyrmion and derived the corresponding first anharmonic terms in the collective Hamiltonian
A mean field calculation of the equation of state of supernova matter
The equation of state for hot dense matter occuring in stellar collapse is calculated using the Hartree-Fock approximation at finite temperature. The effective nucleon-nucleon interaction is a modified Skyrme force which gives a rather good value of the compression modulus in nuclear matter. Results are presented for the adiabat S=1 per baryon, with a fixed value of the electron fraction Ysub(e)=0.25, in the density range rho=0.02 to 0.07 baryons per fm3. We find that nuclei are still present in the medium. As a consequence the adiabatic index is slightly less than 4/3. We also discuss the presence of a transition, around half nuclear matter density, towards a phase made of bubbles
Carrier, Pierre; Tang, Jok M.; Saad, Yousef; Freericks, James K.
Inhomogeneous dynamical mean-field theory has been employed to solve many interesting strongly interacting problems from transport in multilayered devices to the properties of ultracold atoms in a trap. The main computational step, especially for large systems, is the problem of calculating the inverse of a large sparse matrix to solve Dyson's equation and determine the local Green's function at each lattice site from the corresponding local self-energy. We present a new e_cient algorithm, the Lanczos-based low-rank algorithm, for the calculation of the inverse of a large sparse matrix which yields this local (imaginary time) Green's function. The Lanczos-based low-rank algorithm is based on a domain decomposition viewpoint, but avoids explicit calculation of Schur complements and relies instead on low-rank matrix approximations derived from the Lanczos algorithm, for solving the Dyson equation. We report at least a 25-fold improvement of performance compared to explicit decomposition (such as sparse LU) of the matrix inverse. We also report that scaling relative to matrix sizes, of the low-rank correction method on the one hand and domain decomposition methods on the other, are comparable.
The neutron star matter equation of state is considered in the framework of relativistic mean-field theory, when also the scalar-isovector δ -meson effective field is taken into account. The constants of the theory are numerically determined in a way to reproduce the empirically known characteristics of symmetric nuclear matter at saturation density. The thermodynamic characteristics of both asymmetric nucleonic matter and a β -equilibrium hadron-electronic npe-plasma are studied. In the assumption that the transition to strange quark matter is a usual first order phase transition described by Maxwell's construction, the phase transition parameters changes caused by presence of δ -meson field are investigated in details. The advanced version of MIT bag model for the description of a quark phase is used, in which the interactions between quarks are taken into account in one-gluon exchange approach. The phase transition parameters for different values of bag constant in an interval B [60,120] MeV/fm3 are determined and is shown that the account of a δ -meson field results in reduction of pressure of phase transition, Po and of concentrations nN and nQ at phase transition point
Dickman, Ronald
1989-07-01
A recently devised method for determining the pressure in lattice simulations is applied to two-dimensional, athermal chains of 40, 80, and 160 segments, over the full range of fluid densities, from dilute solution to dense melt. The results are used to test Bawendi and Freed's correction to Flory-Huggins mean-field theory, and the des Cloizeaux scaling law. The scaling of the mean-square end-to-end distance with density is also discussed.
Hinschberger, Y.; Dixit, A.; Manfredi, G.; Hervieux, P.-A.
2015-01-01
We demonstrate the equivalence between (i) the semirelativistic limit (up to second order in the inverse of the speed of light) of the self-consistent Dirac-Maxwell equations and (ii) the Breit-Pauli equations in the mean-field (Hartree-like) approximation. We explain how the charge and current densities that act as sources in the Dirac-Maxwell equations are related to the microscopic two-electron interactions of the Breit-Pauli model (spin orbit, spin-other-orbit, and spin-spin). The key role played by the second-order corrections to the charge density is clarified.
Kraaij, Richard
2016-07-01
We prove the large deviation principle (LDP) for the trajectory of a broad class of finite state mean-field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the LDP, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the LDP provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation.
Malik, G P
2016-01-01
Given the Debye temperature of an elemental superconductor (SC) and its Tc, BCS theory enables one to predict the value of its gap 0 at T = 0, or vice versa. This monograph shows that non-elemental SCs can be similarly dealt with via the generalized BCS equations (GBCSEs) which, given any two parameters of the set {Tc, 10, 20 > 10}, enable one to predict the third. Also given herein are new equations for the critical magnetic field and critical current density of an elemental and a non-elemental SC — equations that are derived directly from those that govern pairing in them. The monograph includes topics that are usually not covered in any one text on superconductivity, e.g., BCS-BEC crossover physics, the long-standing puzzle posed by SrTiO3, and heavy-fermion superconductors — all of which are still imperfectly understood and therefore continue to avidly engage theoreticians. It suggests that addressing the Tcs, s and other properties (e.g., number densities of charge carriers) of high-Tc SCs via GBCSE...
Bi-Local Fields in AdS${}_5$ Spacetime
Aouda, Kenichi; Toyoda, Haruki
2016-01-01
Recently, the bi-local fields attracts the interest in studying the duality between $O(N)$ vector model and a higher-spin gauge theory in $AdS$ spacetime. In those theories, the bi-local fields are realized as collective one's of the $O(N)$ vector fields, which are the source of higher spin bulk fields. Historically, the bi-local fields are introduced as a candidate of non-local fields by Yukawa. Today, Yukawa's bi-local fields are understood from a viewpoint of relativistic two-particle bound systems, the bi-local systems. We study the relation between the collective bi-local fields out of HS bulk fields and the fields out of bi-local systems embedded in AdS${}_5$ spacetime with warped metric. It is shown that the effective spring tension of the bi-local system depends on the brane, on which the bi-local system is located. In paticular, a tensionless bi-local sytem, being similar to the collective bi-local fields, can be realized on a low-energy visible brane.
Bilocal Dynamics for Self-Avoiding Walks
Caracciolo, Sergio; Ferraro, G; Papinutto, Mauro; Pelissetto, A; Caracciolo, Sergio; Causo, Maria Serena; Ferraro, Giovanni; Papinutto, Mauro; Pelissetto, Andrea
2000-01-01
We introduce several bilocal algorithms for lattice self-avoiding walks that provide reasonable models for the physical kinetics of polymers in the absence of hydrodynamic effects. We discuss their ergodicity in different confined geometries, for instance in strips and in slabs. A short discussion of the dynamical properties in the absence of interactions is given.
Bilocal Dynamics for Self-Avoiding Walks
Caracciolo, Sergio; Causo, Maria Serena; Ferraro, Giovanni; Papinutto, Mauro; Pelissetto, Andrea
1999-01-01
We introduce several bilocal algorithms for lattice self-avoiding walks that provide reasonable models for the physical kinetics of polymers in the absence of hydrodynamic effects. We discuss their ergodicity in different confined geometries, for instance in strips and in slabs. A short discussion of the dynamical properties in the absence of interactions is given.
The su(3) mean field approximation describes collective nuclear rotation in a density matrix formalism. The densities ρ=q-i l/2 are 3x3 Hermitian matrices in the su(3) dual space, where q is the expectation of the quadrupole moment and l is the expectation of the angular momentum. The mean field approximation restricts these densities to a level surface of the su(3) Casimirs. Each level surface is a coadjoint orbit of the canonical transformation group SU(3). For each density ρ, the su(3) mean field Hamiltonian h[ρ] is an element of the su(3) Lie algebra. A model su(3) energy functional and the symplectic structure on the coadjoint orbit determine uniquely the su(3) mean field Hamiltonian. The densities in time-dependent su(3) mean field theory obey the dynamical equation i ρ radical = [h[ρ],ρ] on a coadjoint orbit. The cranked mean field Hamiltonian is hΩ=h+iΩ, where Ω is the angular velocity of the rotating principal axis frame. A rotating equilibrium density ρ-tilde in the body-fixed frame is a self-consistent solution to the equation [hΩ[ρ-tilde],ρ-tilde]=0. (author)
Risk-sensitive mean-field games
Tembine, Hamidou
2014-04-01
In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton -Jacobi- Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics. © 1963-2012 IEEE.
Extended Deterministic Mean-Field Games
Gomes, Diogo A.
2016-04-21
In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.
Continuous Time Finite State Mean Field Games
In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games
Continuous Time Finite State Mean Field Games
Gomes, Diogo A., E-mail: dgomes@math.ist.utl.pt [Instituto Superior Tecnico, Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matematica (Portugal); Mohr, Joana, E-mail: joana.mohr@ufrgs.br; Souza, Rafael Rigao, E-mail: rafars@mat.ufrgs.br [UFRGS, Instituto de Matematica (Brazil)
2013-08-01
In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N{yields}{infinity} of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.
Continuous time finite state mean field games
Gomes, Diogo A.
2013-04-23
In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games. © 2013 Springer Science+Business Media New York.
Pais, Helena
2016-01-01
The Vlasov formalism is extended to relativistic mean-field hadron models with non-linear terms up to fourth order and applied to the calculation of the crust-core transition density. The effect of the nonlinear $\\omega\\rho$ and $\\sigma\\rho$ coupling terms on the crust-core transition density and pressure, and on the macroscopic properties of some families of hadronic stars is investigated. For that purpose, six families of relativistic mean field models are considered. Within each family, the members differ in the symmetry energy behavior. For all the models, the dynamical spinodals are calculated, and the crust-core transition density and pressure, and the neutron star mass-radius relations are obtained. The effect on the star radius of the inclusion of a pasta calculation in the inner crust is discussed. The set of six models that best satisfy terrestrial and observational constraints predicts a radius of 13.6$\\pm$0.3 km and a crust thickness of $1.36\\pm 0.06$km for a 1.4 $M_\\odot$ star.
Ablakulov, Kh., E-mail: ablakulov@inp.uz; Narzikulov, Z., E-mail: narzikulov@inp.uz [Uzbek Academy of Sciences, Institute of Nuclear Physics (Uzbekistan)
2015-01-15
A phenomenological model is developed in terms of bilocal meson fields in order to describe a vector meson and its leptonic decays. A new Salpeter equation for this particle and the Schwinger-Dyson equation allowing for the presence of an arbitrary potential and for a modification associated with the renormalization of the quark (antiquark ) wave function within the meson are given. An expression for the constant of the leptonic decay of the charged rho meson is obtained from an analysis of the decay process τ → ρν via parametrizing in it the hadronization of intermediate charged weak W bosons into a bilocal vector meson. The potential is chosen in the form of the sum of harmonic-oscillator and Coulomb potentials, and the respective boundary-value problem is formulated. It is shown that the solutions to this problem describe both the mass spectrum of vector mesons and their leptonic-decay constants.
A phenomenological model is developed in terms of bilocal meson fields in order to describe a vector meson and its leptonic decays. A new Salpeter equation for this particle and the Schwinger-Dyson equation allowing for the presence of an arbitrary potential and for a modification associated with the renormalization of the quark (antiquark ) wave function within the meson are given. An expression for the constant of the leptonic decay of the charged rho meson is obtained from an analysis of the decay process τ → ρν via parametrizing in it the hadronization of intermediate charged weak W bosons into a bilocal vector meson. The potential is chosen in the form of the sum of harmonic-oscillator and Coulomb potentials, and the respective boundary-value problem is formulated. It is shown that the solutions to this problem describe both the mass spectrum of vector mesons and their leptonic-decay constants
Moghrabi, Kassem; Roca-Maza, Xavier; Colo', Gianluca
2012-01-01
In a quantum Fermi system the energy per particle calculated at the second order beyond the mean-field approximation diverges if a zero-range interaction is employed. We have previously analyzed this problem in symmetric nuclear matter by using a simplified nuclear Skyrme interaction, and proposed a strategy to treat such a divergence. In the present work, we extend the same strategy to the case of the full nuclear Skyrme interaction. Moreover we show that, in spite of the strong divergence ($\\sim$ $\\Lambda^5$, where $\\Lambda$ is the momentum cutoff) related to the velocity-dependent terms of the interaction, the adopted cutoff regularization can be always simultaneously performed for both symmetric and nuclear matter with different neutron-to-proton ratio. This paves the way to applications to finite nuclei.
Obstacle mean-field game problem
Gomes, Diogo A.
2015-01-01
In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions. © European Mathematical Society 2015.
Bi-Local Holography in the SYK Model: Perturbations
Jevicki, Antal
2016-01-01
We continue the study of the Sachdev-Ye-Kitaev model in the Large $N$ limit. Following our formulation in terms of bi-local collective fields with dynamical reparametrization symmetry, we perform perturbative calculations around the conformal IR point. These are based on an $\\varepsilon$ expansion which allows for analytical evaluation of correlators and finite temperature quantities.
Weak interactions in a bilocal chiral theory
Relativistic covariant equations for quarkonia in the framework of a biolocal description are obtained. These equations can be used to find the solutions for the bound state functions for any given angular momentum. 13 refs.; 1 tab
Mean field games for cognitive radio networks
Tembine, Hamidou
2012-06-01
In this paper we study mobility effect and power saving in cognitive radio networks using mean field games. We consider two types of users: primary and secondary users. When active, each secondary transmitter-receiver uses carrier sensing and is subject to long-term energy constraint. We formulate the interaction between primary user and large number of secondary users as an hierarchical mean field game. In contrast to the classical large-scale approaches based on stochastic geometry, percolation theory and large random matrices, the proposed mean field framework allows one to describe the evolution of the density distribution and the associated performance metrics using coupled partial differential equations. We provide explicit formulas and algorithmic power management for both primary and secondary users. A complete characterization of the optimal distribution of energy and probability of success is given.
Generalized Derivations and Bilocal Jordan Derivations of Nest Algebras
Dangui Yan
2011-01-01
Full Text Available Let H be a complex Hilbert space and B(H the collection of all linear bounded operators, A is the closed subspace lattice including 0 an H, then A is a nest, accordingly alg A={T∈B(H:TN⊆N, ∀N∈A} is a nest algebra. It will be shown that of nest algebra, generalized derivations are generalized inner derivations, and bilocal Jordan derivations are inner derivations.
On Mean Field Limits for Dynamical Systems
Boers, Niklas; Pickl, Peter
2016-07-01
We present a purely probabilistic proof of propagation of molecular chaos for N-particle systems in dimension 3 with interaction forces scaling like 1/\\vert q\\vert ^{3λ - 1} with λ smaller but close to one and cut-off at q = N^{-1/3}. The proof yields a Gronwall estimate for the maximal distance between exact microscopic and approximate mean-field dynamics. This can be used to show weak convergence of the one-particle marginals to solutions of the respective mean-field equation without cut-off in a quantitative way. Our results thus lead to a derivation of the Vlasov equation from the microscopic N-particle dynamics with force term arbitrarily close to the physically relevant Coulomb- and gravitational forces.
A mean field approach to watershed hydrology
Bartlett, Mark; Porporato, Amilcare
2016-04-01
Mean field theory (also known as self-consistent field theory) is commonly used in statistical physics when modeling the space-time behavior of complex systems. The mean field theory approximates a complex multi-component system by considering a lumped (or average) effect for all individual components acting on a single component. Thus, the many body problem is reduced to a one body problem. For watershed hydrology, a mean field theory reduces the numerous point component effects to more tractable watershed averages, resulting in a consistent method for linking the average watershed fluxes to the local fluxes at each point. We apply this approach to the spatial distribution of soil moisture, and as a result, the numerous local interactions related to lateral fluxes of soil water are parameterized in terms of the average soil moisture. The mean field approach provides a basis for unifying and extending common event-based models (e.g. Soil Conservation Service curve number (SCS-CN) method) with more modern semi-distributed models (e.g. Variable Infiltration Capacity (VIC) model, the Probability Distributed (PDM) model, and TOPMODEL). We obtain simple equations for the fractions of the different source areas of runoff, the spatial variability of runoff, and the average runoff value (i.e., the so-called runoff curve). The resulting space time distribution of soil moisture offers a concise description of the variability of watershed fluxes.
Nonasymptotic mean-field games
Tembine, Hamidou
2014-12-01
Mean-field games have been studied under the assumption of very large number of players. For such large systems, the basic idea consists of approximating large games by a stylized game model with a continuum of players. The approach has been shown to be useful in some applications. However, the stylized game model with continuum of decision-makers is rarely observed in practice and the approximation proposed in the asymptotic regime is meaningless for networks with few entities. In this paper, we propose a mean-field framework that is suitable not only for large systems but also for a small world with few number of entities. The applicability of the proposed framework is illustrated through various examples including dynamic auction with asymmetric valuation distributions, and spiteful bidders.
Nonasymptotic mean-field games
Tembine, Hamidou
2014-12-01
Mean-field games have been studied under the assumption of very large number of players. For such large systems, the basic idea consists to approximate large games by a stylized game model with a continuum of players. The approach has been shown to be useful in some applications. However, the stylized game model with continuum of decision-makers is rarely observed in practice and the approximation proposed in the asymptotic regime is meaningless for networked systems with few entities. In this paper we propose a mean-field framework that is suitable not only for large systems but also for a small world with few number of entities. The applicability of the proposed framework is illustrated through a dynamic auction with asymmetric valuation distributions.
Pedestrian Flow in the Mean Field Limit
Haji Ali, Abdul Lateef
2012-11-01
We study the mean-field limit of a particle-based system modeling the behavior of many indistinguishable pedestrians as their number increases. The base model is a modified version of Helbing\\'s social force model. In the mean-field limit, the time-dependent density of two-dimensional pedestrians satisfies a four-dimensional integro-differential Fokker-Planck equation. To approximate the solution of the Fokker-Planck equation we use a time-splitting approach and solve the diffusion part using a Crank-Nicholson method. The advection part is solved using a Lax-Wendroff-Leveque method or an upwind Backward Euler method depending on the advection speed. Moreover, we use multilevel Monte Carlo to estimate observables from the particle-based system. We discuss these numerical methods, and present numerical results showing the convergence of observables that were calculated using the particle-based model as the number of pedestrians increases to those calculated using the probability density function satisfying the Fokker-Planck equation.
Hassani, S Hamed; Urbanke, Ruediger
2011-01-01
We consider a collection of mean field spin systems, each of which is placed on the positions of a one-dimensional chain, coupled together by a Kac-type interaction along the chain. We analyze the simplest possible cases where the individual system is a Curie-Weiss model, possibly with a random field. We are interested in the regime where the size of each mean field model tends to infinity and, the length of the chain and range of the Kac interaction are large but finite. Below the critical temperature, there appears a series of equilibrium states representing kink-like interfaces between the two equilibrium states of the individual system. The van der Waals curve oscillates periodically around the Maxwell plateau. These oscillations have a period inversely proportional to the chain length and an amplitude exponentially small in the range of the interaction; in other words the spinodal points of the chain model lie exponentially close to the phase transition threshold. The amplitude of the oscillations is clo...
Harmonic bilocal fields generated by globally conformal invariant scalar fields
The twist two contribution in the operator product expansion of φ1(x1) φ2(x2) for a pair of globally conformal invariant, scalar fields of equal scaling dimension d in four space-time dimensions is a field V1(x1, x2) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of V1 can be equivalently characterized by a 'single-pole property' concerning the pole structure of the (rational) correlation functions involving the product φ1(x1) φ2(x2). This property is established for the dimension d = 2 of φ1, φ2. As an application we prove that any GCI scalar field of conformal dimension 2 (in four space-time dimensions) can be written as a (possibly infinite) superposition of products of free massless fields. (author)
Bilocal bosonization of QCD and electroweak properties of light pseudoscalar mesons
Quantum chromodynamic based analysis of the low energy electroweak properties of light pseudo-scalars is studied using an approximate bilocal bosonization technique. Particular attention is given to the problem of maintaining electroweak gauge invariance, and a bilocal Wilson-line technique is introduced to address this problem. The decay constants FK and Fπ and the π± charge radius are discussed in detail. 29 refs., 9 figs
The bilocated mind: new perspectives on self-localization and self-identification
Furlanetto, Tiziano; Bertone, Cesare; Becchio, Cristina
2013-01-01
Does the human mind allow for self-locating at more than one place at a time? Evidence from neurology, cognitive neuroscience, and experimental psychology suggests that mental bilocation is a complex, but genuine experience, occurring more frequently than commonly thought. In this article, we distinguish between different components of bilocated self-representation: self-localization in two different places at the same time, self-identification with another body, reduplication of first-person...
The bilocated mind: New perspectives on self-localization and self-identification
Cesare Bertone; Cristina Becchio
2013-01-01
Does the human mind allow for self-locating at more than one place at a time? Evidence from neurology, cognitive neuroscience, and experimental psychology suggests that mental bilocation is a complex, but genuine experience, occurring more frequently than commonly thought. In this article, we distinguish between different components of bilocated self-representation: self-localization in two different places at the same time, self-identification with another body, reduplication of first-person...
The bilocated mind: New perspectives on self-localization and self-identification
Tiziano eFurlanetto
2013-03-01
Full Text Available Does the human mind allow for self-locating at more than one place at a time? Evidence from neurology, cognitive neuroscience, and experimental psychology suggests that mental bilocation is a complex, but genuine experience, occurring more frequently than commonly thought. In this article, we distinguish between different components of bilocated self-representation: self-localization in two different places at the same time, self-identification with another body, reduplication of first-person perspective. We argue that different forms of mental bilocation may result from the combination of these components. To illustrate this, we discuss evidence of mental bilocation in pathological conditions such as heautoscopy, during immersion in virtual environments, and in everyday life, during social interaction. Finally, we consider the conditions for mental bilocation and speculate on the possible role of mental bilocation in the context of social interaction, suggesting that self-localization at two places at the same time may prove advantageous for the construction of a shared space.
Mean-field models for disordered crystals
Cancès, Eric; Lewin, Mathieu
2012-01-01
In this article, we set up a functional setting for mean-field electronic structure models of Hartree-Fock or Kohn-Sham types for disordered crystals. The electrons are quantum particles and the nuclei are classical point-like articles whose positions and charges are random. We prove the existence of a minimizer of the energy per unit volume and the uniqueness of the ground state density of such disordered crystals, for the reduced Hartree-Fock model (rHF). We consider both (short-range) Yukawa and (long-range) Coulomb interactions. In the former case, we prove in addition that the rHF ground state density matrix satisfies a self-consistent equation, and that our model for disordered crystals is the thermodynamic limit of the supercell model.
Mean field interaction in biochemical reaction networks
Tembine, Hamidou
2011-09-01
In this paper we establish a relationship between chemical dynamics and mean field game dynamics. We show that chemical reaction networks can be studied using noisy mean field limits. We provide deterministic, noisy and switching mean field limits and illustrate them with numerical examples. © 2011 IEEE.
Mean Field Games Models-A Brief Survey
Gomes, Diogo A.
2013-11-20
The mean-field framework was developed to study systems with an infinite number of rational agents in competition, which arise naturally in many applications. The systematic study of these problems was started, in the mathematical community by Lasry and Lions, and independently around the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhamé. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this paper we present a brief survey of mean-field models as well as recent results and techniques. In the first part of this paper, we study reduced mean-field games, that is, mean-field games, which are written as a system of a Hamilton-Jacobi equation and a transport or Fokker-Planck equation. We start by the derivation of the models and by describing some of the existence results available in the literature. Then we discuss the uniqueness of a solution and propose a definition of relaxed solution for mean-field games that allows to establish uniqueness under minimal regularity hypothesis. A special class of mean-field games that we discuss in some detail is equivalent to the Euler-Lagrange equation of suitable functionals. We present in detail various additional examples, including extensions to population dynamics models. This section ends with a brief overview of the random variables point of view as well as some applications to extended mean-field games models. These extended models arise in problems where the costs incurred by the agents depend not only on the distribution of the other agents, but also on their actions. The second part of the paper concerns mean-field games in master form. These mean-field games can be modeled as a partial differential equation in an infinite dimensional space. We discuss both deterministic models as well as problems where the agents are correlated. We end the paper with a mean-field model for price impact. © 2013 Springer Science+Business Media New York.
Back-reaction beyond the mean field approximation
A method for solving an initial value problem of a closed system consisting of an electromagnetic mean field and its quantum fluctuations coupled to fermions is presented. By tailoring the large Nf expansion method to the Schwinger-Keldysh closed time path (CTP) formulation of the quantum effective action, causality of the resulting equations of motion is ensured, and a systematic energy conserving and gauge invariant expansion about the electromagnetic mean field in powers of 1/Nf is developed. The resulting equations may be used to study the quantum nonequilibrium effects of pair creation in strong electric fields and the scattering and transport processes of a relativistic e+e- plasma. Using the Bjorken ansatz of boost invariance initial conditions in which the initial electric mean field depends on the proper time only, we show numerical results for the case in which the Nf expansion is truncated in the lowest order, and compare them with those of a phenomenological transport equation
A Maximum Principle for SDEs of Mean-Field Type
We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.
Nonequilibrium dynamical mean-field theory
Eckstein, Martin
2009-12-21
The aim of this thesis is the investigation of strongly interacting quantum many-particle systems in nonequilibrium by means of the dynamical mean-field theory (DMFT). An efficient numerical implementation of the nonequilibrium DMFT equations within the Keldysh formalism is provided, as well a discussion of several approaches to solve effective single-site problem to which lattice models such as the Hubbard-model are mapped within DMFT. DMFT is then used to study the relaxation of the thermodynamic state after a sudden increase of the interaction parameter in two different models: the Hubbard model and the Falicov-Kimball model. In the latter case an exact solution can be given, which shows that the state does not even thermalize after infinite waiting times. For a slow change of the interaction, a transition to adiabatic behavior is found. The Hubbard model, on the other hand, shows a very sensitive dependence of the relaxation on the interaction, which may be called a dynamical phase transition. Rapid thermalization only occurs at the interaction parameter which corresponds to this transition. (orig.)
Nonequilibrium dynamical mean-field theory
The aim of this thesis is the investigation of strongly interacting quantum many-particle systems in nonequilibrium by means of the dynamical mean-field theory (DMFT). An efficient numerical implementation of the nonequilibrium DMFT equations within the Keldysh formalism is provided, as well a discussion of several approaches to solve effective single-site problem to which lattice models such as the Hubbard-model are mapped within DMFT. DMFT is then used to study the relaxation of the thermodynamic state after a sudden increase of the interaction parameter in two different models: the Hubbard model and the Falicov-Kimball model. In the latter case an exact solution can be given, which shows that the state does not even thermalize after infinite waiting times. For a slow change of the interaction, a transition to adiabatic behavior is found. The Hubbard model, on the other hand, shows a very sensitive dependence of the relaxation on the interaction, which may be called a dynamical phase transition. Rapid thermalization only occurs at the interaction parameter which corresponds to this transition. (orig.)
Mean Field Theory for Sigmoid Belief Networks
Saul, L. K.; Jaakkola, T.; Jordan, M. I.
1996-01-01
We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition---the classification of handwritten digits.
Mean-field models and exotic nuclei
Bender, M.; Buervenich, T.; Maruhn, J.A.; Greiner, W. [Inst. fuer Theoretische Physik, Univ. Frankfurt (Germany); Rutz, K. [Inst. fuer Theoretische Physik, Univ. Frankfurt (Germany)]|[Gesellschaft fuer Schwerionenforschung mbH, Darmstadt (Germany); Reinhard, P.G. [Inst. fuer Theoretische Physik, Univ. Erlangen (Germany)
1998-06-01
We discuss two widely used nuclear mean-field models, the relativistic mean-field model and the (nonrelativistic) Skyrme-Hartree-Fock model, and their capability to describe exotic nuclei. Test cases are superheavy nuclei and neutron-rich Sn isotopes. New information in this regime helps to fix hitherto loosely determined aspects of the models. (orig.)
Condensates beyond mean field theory: quantum backreaction as decoherence
Vardi, Amichay; Anglin, James R.
2000-01-01
We propose an experiment to measure the slow log(N) convergence to mean-field theory (MFT) around a dynamical instability. Using a density matrix formalism, we derive equations of motion which go beyond MFT and provide accurate predictions for the quantum break-time. The leading quantum corrections appear as decoherence of the reduced single-particle quantum state.
Incorporating spatial correlations into multispecies mean-field models
Markham, Deborah C.; Simpson, Matthew J.; Maini, Philip K.; Gaffney, Eamonn A.; Baker, Ruth E.
2013-11-01
In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modeling interactions between such species, we often make use of the mean-field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean-field approximation is only used in appropriate settings. In circumstances where the mean-field approximation is unsuitable, we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper, we provide a method that overcomes many of the failures of the mean-field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multispecies case and show results specific to a two-species problem. We compare averaged discrete results to both the mean-field approximation and our improved method, which incorporates spatial correlations. We note that the mean-field approximation fails dramatically in some cases, predicting very different behavior from that seen upon averaging multiple realizations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behavior in all cases, thus providing a more reliable modeling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques.
Large amplitude motion with a stochastic mean-field approach
Yilmaz Bulent
2012-12-01
Full Text Available In the stochastic mean-field approach, an ensemble of initial conditions is considered to incorporate correlations beyond the mean-field. Then each starting point is propagated separately using the Time-Dependent Hartree-Fock equation of motion. This approach provides a rather simple tool to better describe fluctuations compared to the standard TDHF. Several illustrations are presented showing that this theory can be rather effective to treat the dynamics close to a quantum phase transition. Applications to fusion and transfer reactions demonstrate the great improvement in the description of mass dispersion.
General Relativistic Mean Field Theory for rotating nuclei
Madokoro, Hideki [Kyushu Univ., Fukuoka (Japan). Dept. of Physics; Matsuzaki, Masayuki
1998-03-01
The {sigma}-{omega} model Lagrangian is generalized to an accelerated frame by using the technique of general relativity which is known as tetrad formalism. We apply this model to the description of rotating nuclei within the mean field approximation, which we call General Relativistic Mean Field Theory (GRMFT) for rotating nuclei. The resulting equations of motion coincide with those of Munich group whose formulation was not based on the general relativistic transformation property of the spinor fields. Some numerical results are shown for the yrast states of the Mg isotopes and the superdeformed rotational bands in the A {approx} 60 mass region. (author)
Socio-economic applications of finite state mean field games
Gomes, Diogo
2014-10-06
In this paper, we present different applications of finite state mean field games to socio-economic sciences. Examples include paradigm shifts in the scientific community or consumer choice behaviour in the free market. The corresponding finite state mean field game models are hyperbolic systems of partial differential equations, for which we present and validate different numerical methods. We illustrate the behaviour of solutions with various numerical experiments,which show interesting phenomena such as shock formation. Hence, we conclude with an investigation of the shock structure in the case of two-state problems.
The finite temperature QED vacuum, within the mean field approximation
This report is made of two largely independent parts. The first one deals with a finite temperature generalization of a mean field theory which we already used for a study of the QED vacuum stability against the 2-body coulomb interaction. We show that temperature does not change the stability conditions, and we study the solutions of the gap equations, especially in the high temperature domain. The second part is devoted to photons rather than electrons; we evaluate the first vacuum polarization corrections to the properties of back-body radiation, considering the Euler-Heisenberg photon-photon interaction within the mean-field approximation
Bauso, Dario
2014-05-07
This article examines mean-field games for marriage. The results support the argument that optimizing the long-term well-being through effort and social feeling state distribution (mean-field) will help to stabilize marriage. However, if the cost of effort is very high, the couple fluctuates in a bad feeling state or the marriage breaks down. We then examine the influence of society on a couple using mean-field sentimental games. We show that, in mean-field equilibrium, the optimal effort is always higher than the one-shot optimal effort. We illustrate numerically the influence of the couple\\'s network on their feeling states and their well-being. © 2014 Bauso et al.
Mean field theory for long chain molecules
Pereira, Gerald G.
1996-06-01
We provide a mathematical formalism for a self-consistent mean field treatment of long chain molecules. The formalism is applied to the case of a neutral polymer under the excluded volume interaction. Upon scaling the problem in the N→∞ limit we find the natural scaling length RN, of the polymer, which is made up of (N+1) monomers or beads, is RN˜N3/5, the well known Flory result. The asymptotics of the problem is dominated by the neighborhood of the turning point, so that a uniformly valid Green's function solution of the differential equations is necessary. In the neighborhood of a point y* the scaled polymer density fN(x), is found to decay sharply. If we let x denote the scaled distance from one end of the chain to a point in space we obtain, for y*-x≳O(N-2/15), a closed form expression for the polymer density viz., fN(x)˜{1/2x2[fN(x)-fN(y*)]1/2} while for x-y*≳O(N-2/15) the density is shown to be, to leading order, zero. Although our results imply the rate of decay of the density at y* is O(N1/5) we are unable to verify this explicitly by calculating fN'(y*). We believe this is due to the inability of the WKB theory to correctly approximate solutions in regions of rapid variation. We suggest remedies for this, so that a complete self-consistent solution may be obtained.
Local excitations in mean field spin glasses
Krzakala, F.; G.PARISI()
2003-01-01
We address the question of geometrical as well as energetic properties of local excitations in mean field Ising spin glasses. We study analytically the Random Energy Model and numerically a dilute mean field model, first on tree-like graphs, equivalent to a replica symmetric computation, and then directly on finite connectivity random lattices. In the first model, characterized by a discontinuous replica symmetry breaking, we found that the energy of finite volume excitation is infinite where...
Mean-field approximation minimizes relative entropy
The authors derive the mean-field approximation from the information-theoretic principle of minimum relative entropy instead of by minimizing Peierls's inequality for the Weiss free energy of statistical physics theory. They show that information theory leads to the statistical mechanics procedure. As an example, they consider a problem in binary image restoration. They find that mean-field annealing compares favorably with the stochastic approach
Microscopically constrained mean-field models from chiral nuclear thermodynamics
Rrapaj, Ermal; Roggero, Alessandro; Holt, Jeremy W.
2016-06-01
We explore the use of mean-field models to approximate microscopic nuclear equations of state derived from chiral effective field theory across the densities and temperatures relevant for simulating astrophysical phenomena such as core-collapse supernovae and binary neutron star mergers. We consider both relativistic mean-field theory with scalar and vector meson exchange as well as energy density functionals based on Skyrme phenomenology and compare to thermodynamic equations of state derived from chiral two- and three-nucleon forces in many-body perturbation theory. Quantum Monte Carlo simulations of symmetric nuclear matter and pure neutron matter are used to determine the density regimes in which perturbation theory with chiral nuclear forces is valid. Within the theoretical uncertainties associated with the many-body methods, we find that select mean-field models describe well microscopic nuclear thermodynamics. As an additional consistency requirement, we study as well the single-particle properties of nucleons in a hot/dense environment, which affect e.g., charged-current weak reactions in neutron-rich matter. The identified mean-field models can be used across a larger range of densities and temperatures in astrophysical simulations than more computationally expensive microscopic models.
Band mixing effects in mean field theories
The 1/N expansion method, which is an angular momentum projected mean field theory, is used to investigate the nature of electromagnetic transitions in the interacting boson model (IBM). Conversely, comparison with the exact IBM results sheds light on the range of validity of the mean field theory. It is shown that the projected mean field results for the E2 transitions among the ground, β and γ bands are incomplete for the spin dependent terms and it is essential to include band mixing effect for a correct (Mikhailov) analysis of E2 data. The algebraic expressions derived are general and will be useful in the analysis of experimental data in terms of both the sd and sdg boson models. 17 refs., 7 figs., 8 tabs
Pion mean fields and heavy baryons
Yang, Ghil-Seok; Polyakov, Maxim V; Praszałowicz, Michał
2016-01-01
We show that the masses of the lowest-lying heavy baryons can be very well described in a pion mean-field approach. We consider a heavy baryon as a system consisting of the $N_c-1$ light quarks that induce the pion mean field, and a heavy quark as a static color source under the influence of this mean field. In this approach we derive a number of \\textit{model-independent} relations and calculate the heavy baryon masses using those of the lowest-lying light baryons as input. The results are in remarkable agreement with the experimental data. In addition, the mass of the $\\Omega_b^*$ baryon is predicted.
Mean-field magnetohydrodynamics and dynamo theory
Krause, F
2013-01-01
Mean-Field Magnetohydrodynamics and Dynamo Theory provides a systematic introduction to mean-field magnetohydrodynamics and the dynamo theory, along with the results achieved. Topics covered include turbulence and large-scale structures; general properties of the turbulent electromotive force; homogeneity, isotropy, and mirror symmetry of turbulent fields; and turbulent electromotive force in the case of non-vanishing mean flow. The turbulent electromotive force in the case of rotational mean motion is also considered. This book is comprised of 17 chapters and opens with an overview of the gen
Mean-field models and superheavy elements
We discuss the performance of two widely used nuclear mean-field models, the relativistic mean-field theory (RMF) and the non-relativistic Skyrme-Hartree-Fock approach (SHF), with particular emphasis on the description of superheavy elements (SHE). We provide a short introduction to the SHF and RMF, the relations between these two approaches and the relations to other nuclear structure models, briefly review the basic properties with respect to normal nuclear observables, and finally present and discuss recent results on the binding properties of SHE computed with a broad selection of SHF and RMF parametrisations. (orig.)
"Phase diagram" of a mean field game
Swiecicki, Igor; Gobron, Thierry; Ullmo, Denis
2016-01-01
Mean field games were introduced by J-M. Lasry and P-L. Lions in the mathematical community, and independently by M. Huang and co-workers in the engineering community, to deal with optimization problems when the number of agents becomes very large. In this article we study in detail a particular example called the "seminar problem" introduced by O. Guéant, J-M. Lasry, and P-L. Lions in 2010. This model contains the main ingredients of any mean field game but has the particular feature that all agents are coupled only through a simple random event (the seminar starting time) that they all contribute to form. In the mean field limit, this event becomes deterministic and its value can be fixed through a self consistent procedure. This allows for a rather thorough understanding of the solutions of the problem, through both exact results and a detailed analysis of various limiting regimes. For a sensible class of initial configurations, distinct behaviors can be associated to different domains in the parameter space. For this reason, the "seminar problem" appears to be an interesting toy model on which both intuition and technical approaches can be tested as a preliminary study toward more complex mean field game models.
A regularized stationary mean-field game
Yang, Xianjin
2016-04-19
In the thesis, we discuss the existence and numerical approximations of solutions of a regularized mean-field game with a low-order regularization. In the first part, we prove a priori estimates and use the continuation method to obtain the existence of a solution with a positive density. Finally, we introduce the monotone flow method and solve the system numerically.
Mean field methods for cortical network dynamics
Hertz, J.; Lerchner, Alexander; Ahmadi, M.
2004-01-01
We review the use of mean field theory for describing the dynamics of dense, randomly connected cortical circuits. For a simple network of excitatory and inhibitory leaky integrate- and-fire neurons, we can show how the firing irregularity, as measured by the Fano factor, increases with the...
Instabilities in the Mean Field Limit
Han-Kwan, Daniel; Nguyen, Toan T.
2016-03-01
Consider a system of N particles interacting through Newton's second law with Coulomb interaction potential in one spatial dimension or a {C}^2 smooth potential in any dimension. We prove that in the mean field limit N → + ∞, the N particles system displays instabilities in times of order log N, for some configurations approximately distributed according to unstable homogeneous equilibria.
Noise-induced behaviors in neural mean field dynamics
Touboul, Jonathan; Faugeras, Olivier
2011-01-01
The collective behavior of cortical neurons is strongly affected by the presence of noise at the level of individual cells. In order to study these phenomena in large-scale assemblies of neurons, we consider networks of firing-rate neurons with linear intrinsic dynamics and nonlinear coupling, belonging to a few types of cell populations and receiving noisy currents. Asymptotic equations as the number of neurons tends to infinity (mean field equations) are rigorously derived based on a probabilistic approach. These equations are implicit on the probability distribution of the solutions which generally makes their direct analysis difficult. However, in our case, the solutions are Gaussian, and their moments satisfy a closed system of nonlinear ordinary differential equations (ODEs), which are much easier to study than the original stochastic network equations, and the statistics of the empirical process uniformly converge towards the solutions of these ODEs. Based on this description, we analytically and numer...
Mean-field games with logistic population dynamics
Gomes, Diogo A.
2013-12-01
In its standard form, a mean-field game can be defined by coupled system of equations, a Hamilton-Jacobi equation for the value function of agents and a Fokker-Planck equation for the density of agents. Traditionally, the latter equation is adjoint to the linearization of the former. Since the Fokker-Planck equation models a population dynamic, we introduce natural features such as seeding and birth, and nonlinear death rates. In this paper we analyze a stationary meanfield game in one dimension, illustrating various techniques to obtain regularity of solutions in this class of systems. In particular we consider a logistic-type model for birth and death of the agents which is natural in problems where crowding affects the death rate of the agents. The introduction of these new terms requires a number of new ideas to obtain wellposedness. In a forthcoming publication we will address higher dimensional models. ©2013 IEEE.
Time-Dependent Mean-Field Games with Logarithmic Nonlinearities
Gomes, Diogo A.
2015-10-06
In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.
Ostwald ripening in Two Dimensions: Correlations and Scaling Beyond Mean Field
Levitan, Boris; Domany, Eytan
1997-01-01
We present a systematic quasi-mean field model of the Ostwald ripening process in two dimensions. Our approach yields a set of dynamic equations for the temporal evolution of the minority phase droplets' radii. The equations contain only pairwise interactions between the droplets; these interactions are evaluated in a mean- field type manner. We proceed to solve numerically the dynamic equations for systems of tens of thousands of interacting droplets. The numerical results are compared with ...
Mean field and collisions in hot nuclei
Collisions between heavy nuclei produce nuclear matter of high density and excitation. Brueckner methods are used to calculate the momentum and temperature dependent mean field for nucleons propagating through nuclear matter during these collisions. The mean field is complex and the imaginary part is related to the ''two-body'' collision, while the real part relates to ''one-body'' collisions. A potential model for the N-N interactions is avoided by calculating the Reaction matrix directly from the T-matrix (i.e., N-N phase shifts) using a version of Brueckner theory previously published by the author. Results are presented for nuclear matter at normal and twice normal density and for temperatures up to 50 MeV. 23 refs., 7 figs
Mean field methods for cortical network dynamics
Hertz, J.; Lerchner, Alexander; Ahmadi, M.
2004-01-01
We review the use of mean field theory for describing the dynamics of dense, randomly connected cortical circuits. For a simple network of excitatory and inhibitory leaky integrate- and-fire neurons, we can show how the firing irregularity, as measured by the Fano factor, increases with the stren......We review the use of mean field theory for describing the dynamics of dense, randomly connected cortical circuits. For a simple network of excitatory and inhibitory leaky integrate- and-fire neurons, we can show how the firing irregularity, as measured by the Fano factor, increases...... with the strength of the synapses in the network and with the value to which the membrane potential is reset after a spike. Generalizing the model to include conductance-based synapses gives insight into the connection between the firing statistics and the high- conductance state observed experimentally in visual...
Mean-field learning for satisfactory solutions
Tembine, Hamidou
2013-12-01
One of the fundamental challenges in distributed interactive systems is to design efficient, accurate, and fair solutions. In such systems, a satisfactory solution is an innovative approach that aims to provide all players with a satisfactory payoff anytime and anywhere. In this paper we study fully distributed learning schemes for satisfactory solutions in games with continuous action space. Considering games where the payoff function depends only on own-action and an aggregate term, we show that the complexity of learning systems can be significantly reduced, leading to the so-called mean-field learning. We provide sufficient conditions for convergence to a satisfactory solution and we give explicit convergence time bounds. Then, several acceleration techniques are used in order to improve the convergence rate. We illustrate numerically the proposed mean-field learning schemes for quality-of-service management in communication networks. © 2013 IEEE.
Mean Field Studies of Exotic Nuclei}
Chinn, C. R.; Umar, A. S.; Vallières, M.; Strayer, M. R.
1994-01-01
{Full three dimensional static and dynamic mean field calculations using collocation basis splines with a Skyrme type Hamiltonian are described. This program is developed to address the difficult theoretical challenges offered by exotic nuclei. Ground state and deformation properties are calculated using static Hartree-Fock, Hartree-Fock+BCS and constrained Hartree-Fock models. Collective properties, such as reaction rates and resonances, are described using a new alternate method for evaluat...
Mean-field cooperativity in chemical kinetics
Di Biasio, Aldo; Agliari, Elena; Barra, Adriano; Burioni, Raffaella
2011-01-01
We consider cooperative reactions and we study the effects of the interaction strength among the system components on the reaction rate, hence realizing a connection between microscopic and macroscopic observables. Our approach is based on statistical mechanics models and it is developed analytically via mean-field techniques. First of all, we show that, when the coupling strength is set positive, the model is able to consistently recover all the various cooperative measures previously introd...
'Phase diagram' of a mean field game
Swiecicki, Igor; Ullmo, Denis
2015-01-01
Mean field games were introduced by J-M.Lasry and P-L. Lions in the mathematical community, and independently by M. Huang and co-workers in the engineering community, to deal with optimization problems when the number of agents becomes very large. In this article we study in detail a particular example called the 'seminar problem' introduced by O.Gu\\'eant, J-M Lasry, and P-L. Lions in 2010. This model contains the main ingredients of any mean field game but has the particular feature that all agent are coupled only through a simple random event (the seminar starting time) that they all contribute to form. In the mean field limit, this event becomes deterministic and its value can be fixed through a self consistent procedure. This allows for a rather thorough understanding of the solutions of the problem, through both exact results and a detailed analysis of various limiting regimes. For a sensible class of initial configurations, distinct behaviors can be associated to different domains in the parameter space...
Mean-field theory of a recurrent epidemiological model
Nagy, Viktor
2009-06-01
Our purpose is to provide a mean-field theory for the discrete time-step susceptible-infected-recovered-susceptible (SIRS) model on uncorrelated networks with arbitrary degree distributions. The effect of network structure, time delays, and infection rate on the stability of oscillating and fixed point solutions is examined through analysis of discrete time mean-field equations. Consideration of two scenarios for disease contagion demonstrates that the manner in which contagion is transmitted from an infected individual to a contacted susceptible individual is of primary importance. In particular, the manner of contagion transmission determines how the degree distribution affects model behavior. We find excellent agreement between our theoretical results and numerical simulations on networks with large average connectivity.
Mean-field theory of echo state networks
Massar, Marc; Massar, Serge
2013-04-01
Dynamical systems driven by strong external signals are ubiquitous in nature and engineering. Here we study “echo state networks,” networks of a large number of randomly connected nodes, which represent a simple model of a neural network, and have important applications in machine learning. We develop a mean-field theory of echo state networks. The dynamics of the network is captured by the evolution law, similar to a logistic map, for a single collective variable. When the network is driven by many independent external signals, this collective variable reaches a steady state. But when the network is driven by a single external signal, the collective variable is non stationary but can be characterized by its time averaged distribution. The predictions of the mean-field theory, including the value of the largest Lyapunov exponent, are compared with the numerical integration of the equations of motion.
Derivation of mean-field dynamics for fermions
In this work, we derive the time-dependent Hartree(-Fock) equations as an effective dynamics for fermionic many-particle systems. Our main results are the first for a quantum mechanical mean-field dynamics for fermions; in previous works, the mean-field limit is usually either coupled to a semiclassical limit, or the interaction is scaled down so much, that the system behaves freely for large particle number N. We mainly consider systems with total kinetic energy bounded by const.N and long-range interaction potentials, e.g., Coulomb interaction. Examples for such systems are large molecules or certain solid states. Our analysis also applies to attractive interactions, as, e.g., in fermionic stars. The fermionic Hartree(-Fock) equations are a standard tool to describe, e.g., excited states or chemical reactions of large molecules (like proteins). A deeper understanding of these equations as an approximation to the time evolution of a many body quantum system is thus highly relevant. We consider the fermionic Hartree equations (i.e., the Hartree-Fock equations without exchange term) in this work, since the exchange term is subleading in our setting. The main result is that the fermionic Hartree dynamics approximates the Schroedinger dynamics well for large N. This statement becomes exact in the thermodynamic limit N→∞. We give explicit values for the rates of convergence. We prove two types of results. The first type is very general and concerns arbitrary free Hamiltonians (e.g., relativistic, non-relativistic, with external fields) and arbitrary interactions. The theorems give explicit conditions on the solutions to the fermionic Hartree equations under which a derivation of the mean-field dynamics succeeds. The second type of results scrutinizes situations where the conditions are fulfilled. These results are about non-relativistic free Hamiltonians with external fields, systems with total kinetic energy bounded by const.N and with long-range interactions of
Numerical accuracy of mean-field calculations in coordinate space
Ryssens, W; Heenen, P -H
2015-01-01
Background: Mean-field methods based on an energy density functional (EDF) are powerful tools used to describe many properties of nuclei in the entirety of the nuclear chart. The accuracy required on energies for nuclear physics and astrophysics applications is of the order of 500 keV and much effort is undertaken to build EDFs that meet this requirement. Purpose: The mean-field calculations have to be accurate enough in order to preserve the accuracy of the EDF. We study this numerical accuracy in detail for a specific numerical choice of representation for the mean-field equations that can accommodate any kind of symmetry breaking. Method: The method that we use is a particular implementation of 3-dimensional mesh calculations. Its numerical accuracy is governed by three main factors: the size of the box in which the nucleus is confined, the way numerical derivatives are calculated and the distance between the points on the mesh. Results: We have examined the dependence of the results on these three factors...
Control and Nash Games with Mean Field Effect
Alain BENSOUSSAN; Jens FREHSE
2013-01-01
Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007,of Gueant-Lasry-Lions in 2011,of HuangCaines-Malham in 2007 and many others).There are a lot of applications.In general,the applications concern approximating an infinite number of players with common behavior by a representative agent.This agent has to solve a control problem perturbed by a field equation,representing in some way the behavior of the average infinite number of agents.This approach does not lead easily to the problems of Nash equilibrium for a finite number of players,perturbed by field equations,unless one considers averaging within different groups,which has not been done in the literature,and seems quite challenging.In this paper,the authors approach similar problems with a different motivation which makes sense for control and also for differential games.Thus the systems of nonlinear partial differential equations with mean field terms,which have not been addressed in the literature so far,are considered here.
Mean-field theory and self-consistent dynamo modeling
Mean-field theory of dynamo is discussed with emphasis on the statistical formulation of turbulence effects on the magnetohydrodynamic equations and the construction of a self-consistent dynamo model. The dynamo mechanism is sought in the combination of the turbulent residual-helicity and cross-helicity effects. On the basis of this mechanism, discussions are made on the generation of planetary magnetic fields such as geomagnetic field and sunspots and on the occurrence of flow by magnetic fields in planetary and fusion phenomena. (author)
Dynamic Programming for Mean-field type Control
Laurière, Mathieu; Pironneau, Olivier
2014-01-01
For mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF $\\rho$ of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function $V(\\tau,\\rho_\\tau)$ where the dependency on $\\rho_\\tau$ is functional as in P.L. Lions' analysis of mean-filed games (2007). We derive HJB equations and apply them to two examples, a...
Spin and orbital exchange interactions from Dynamical Mean Field Theory
Secchi, A.; Lichtenstein, A. I.; Katsnelson, M. I.
2016-02-01
We derive a set of equations expressing the parameters of the magnetic interactions characterizing a strongly correlated electronic system in terms of single-electron Green's functions and self-energies. This allows to establish a mapping between the initial electronic system and a spin model including up to quadratic interactions between the effective spins, with a general interaction (exchange) tensor that accounts for anisotropic exchange, Dzyaloshinskii-Moriya interaction and other symmetric terms such as dipole-dipole interaction. We present the formulas in a format that can be used for computations via Dynamical Mean Field Theory algorithms.
Mean-field theory and self-consistent dynamo modeling
Yoshizawa, Akira; Yokoi, Nobumitsu [Tokyo Univ. (Japan). Inst. of Industrial Science; Itoh, Sanae-I [Kyushu Univ., Fukuoka (Japan). Research Inst. for Applied Mechanics; Itoh, Kimitaka [National Inst. for Fusion Science, Toki, Gifu (Japan)
2001-12-01
Mean-field theory of dynamo is discussed with emphasis on the statistical formulation of turbulence effects on the magnetohydrodynamic equations and the construction of a self-consistent dynamo model. The dynamo mechanism is sought in the combination of the turbulent residual-helicity and cross-helicity effects. On the basis of this mechanism, discussions are made on the generation of planetary magnetic fields such as geomagnetic field and sunspots and on the occurrence of flow by magnetic fields in planetary and fusion phenomena. (author)
Mean-field approach for diffusion of interacting particles.
Suárez, G; Hoyuelos, M; Mártin, H
2015-12-01
A nonlinear Fokker-Planck equation is obtained in the continuous limit of a one-dimensional lattice with an energy landscape of wells and barriers. Interaction is possible among particles in the same energy well. A parameter γ, related to the barrier's heights, is introduced. Its value is determinant for the functional dependence of the mobility and diffusion coefficient on particle concentration, but has no influence on the equilibrium solution. A relation between the mean-field potential and the microscopic interaction energy is derived. The results are illustrated with classical particles with interactions that reproduce fermion and boson statistics. PMID:26764643
Superheavy Nuclei: Relativistic Mean Field Outlook
Afanasjev, A V
2006-01-01
The analysis of quasiparticle spectra in heaviest $A\\sim 250$ nuclei with spectroscopic data provides an additional constraint for the choice of effective interaction for the description of superheavy nuclei. It strongly suggest that only the parametrizations of the relativistic mean field Lagrangian which predict Z=120 and N=172 as shell closures are reliable for superheavy nuclei. The influence of the central depression in the density distribution of spherical superheavy nuclei on the shell structure is studied. Large central depression produces large shell gaps at Z=120 and N=172. The shell gaps at Z=126 and N=184 are favored by a flat density distribution in the central part of nucleus. It is shown that approximate particle number projection (PNP) by means of the Lipkin-Nogami method removes pairing collapse seen at these gaps in the calculations without PNP.
Superheavy nuclei: a relativistic mean field outlook
The analysis of quasi-particle spectra in the heaviest A∼250 nuclei with spectroscopic data provides an additional constraint for the choice of effective interaction for the description of superheavy nuclei. It strongly suggests that only the parametrizations which predict Z = 120 and N = 172 as shell closures are reliable for superheavy nuclei within the relativistic mean field theory. The influence of the central depression in the density distribution of spherical superheavy nuclei on the shell structure is studied. A large central depression produces large shell gaps at Z = 120 and N = 172. The shell gaps at Z = 126 and N = 184 are favoured by a flat density distribution in the central part of the nucleus. It is shown that approximate particle number projection (PNP) by means of the Lipkin-Nogami (LN) method removes pairing collapse seen at these gaps in the calculations without PNP
Mass Predictions from Mean-Field Calculations
Several methods based on effective interactions or Lagrangians are available today. Although different in many respects (use of zero range or finite range interactions, relativistic or non relativistic framework, different treatments of pairing correlations), their applications to nuclei far from stability have shown converging results which still have to be incorporated in macroscopic approaches. Many efforts are also actually devoted to the improvements of the effective interactions, especially of the pairing force. Finally, developments are performed to include in a microscopic framework correlations beyond a mean-field (in particular, the correlations generated by rotation and vibration in the deformed nuclear potential). I shall review some key aspects of these developments and show how they affect the determination of nuclear masses in particular at the limits of stability
Invisible dynamo in mean-field models
Reshetnyak, M. Yu.
2016-07-01
The inverse problem in a spherical shell to find the two-dimensional spatial distributions of the α-effect and differential rotation in a mean-field dynamo model has been solved. The derived distributions lead to the generation of a magnetic field concentrated inside the convection zone. The magnetic field is shown to have no time to rise from the region of maximum generation located in the lower layers to the surface in the polarity reversal time due to magnetic diffusion. The ratio of the maximum magnetic energy in the convection zone to its value at the outer boundary reaches two orders of magnitude or more. This result is important in interpreting the observed stellar and planetary magnetic fields. The proposed method of solving the inverse nonlinear dynamo problem is easily adapted for a wide class of mathematical-physics problems.
Kinetic and mean field description of Gibrat's law
Toscani, Giuseppe
2016-01-01
We introduce and analyze a linear kinetic model that describes the evolution of the probability density of the number of firms in a society, in which the microscopic rate of change obeys to the so-called law of proportional effect proposed by Gibrat. Despite its apparent simplicity, the possible mean field limits of the kinetic model are varied. In some cases, the asymptotic limit can be described by a first-order partial differential equation. In other cases, the mean field equation is a linear diffusion with a non constant diffusion coefficient that models also the geometric Brownian motion and can be studied analytically. In this case, it is shown that the large-time behavior of the solution is represented, for a large class of initial data, by a lognormal distribution with constant mean value and variance increasing exponentially in time at a precise rate. The relationship between the kinetic and the diffusion models allow to introduce an easy-to- implement expression for computing the Fourier transform o...
Nonlinear regimes in mean-field full-sphere dynamo
Pipin, V V
2016-01-01
The mean-field dynamo model is employed to study the non-linear dynamo regimes in a fully convective star of mass 0.3$M_{\\odot}$ rotating with period of 10 days. The differential rotation law was estimated using the mean-field hydrodynamic and heat transport equations. For the intermediate parameter of the turbulent magnetic Reynolds number, $Pm_{T}=3$ we found the oscillating dynamo regimes with period about 40Yr. The higher $Pm_{T}$ results to longer dynamo periods. The meridional circulation has one cell per hemisphere. It is counter-clockwise in the Northen hemisphere. The amplitude of the flow at the surface around 1 m/s. Tne models with regards for meridional circulation show the anti-symmetric relative to equator magnetic field. If the large-scale flows is fixed we find that the dynamo transits from axisymmetric to non-axisymmetric regimes for the overcritical parameter of the $\\alpha$effect. The change of dynamo regime occurs because of the non-axisymmetric non-linear $\\alpha$-effect. The situation pe...
Deterministic Mean-Field Ensemble Kalman Filtering
Law, Kody J. H.
2016-05-03
The proof of convergence of the standard ensemble Kalman filter (EnKF) from Le Gland, Monbet, and Tran [Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford University Press, Oxford, UK, 2011, pp. 598--631] is extended to non-Gaussian state-space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence k between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for dimension d<2k. The fidelity of approximation of the true distribution is also established using an extension of the total variation metric to random measures. This is limited by a Gaussian bias term arising from nonlinearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory.
A Mean-Field Description for AdS Black Hole
Dutta, Suvankar; P, Sachin Shain
2016-01-01
In this paper we find an equivalent mean-field description for asymptotically $AdS$ black hole in high temperature limit and in arbitrary dimensions. We obtain a class of mean-field potential for which the description is valid. We explicitly show that there is an one to one correspondence between the thermodynamics of a gas of interacting particles moving under a mean-field potential and an $AdS$ black hole, namely the equation of state, temperature, pressure, entropy and enthalpy of both the...
Electrical Vehicles in the Smart Grid: A Mean Field Game Analysis
Couillet, Romain; Medina Perlaza, Samir; Tembine, Hamidou; Debbah, Mérouane
2012-01-01
In this article, we investigate the competitive interaction between electrical vehicles or hybrid oil-electricity vehicles in a Cournot market consisting of electricity transactions to or from an underlying electricity distribution network. We provide a mean field game formulation for this competition, and introduce the set of fundamental differential equations ruling the behavior of the vehicles at the feedback Nash equilibrium, referred here to as the mean field equilibrium. This framework ...
Mean field theory for non-abelian gauge theories and fluid dynamics. A brief progress report
We review the long standing problem of 'mean field theory' for non-abelian gauge theories. As a consequence of the AdS/CFT correspondence, in the large N limit, at strong coupling, and high temperatures and density, the 'mean field theory' is described by the Navier-Stokes equations of fluid dynamics. We also discuss and present results on the non-conformal fluid dynamics of the D1 brane in 1+1 dim. (author)
Mean Field Variational Approximation for Continuous-Time Bayesian Networks
Cohn, Ido; Friedman, Nir; Kupferman, Raz
2012-01-01
Continuous-time Bayesian networks is a natural structured representation language for multicomponent stochastic processes that evolve continuously over time. Despite the compact representation, inference in such models is intractable even in relatively simple structured networks. Here we introduce a mean field variational approximation in which we use a product of inhomogeneous Markov processes to approximate a distribution over trajectories. This variational approach leads to a globally consistent distribution, which can be efficiently queried. Additionally, it provides a lower bound on the probability of observations, thus making it attractive for learning tasks. We provide the theoretical foundations for the approximation, an efficient implementation that exploits the wide range of highly optimized ordinary differential equations (ODE) solvers, experimentally explore characterizations of processes for which this approximation is suitable, and show applications to a large-scale realworld inference problem.
Thermal Effects in Dense Matter Beyond Mean Field Theory
Constantinou, Constantinos; Prakash, Madappa
2016-01-01
The formalism of next-to-leading order Fermi Liquid Theory is employed to calculate the thermal properties of symmetric nuclear and pure neutron matter in a relativistic many-body theory beyond the mean field level which includes two-loop effects. For all thermal variables, the semi-analytical next-to-leading order corrections reproduce results of the exact numerical calculations for entropies per baryon up to 2. This corresponds to excellent agreement down to sub-nuclear densities for temperatures up to $20$ MeV. In addition to providing physical insights, a rapid evaluation of the equation of state in the homogeneous phase of hot and dense matter is achieved through the use of the zero-temperature Landau effective mass function and its derivatives.
Classification of networks of automata by dynamical mean field theory
Dynamical mean field theory is used to classify the 224=65,536 different networks of binary automata on a square lattice with nearest neighbour interactions. Application of mean field theory gives 700 different mean field classes, which fall in seven classes of different asymptotic dynamics characterized by fixed points and two-cycles. (orig.)
Dynamical mean field model of a neural-glial mass.
Sotero, Roberto C; Martínez-Cancino, Ramón
2010-04-01
Our goal is to model the behavior of an ensemble of interacting neurons and astrocytes (the neural-glial mass). For this, a model describing N tripartite synapses is proposed. Each tripartite synapse consists of presynaptic and postsynaptic nerve terminals, as well as the synaptically associated astrocytic microdomain, and is described by a system of 13 stochastic differential equations. Then, by applying the dynamical mean field approximation (DMA) (Hasegawa, 2003a , 2003b ) the system of 13N equations is reduced to 13(13 + 2) = 195 deterministic differential equations for the means and the second-order moments of local and global variables. Simulations are carried out for studying the response of the neural-glial mass to external inputs applied to either the presynaptic terminals or the astrocytes. Three cases were considered: the astrocytes influence only the presynaptic terminal, only the postsynaptic terminal, or both the presynaptic and postsynaptic terminals. As a result, a wide range of responses varying from singles spikes to train of spikes was evoked on presynaptic and postsynaptic terminals. The experimentally observed phenomenon of spontaneous activity in astrocytes was replicated on the neural-glial mass. The model predicts that astrocytes can have a strong and activity-dependent influence on synaptic transmission. Finally, simulations show that the dynamics of astrocytes influences the synchronization ratio between neurons, predicting a peak in the synchronization for specific values of the astrocytes' parameters. PMID:20028223
Evolution of primordial magnetic fields in mean-field approximation
Campanelli, Leonardo
2013-01-01
We study the evolution of phase-transition-generated cosmic magnetic fields coupled to the primeval cosmic plasma in turbulent and viscous free-streaming regimes. The evolution laws for the magnetic energy density and correlation length, both in helical and non-helical cases, are found by solving the autoinduction and Navier-Stokes equations in mean-field approximation. Analytical results are derived in Minkowski spacetime and then extended to the case of a Friedmann universe with zero spatial curvature, both in radiation and matter dominated eras. The three possible viscous free-streaming phases are characterized by a drag term in the Navier-Stokes equation which depends on the free-steaming properties of neutrinos, photons, or hydrogen atoms, respectively. In the case of non-helical magnetic fields, the magnetic intensity $B$ and the magnetic correlation length $\\xi_B$ evolve asymptotically with the temperature $T$ as $B(T) \\simeq \\kappa_B (N_i v_i)^{\\varrho_1} (T/T_i)^{\\varrho_2}$ and $\\xi_B(T) \\simeq \\kap...
A Mean-Field Description for AdS Black Hole
Dutta, Suvankar
2016-01-01
In this paper we find an equivalent mean-field description for asymptotically $AdS$ black hole in high temperature limit and in arbitrary dimensions. We obtain a class of mean-field potential for which the description is valid. We explicitly show that there is an one to one correspondence between the thermodynamics of a gas of interacting particles moving under a mean-field potential and an $AdS$ black hole, namely the equation of state, temperature, pressure, entropy and enthalpy of both the systems match. In $3+1$ dimensions, in particular, the mean-field description can be thought of as an ensemble of tiny interacting {\\it asymptotically flat} black holes moving in volume $V$ and at temperature $T$. This motivates us to identify these asymptotically flat black holes as microstructure of asymptotically $AdS$ black holes in $3+1$ dimensions.
Self-consistent mean field forces in turbulent plasmas: Current and momentum relaxation
The properties of turbulent plasmas are described using the two-fluid equations. Under some modest assumptions, global constraints for the turbulent mean field forces that act on the ion and electron fluids are derived. These constraints imply a functional form for the parallel mean field forces in the Ohm's law and the momentum balance equation. These forms suggest that the fluctuations attempt to relax the plasma to a state where both the current and the bulk plasma momentum are aligned along the mean magnetic field with proportionality constants that are global constants. Observations of flow profile evolution during discrete dynamo activity in reversed field pinch experiments are interpreted
Time-Dependent Mean-Field Games in the Subquadratic Case
Gomes, Diogo A.
2014-10-14
In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of time-dependent mean-field games.
Mean-field theory of atomic self-organization in optical cavities
Jäger, Simon B.; Schütz, Stefan; Morigi, Giovanna
2016-08-01
Photons mediate long-range optomechanical forces between atoms in high-finesse resonators, which can induce the formation of ordered spatial patterns. When a transverse laser drives the atoms, the system undergoes a second-order phase transition that separates a uniform spatial density from a Bragg grating maximizing scattering into the cavity and is controlled by the laser intensity. Starting from a Fokker-Planck equation describing the semiclassical dynamics of the N -atom distribution function, we systematically develop a mean-field model and analyze its predictions for the equilibrium and out-of-equilibrium dynamics. The validity of the mean-field model is tested by comparison with the numerical simulations of the N -body Fokker-Planck equation and by means of a Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The mean-field theory predictions well reproduce several results of the N -body Fokker-Planck equation for sufficiently short times and are in good agreement with existing theoretical approaches based on field-theoretical models. The mean field, on the other hand, predicts thermalization time scales which are at least one order of magnitude shorter than the ones predicted by the N -body dynamics. We attribute this discrepancy to the fact that the mean-field ansatz discards the effects of the long-range incoherent forces due to cavity losses.
Beyond mean field theory: statistical field theory for neural networks
Mean field theories have been a stalwart for studying the dynamics of networks of coupled neurons. They are convenient because they are relatively simple and possible to analyze. However, classical mean field theory neglects the effects of fluctuations and correlations due to single neuron effects. Here, we consider various possible approaches for going beyond mean field theory and incorporating correlation effects. Statistical field theory methods, in particular the Doi–Peliti–Janssen formalism, are particularly useful in this regard. (paper)
Lerchner, A; Hertz, J; Ahmadi, M
2004-01-01
We present a complete mean field theory for a balanced state of a simple model of an orientation hypercolumn. The theory is complemented by a description of a numerical procedure for solving the mean-field equations quantitatively. With our treatment, we can determine self-consistently both the firing rates and the firing correlations, without being restricted to specific neuron models. Here, we solve the analytically derived mean-field equations numerically for integrate-and-fire neurons. Several known key properties of orientation selective cortical neurons emerge naturally from the description: Irregular firing with statistics close to -- but not restricted to -- Poisson statistics; an almost linear gain function (firing frequency as a function of stimulus contrast) of the neurons within the network; and a contrast-invariant tuning width of the neuronal firing. We find that the irregularity in firing depends sensitively on synaptic strengths. If Fano factors are bigger than 1, then they are so for all stim...
Lerchner, Alexander; Sterner, G.; Hertz, J.;
2006-01-01
We present a complete mean field theory for a balanced state of a simple model of an orientation hypercolumn, with a numerical procedure for solving the mean-field equations quantitatively. With our treatment, one can determine self-consistently both the firing rates and the firing correlations......, without being restricted to specific neuron models. Here, we solve the mean-field equations numerically for integrate-and-fire neurons. Several known key properties of orientation selective cortical neurons emerge naturally from the description: Irregular firing with statistics close to - but not...... restricted to Poisson statistics; an almost linear gain function (firing frequency as a function of stimulus contrast) of the neurons within the network; and a contrast-invariant tuning width of the neuronal firing. We find that the irregularity in firing depends sensitively on synaptic strengths. If the...
Exact mean field dynamics for epidemic-like processes on heterogeneous networks
Lucas, Andrew
2012-01-01
We show that the mean field equations for the SIR epidemic can be exactly solved for a network with arbitrary degree distribution. Our exact solution consists of reducing the dynamics to a lone first order differential equation, which has a solution in terms of an integral over functions dependent on the degree distribution of the network, and reconstructing all mean field functions of interest from this integral. Irreversibility of the SIR epidemic is crucial for the solution. We also find exact solutions to the sexually transmitted disease SI epidemic on bipartite graphs, to a simplified rumor spreading model, and to a new model for recommendation spreading, via similar techniques. Numerical simulations of these processes on scale free networks demonstrate the qualitative validity of mean field theory in most regimes.
Development of mean field theories in nuclear physics and in desordered media
This work, in two parts, deals with the development of mean field theories in nuclear physics (nuclei in balance and collisions of heavy ions) as well as in disordered media. In the first part, two different ways of tackling the problem of developments around mean field theories are explained. Possessing an approach wave function for the system, the natural idea for including the correlations is to develop the exact wave function of the system around the mean field wave function. The first two chapters show two different ways of dealing with this problem: the perturbative approach - Hartree-Fock equations with two body collisions and functional methods. In the second part: mean field theory for spin glasses. The problem for spin glasses is to construct a physically acceptable mean field theory. The importance of this problem in statistical mechanics is linked to the fact that the mean field theory provides a qualitative description of the low temperature phase and is the starting point needed for using more sophisticated methods (renormalization group). Two approaches to this problem are presented, one based on the Sherrington-Kirkpatrick model and the other based on a model of spins with purely local disorder and competitive interaction between the spins
Hot and dense matter beyond relativistic mean field theory
Zhang, Xilin
2016-01-01
Properties of hot and dense matter are calculated in the framework of quantum hadro-dynamics by including contributions from two-loop (TL) diagrams arising from the exchange of iso-scalar and iso-vector mesons between nucleons. Our extension of mean-field theory (MFT) employs the same five density-independent coupling strengths which are calibrated using the empirical properties at the equilibrium density of iso-spin symmetric matter. Results of calculations from the MFT and TL approximations are compared for conditions of density, temperature, and proton fraction encountered in astrophysics applications involving compact objects. The TL results for the equation of state (EOS) of cold pure neutron matter at sub- and near-nuclear densities agree well with those of modern quantum Monte Carlo and effective field-theoretical approaches. Although the high-density EOS in the TL approximation for neutron-star matter is substantially softer than its MFT counterpart, it is able to support a $2M_\\odot$ neutron star req...
Mean-field theory of four species in cyclic competition
Durney, C. H.; Case, S. O.; Pleimling, M.; Zia, R. K. P.
2011-03-01
We consider a simple model of cyclic competition of M species: When a pair of individuals from species k and k + 1 interact, the latter transforms into the former. Even with no spatial structure, such systems often display interesting and counterintuitive behavior. With possible applications in both biological systems (e.g., Min proteins, E. Coli, lizards) and game theory (e.g., rock-paper-scissors), the M = 3 case has attracted considerable recent attention. We study a M = 4 system (with no spatial structure) and find major differences, e.g., (1) the presence of macroscopically many absorbing states, (2) coexistence of species, and (3) violation of the ``law'' of survival of the weakest - a central theme in the M = 3 case. Like the game of Bridge, the system typically ends with ``partner pairs.'' After describing the full stochastic model and its master equation, we present the mean-field approximation. Several exact, analytic predictions will be shown. Their limitations and implications for the stochastic system will also be discussed. Supported in part by NSF-DMR-0705152, 0904999, 1005417.
Mean field theory of self-organized critical phenomena
A mean field theory is presented for the recently discovered self-organized critical phenomena. The critical exponents are calculated and found to be the same as the mean field values for percolation. The power spectrum has 1/f behavior with exponent Phi = 1
Verbalization of Mean Field Utterances in German Instructions
Tayupova O. I.
2013-01-01
Full Text Available The article investigates ways of actualization of mean field utterances used in modern German instructions considering the type of the text. The author determines and analyzes similarities and differences in linguistic means used in mean field utterances in the context of such text subtypes as instructions to household appliances, cosmetic products directions and prescribing information for pharmaceutical drugs use.
Evolution of primordial magnetic fields in mean-field approximation
Campanelli, Leonardo
2014-01-01
We study the evolution of phase-transition-generated cosmic magnetic fields coupled to the primeval cosmic plasma in the turbulent and viscous free-streaming regimes. The evolution laws for the magnetic energy density and the correlation length, both in the helical and the non-helical cases, are found by solving the autoinduction and Navier-Stokes equations in the mean-field approximation. Analytical results are derived in Minkowski spacetime and then extended to the case of a Friedmann universe with zero spatial curvature, both in the radiation- and the matter-dominated era. The three possible viscous free-streaming phases are characterized by a drag term in the Navier-Stokes equation which depends on the free-streaming properties of neutrinos, photons, or hydrogen atoms, respectively. In the case of non-helical magnetic fields, the magnetic intensity and the magnetic correlation length evolve asymptotically with the temperature, , as and . Here, , , and are, respectively, the temperature, the number of magnetic domains per horizon length, and the bulk velocity at the onset of the particular regime. The coefficients , , , , , and , depend on the index of the assumed initial power-law magnetic spectrum, , and on the particular regime, with the order-one constants and depending also on the cutoff adopted for the initial magnetic spectrum. In the helical case, the quasi-conservation of the magnetic helicity implies, apart from logarithmic corrections and a factor proportional to the initial fractional helicity, power-like evolution laws equal to those in the non-helical case, but with equal to zero.
Statistical thermodynamics and mean-field theory for the alloy under irradiation model
A generalization of statistical thermodynamics to the open systems case, is discussed, using as an example the alloy-under-irradiation model. The statistical properties of stationary states are described with the use of generalized thermodynamic potentials and 'quasi-interactions' determined from the master equation for micro-configuration probabilities. Methods for resolving this equation are illustrated by the mean-field type calculations of correlators, thermodynamic potentials and phase diagrams for disordered alloys
Expansion Around the Mean-Field Solution of the Bak-Sneppen Model
Marsili, Matteo; Rios, Paolo De Los; Maslov, Sergei
1997-01-01
We study a recently proposed equation for the avalanche distribution in the Bak-Sneppen model. We demonstrate that this equation indirectly relates $\\tau$,the exponent for the power law distribution of avalanche sizes, to $D$, the fractal dimension of an avalanche cluster.We compute this relation numerically and approximate it analytically up to the second order of expansion around the mean field exponents. Our results are consistent with Monte Carlo simulations of Bak-Sneppen model in one an...
An [imaginary time] Schr\\"odinger approach to mean field games
Swiecicki, Igor; Ullmo, Denis
2015-01-01
Mean Field Games (MFG) provide a theoretical frame to model socio-economic systems. In this letter, we study a particular class of MFG which shows strong analogies with the {\\em non-linear Schr\\"odinger and Gross-Pitaevski equations} introduced in physics to describe a variety of physical phenomena ranging from deep-water waves to interacting bosons. Using this bridge many results and techniques developed along the years in the latter context can be transferred to the former. As an illustration, we study in some details an example in which the "players" in the mean field game are under a strong incentive to coordinate themselves.
Single-particle potential in a relativistic Hartree-Fock mean field approximation
Jaminon, Martine; Mahaux, Claude; Rochus, Pierre
1981-01-01
A relativistic Hartree-Fock mean field approximation is investigated in a model in which the nucléon field interacts with scalar and vector meson fields. The Hartree-Fock potential felt by individual nucléons enters in a relativistic Dirac single-particle equation. It is shown that in the case of symmetric nuclear matter one can always find a potential which is fully equivalent to the most general mean field and which is only the sum of a Lorentz scalar, of one component of a Lorentz tensor a...
Non-equilibrium mean-field theories on scale-free networks
Many non-equilibrium processes on scale-free networks present anomalous critical behavior that is not explained by standard mean-field theories. We propose a systematic method to derive stochastic equations for mean-field order parameters that implicitly account for the degree heterogeneity. The method is used to correctly predict the dynamical critical behavior of some binary spin models and reaction–diffusion processes. The validity of our non-equilibrium theory is further supported by showing its relation with the generalized Landau theory of equilibrium critical phenomena on networks
On the convergence of finite state mean-field games through Γ-convergence
Ferreira, Rita C.
2014-10-01
In this study, we consider the long-term convergence (trend toward an equilibrium) of finite state mean-field games using Γ-convergence. Our techniques are based on the observation that an important class of mean-field games can be viewed as the Euler-Lagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a long-term convergence problem into a Γ-convergence problem. Our results generalize previous results related to long-term convergence for finite state problems. © 2014 Elsevier Inc.
Quantum correlations in nuclear mean field theory through source terms
Lee, S J
1996-01-01
Starting from full quantum field theory, various mean field approaches are derived systematically. With a full consideration of external source dependence, the stationary phase approximation of an action gives a nuclear mean field theory which includes quantum correlation effects (such as particle-hole or ladder diagram) in a simpler way than the Brueckner-Hartree-Fock approach. Implementing further approximation, the result can be reduced to Hartree-Fock or Hartree approximation. The role of the source dependence in a mean field theory is examined.
Mean field theories and dual variation mathematical structures of the mesoscopic model
Suzuki, Takashi
2015-01-01
Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.
Mean-field instabilities and cluster formation in nuclear reactions
Colonna, M; Baran, V
2016-01-01
We review recent results on intermediate mass cluster production in heavy ion collisions at Fermi energy and in spallation reactions. Our studies are based on modern transport theories, employing effective interactions for the nuclear mean-field and incorporating two-body correlations and fluctuations. Namely we will consider the Stochastic Mean Field (SMF) approach and the recently developed Boltzmann-Langevin One Body (BLOB) model. We focus on cluster production emerging from the possible occurrence of low-density mean-field instabilities in heavy ion reactions. Within such a framework, the respective role of one and two-body effects, in the two models considered, will be carefully analysed. We will discuss, in particular, fragment production in central and semi-peripheral heavy ion collisions, which is the object of many recent experimental investigations. Moreover, in the context of spallation reactions, we will show how thermal expansion may trigger the development of mean-field instabilities, leading to...
Extrapolation of mean-field models to superheavy nuclei
The extrapolation of self-consistent nuclear mean-field models to the region of superheavy elements is discussed with emphasis on the extrapolating power of the models. The predictions of modern mean-field models are confronted with recent experimental data. It is shown that a final conclusion about the location of the expected island of spherical doubly-magic superheavy nuclei cannot be drawn on the basis of the available data. (orig.)
Two-level interacting boson models beyond the mean field
Arias, J M; García-Ramos, J E; Vidal, J
2007-01-01
The phase diagram of two-level boson Hamiltonians, including the Interacting Boson Model (IBM), is studied beyond the standard mean field approximation using the Holstein-Primakoff mapping. The limitations of the usual intrinsic state (mean field) formalism concerning finite-size effects are pointed out. The analytic results are compared to numerics obtained from exact diagonalizations. Excitation energies and occupation numbers are studied in different model space regions (Casten triangle for IBM) and especially at the critical points.
Probabilistic data modelling with adaptive TAP mean-field theory
Opper, M.; Winther, Ole
2001-01-01
We demonstrate for the case of single-layer neural networks how an extension of the TAP mean-field approach of disorder physics can be applied to the computation of approximate averages in probabilistic models for real data.......We demonstrate for the case of single-layer neural networks how an extension of the TAP mean-field approach of disorder physics can be applied to the computation of approximate averages in probabilistic models for real data....
Mean-Field Theory of the Solar Dynamo
Schmitt, D.
The generation of the solar magnetic field is generally ascribed to dynamo processes in the convection zone. The dynamo effects, differential rotation (Omega-effect) and helical turbulence (alpha-effect) are explained, and the basic properties of the mean-field dynamo equations are discussed in view of the observed properties of the solar cycle. Problems of the classical picture of a dynamo in the convection zone (fibril state of magnetic flux, field strength, magnetic buoyancy, polarity rules, differential rotation and butterfly diagram) are addressed and some alternatives to overcome these problems are presented. A possibility to make up for the missing radial gradient of rotation in the convection zone is an alpha^2-Omega-dynamo with an anisotropic alpha-tensor. Dynamo solutions then might have the characteristics of the butterfly diagram. Another approach involves meridional circulation as the cause of the migration of a dynamo wave. Another suggestion is that the solar dynamo operates in the overshoot region at the base of the convection zone where strong fields, necessary to explain the polarity rules, can be stored and radial gradients in the angular velocity occur. As an alternative to the turbulent alpha-effect a dynamic alpha-effect based on magnetostrophic waves driven by a magnetic buoyancy instability of a magnetic flux layer is introduced. Model calculations which use the internal rotation of the Sun as deduced from helioseismology only show solar cycle behaviour if the turbulent diffusivity is reduced in the layer and the alpha-effect is concentrated near the equator. Another possibility is a combined model. The non-uniform rotation and most of the azimuthal magnetic flux are confined to a thin layer at the bottom of the convection zone where turbulent diffusion is greatly reduced, with the convective region above containing only weak fields for which the alpha-effect and turbulent diffusion operate in the conventional manner. The dynamo takes on the
Solution of the hyperon puzzle within a relativistic mean-field model
Maslov, K.A. [National Research Nuclear University (MEPhI), 115409 Moscow (Russian Federation); Kolomeitsev, E.E., E-mail: E.Kolomeitsev@gsi.de [Matej Bel University, SK-97401 Banska Bystrica (Slovakia); Voskresensky, D.N. [National Research Nuclear University (MEPhI), 115409 Moscow (Russian Federation)
2015-09-02
The equation of state of cold baryonic matter is studied within a relativistic mean-field model with hadron masses and coupling constants depending on the scalar field. All hadron masses undergo a universal scaling, whereas the couplings are scaled differently. The appearance of hyperons in dense neutron star interiors is accounted for, however the equation of state remains sufficiently stiff if the reduction of the ϕ meson mass is included. Our equation of state matches well the constraints known from analyses of the astrophysical data and particle production in heavy-ion collisions.
Short-time existence of solutions for mean-field games with congestion
Gomes, Diogo A.
2015-11-20
We consider time-dependent mean-field games with congestion that are given by a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. These models are motivated by crowd dynamics in which agents have difficulty moving in high-density areas. The congestion effects make the Hamilton–Jacobi equation singular. The uniqueness of solutions for this problem is well understood; however, the existence of classical solutions was only known in very special cases, stationary problems with quadratic Hamiltonians and some time-dependent explicit examples. Here, we demonstrate the short-time existence of C∞ solutions for sub-quadratic Hamiltonians.
Mean-field games and two-point boundary value problems
Mylvaganam, T.; Bauso, D.; Astolfi, A.
2015-01-01
© 2014 IEEE. A large population of agents seeking to regulate their state to values characterized by a low density is considered. The problem is posed as a mean-field game, for which solutions depend on two partial differential equations, namely the Hamilton-Jacobi-Bellman equation and the Fokker-Plank-Kolmogorov equation. The case in which the distribution of agents is a sum of polynomials and the value function is quadratic is considered. It is shown that a set of ordinary differential equa...
A Stochastic Maximum Principle for a Stochastic Differential Game of a Mean-Field Type
We construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certain form of N-agent stochastic differential game (SDG) of a mean-field type. The information structure considered for the SDG is of a possible asymmetric and partial type. To prove our SMP we take an approach based on spike-variations and adjoint representation techniques, analogous to that of S. Peng (SIAM J. Control Optim. 28(4):966–979, 1990) in the optimal stochastic control context. In our proof we apply adjoint representation procedures at three points. The first-order adjoint processes are defined as solutions to certain mean-field backward stochastic differential equations, and second-order adjoint processes of a first type are defined as solutions to certain backward stochastic differential equations. Second-order adjoint processes of a second type are defined as solutions of certain backward stochastic equations of a type that we introduce in this paper, and which we term conditional mean-field backward stochastic differential equations. From the resulting representations, we show that the terms relating to these second-order adjoint processes of the second type are of an order such that they do not appear in our final SMP equations. A comparable situation exists in an article by R. Buckdahn, B. Djehiche, and J. Li (Appl. Math. Optim. 64(2):197–216, 2011) that constructs a SMP for a mean-field type optimal stochastic control problem; however, the approach we take of using these second-order adjoint processes of a second type to deal with the type of terms that we refer to as the second form of quadratic-type terms represents an alternative to a development, to our setting, of the approach used in their article for their analogous type of term.
A Stochastic Maximum Principle for a Stochastic Differential Game of a Mean-Field Type
Hosking, John Joseph Absalom, E-mail: j.j.a.hosking@cma.uio.no [University of Oslo, Centre of Mathematics for Applications (CMA) (Norway)
2012-12-15
We construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certain form of N-agent stochastic differential game (SDG) of a mean-field type. The information structure considered for the SDG is of a possible asymmetric and partial type. To prove our SMP we take an approach based on spike-variations and adjoint representation techniques, analogous to that of S. Peng (SIAM J. Control Optim. 28(4):966-979, 1990) in the optimal stochastic control context. In our proof we apply adjoint representation procedures at three points. The first-order adjoint processes are defined as solutions to certain mean-field backward stochastic differential equations, and second-order adjoint processes of a first type are defined as solutions to certain backward stochastic differential equations. Second-order adjoint processes of a second type are defined as solutions of certain backward stochastic equations of a type that we introduce in this paper, and which we term conditional mean-field backward stochastic differential equations. From the resulting representations, we show that the terms relating to these second-order adjoint processes of the second type are of an order such that they do not appear in our final SMP equations. A comparable situation exists in an article by R. Buckdahn, B. Djehiche, and J. Li (Appl. Math. Optim. 64(2):197-216, 2011) that constructs a SMP for a mean-field type optimal stochastic control problem; however, the approach we take of using these second-order adjoint processes of a second type to deal with the type of terms that we refer to as the second form of quadratic-type terms represents an alternative to a development, to our setting, of the approach used in their article for their analogous type of term.
Local excitations in mean-field spin glasses
Krzakala, F.; Parisi, G.
2004-06-01
We address the question of geometrical as well as energetic properties of local excitations in mean-field Ising spin glasses. We study analytically the Random Energy Model and numerically a dilute mean-field model, first on tree-like graphs, equivalent to a replica-symmetric computation, and then directly on finite-connectivity random lattices. In the first model, characterized by a discontinuous replica symmetry breaking, we found that the energy of finite-volume excitation is infinite, whereas in the dilute mean-field model, described by a continuous replica symmetry breaking, it slowly decreases with sizes and saturates at a finite value, in contrast with what would be naively expected. The geometrical properties of these excitations are similar to those of lattice animals or branched polymers. We discuss the meaning of these results in terms of replica symmetry breaking and also possible relevance in finite-dimensional systems.
Mean field strategies induce unrealistic nonlinearities in calcium puffs
GuillermoSolovey
2011-08-01
Full Text Available Mean field models are often useful approximations to biological systems, but sometimes, they can yield misleading results. In this work, we compare mean field approaches with stochastic models of intracellular calcium release. In particular, we concentrate on calcium signals generated by the concerted opening of several clustered channels (calcium puffs. To this end we simulate calcium puffs numerically and then try to reproduce features of the resulting calcium distribution using mean field models were all the channels open and close simultaneously. We show that an unrealistic nonlinear relationship between the current and the number of open channels is needed to reproduce the simulated puffs. Furthermore, a single channel current which is five times smaller than the one of the stochastic simulations is also needed. Our study sheds light on the importance of the stochastic kinetics of the calcium release channel activity to estimate the release fluxes.
Dynamical Mean Field Approximation Applied to Quantum Field Theory
Akerlund, Oscar; Georges, Antoine; Werner, Philipp
2013-01-01
We apply the Dynamical Mean Field (DMFT) approximation to the real, scalar phi^4 quantum field theory. By comparing to lattice Monte Carlo calculations, perturbation theory and standard mean field theory, we test the quality of the approximation in two, three, four and five dimensions. The quantities considered in these tests are the critical coupling for the transition to the ordered phase and the associated critical exponents nu and beta. We also map out the phase diagram in four dimensions. In two and three dimensions, DMFT incorrectly predicts a first order phase transition for all bare quartic couplings, which is problematic, because the second order nature of the phase transition of lattice phi^4-theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behaviour away from the phase transition, one can obtain critical couplings and critical exponents. They differ from those of mean field theory and are much closer to the correct values. In four dimensions the transition is sec...
Noisy mean field game model for malware propagation in opportunistic networks
Tembine, Hamidou
2012-01-01
In this paper we present analytical mean field techniques that can be used to better understand the behavior of malware propagation in opportunistic large networks. We develop a modeling methodology based on stochastic mean field optimal control that is able to capture many aspects of the problem, especially the impact of the control and heterogeneity of the system on the spreading characteristics of malware. The stochastic large process characterizing the evolution of the total number of infected nodes is examined with a noisy mean field limit and compared to a deterministic one. The stochastic nature of the wireless environment make stochastic approaches more realistic for such types of networks. By introducing control strategies, we show that the fraction of infected nodes can be maintained below some threshold. In contrast to most of the existing results on mean field propagation models which focus on deterministic equations, we show that the mean field limit is stochastic if the second moment of the number of object transitions per time slot is unbounded with the size of the system. This allows us to compare one path of the fraction of infected nodes with the stochastic trajectory of its mean field limit. In order to take into account the heterogeneity of opportunistic networks, the analysis is extended to multiple types of nodes. Our numerical results show that the heterogeneity can help to stabilize the system. We verify the results through simulation showing how to obtain useful approximations in the case of very large systems. © 2012 ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering.
Hot and dense matter beyond relativistic mean field theory
Zhang, Xilin; Prakash, Madappa
2016-05-01
Properties of hot and dense matter are calculated in the framework of quantum hadrodynamics by including contributions from two-loop (TL) diagrams arising from the exchange of isoscalar and isovector mesons between nucleons. Our extension of mean field theory (MFT) employs the same five density-independent coupling strengths which are calibrated using the empirical properties at the equilibrium density of isospin-symmetric matter. Results of calculations from the MFT and TL approximations are compared for conditions of density, temperature, and proton fraction encountered in the study of core-collapse supernovae, young and old neutron stars, and mergers of compact binary stars. The TL results for the equation of state (EOS) of cold pure neutron matter at sub- and near-nuclear densities agree well with those of modern quantum Monte Carlo and effective field-theoretical approaches. Although the high-density EOS in the TL approximation for cold and β -equilibrated neutron-star matter is substantially softer than its MFT counterpart, it is able to support a 2 M⊙ neutron star required by recent precise determinations. In addition, radii of 1.4 M⊙ stars are smaller by ˜1 km than those obtained in MFT and lie in the range indicated by analysis of astronomical data. In contrast to MFT, the TL results also give a better account of the single-particle or optical potentials extracted from analyses of medium-energy proton-nucleus and heavy-ion experiments. In degenerate conditions, the thermal variables are well reproduced by results of Landau's Fermi-liquid theory in which density-dependent effective masses feature prominently. The ratio of the thermal components of pressure and energy density expressed as Γth=1 +(Pth/ɛth) , often used in astrophysical simulations, exhibits a stronger dependence on density than on proton fraction and temperature in both MFT and TL calculations. The prominent peak of Γth at supranuclear density found in MFT is, however, suppressed in
Suppression of oscillations in mean-field diffusion
Neeraj Kumar Kamal; Pooja Rani Sharma; Manish Dev Shrimali
2015-02-01
We study the role of mean-field diffusive coupling on suppression of oscillations for systems of limit cycle oscillators. We show that this coupling scheme not only induces amplitude death (AD) but also oscillation death (OD) in coupled identical systems. The suppression of oscillations in the parameter space crucially depends on the value of mean-field diffusion parameter. It is also found that the transition from oscillatory solutions to OD in conjugate coupling case is different from the case when the coupling is through similar variable. We rationalize our study using linear stability analysis.
Condition monitoring with Mean field independent components analysis
Pontoppidan, Niels Henrik; Sigurdsson, Sigurdur; Larsen, Jan
2005-01-01
We discuss condition monitoring based on mean field independent components analysis of acoustic emission energy signals. Within this framework it is possible to formulate a generative model that explains the sources, their mixing and also the noise statistics of the observed signals. By using a...... from a large diesel engine is used to demonstrate this approach. The results show that mean field independent components analysis gives a better detection of fault compared to principal components analysis, while at the same time selecting a more compact model...
Shapes and Dynamics from the Time-Dependent Mean Field
Stevenson, P D; Rios, A
2015-01-01
Explaining observed properties in terms of underlying shape degrees of freedom is a well--established prism with which to understand atomic nuclei. Self--consistent mean--field models provide one tool to understand nuclear shapes, and their link to other nuclear properties and observables. We present examples of how the time--dependent extension of the mean--field approach can be used in particular to shed light on nuclear shape properties, particularly looking at the giant resonances built on deformed nuclear ground states, and at dynamics in highly-deformed fission isomers. Example calculations are shown of $^{28}$Si in the first case, and $^{240}$Pu in the latter case.
Relativistic Chiral Mean Field Model for Finite Nuclei
Ogawa, Yoko; Toki, Hiroshi; Tamenaga, Setsuo; Haga, Akihiro
2012-01-01
We present a relativistic chiral mean field (RCMF) model, which is a method for the proper treatment of pion-exchange interaction in the nuclear many-body problem. There the dominant term of the pionic correlation is expressed in two-particle two-hole (2p-2h) states with particle-holes having pionic quantum number, J^{pi}. The charge-and-parity-projected relativistic mean field (CPPRMF) model developed so far treats surface properties of pionic correlation in 2p-2h states with J^{pi} = 0^{-} ...
A mean field theory of coded CDMA systems
We present a mean field theory of code-division multiple-access (CDMA) systems with error-control coding. On the basis of the relation between the free energy and mutual information, we obtain an analytical expression of the maximum spectral efficiency of the coded CDMA system, from which a mean-field description of the coded CDMA system is provided in terms of a bank of scalar Gaussian channels whose variances in general vary at different code symbol positions. Regular low-density parity-check (LDPC)-coded CDMA systems are also discussed as an example of the coded CDMA systems
A mean field theory of coded CDMA systems
Yano, Toru [Graduate School of Science and Technology, Keio University, Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa 223-8522 (Japan); Tanaka, Toshiyuki [Graduate School of Informatics, Kyoto University, Yoshida Hon-machi, Sakyo-ku, Kyoto-shi, Kyoto 606-8501 (Japan); Saad, David [Neural Computing Research Group, Aston University, Birmingham B4 7ET (United Kingdom)], E-mail: yano@thx.appi.keio.ac.jp
2008-08-15
We present a mean field theory of code-division multiple-access (CDMA) systems with error-control coding. On the basis of the relation between the free energy and mutual information, we obtain an analytical expression of the maximum spectral efficiency of the coded CDMA system, from which a mean-field description of the coded CDMA system is provided in terms of a bank of scalar Gaussian channels whose variances in general vary at different code symbol positions. Regular low-density parity-check (LDPC)-coded CDMA systems are also discussed as an example of the coded CDMA systems.
Derivation of mean-field dynamics for fermions
Petrat, Sören
2014-01-01
In dieser Arbeit werden die zeitabhängigen Hartree(-Fock) Gleichungen als effektive Dynamik für fermionische Vielteilchen-Systeme hergeleitet. Die Hauptresultate sind die ersten für eine quantenmechanische Mean-Field Dynamik ("Mittlere-Feld Dynamik") für Fermionen; in vorherigen Arbeiten ist der Mean-Field Limes üblicherweise entweder mit einem semiklassischen Limes gekoppelt oder die Wechselwirkung wird so stark runterskaliert, dass sich das System für große Teilchenzahl N frei verhält. Wir ...
Mean field with corrections in lattice gauge theory
A systematic expansion of the path integral for lattice gauge theory is performed around the mean field solution. In this letter we present the results for the pure gauge groups Z(2), SU(2) and SO(3). The agreement with Monte Carlo calculations is excellent. For the discrete group the calculation is performed with and without gauge fixing, whereas for the continuous groups gauge fixing is mandatory. In the case of SU(2) the absence of a phase transition is correctly signalled by mean field theory. (orig.)
Mean field with corrections in lattice gauge theory
A systematic expansion of the path integral for lattice gauge theory is performed around the mean field solution. In this letter the authors present the results for the pure gauge groups Z(2), SU(2) and SO(3). The agreement with Monte Carlo calculations is excellent. For the discrete group the calculation is performed with and without gauge fixing, whereas for the continuous groups gauge fixing is mandatory. In the case of SU(2) the absence of a phase transition is correctly signalled by mean field theory. (Auth.)
Phase transition in a mean-field model for sympatric speciation
Schwämmle, V.; Luz-Burgoa, K.; Martins, J. S. Sá; de Oliveira, S. Moss
2005-01-01
We introduce an analytical model for population dynamics with intra-specific competition, mutation and assortative mating as basic ingredients. The set of equations that describes the time evolution of population size in a mean-field approximation may be decoupled. We find a phase transition leading to sympatric speciation as a parameter that quantifies competition strength is varied. This transition, previously found in a computational model, occurs to be of first order.
Nuclear magnetic moments and the spin-orbit current in the relativistic mean field theory
The Dirac magnetic moments in the relativistic mean field theory are affected not only by the effective mass, but also by the spin-orbit current related to the spin-orbit force through the continuity equation. Previous arguments on the cancellation of the effective-mass effect in nuclear matter are not simply applied to finite nuclei to obtain the Schmidt values. Effects of the spin-orbit current on (e, e') response functions are also mentioned. (orig.)
GENG Li-Sheng; MENG Jie; Toki Hiroshi
2007-01-01
A reflection asymmetric relativistic mean field (RAS-RMF) approach is developed by expanding the equations of motion for both the nucleons and the mesons on the eigenfunctions of the two-centre harmonic-oscillator potential.The efficiency and reliability of the RAS-RMF approach are demonstrated in its application to the well-known octupole deformed nucleus 226Ra and the available data, including the binding energy and the deformation parameters, are well reproduced.
A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics
Petrat, Sören; Pickl, Peter
2016-03-01
We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151-164 2011) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.
Hydrodynamic mean-field solutions of 1D exclusion processes with spatially varying hopping rates
Lakatos, Greg; O' Brien, John; Chou, Tom [Department of Biomathematics and Institute for Pure and Applied Mathematics, UCLA, Los Angeles, CA 90095 (United States)
2006-03-10
We analyse the open boundary partially asymmetric exclusion process with smoothly varying internal hopping rates in the infinite-size, mean-field limit. The mean-field equations for particle densities are written in terms of Ricatti equations with the steady-state current J as a parameter. These equations are solved both analytically and numerically. Upon imposing the boundary conditions set by the injection and extraction rates, the currents J are found self-consistently. We find a number of cases where analytic solutions can be found exactly or approximated. Results for J from asymptotic analyses for slowly varying hopping rates agree extremely well with those from extensive Monte Carlo simulations, suggesting that mean-field currents asymptotically approach the exact currents in the hydrodynamic limit, as the hopping rates vary slowly over the lattice. If the forward hopping rate is greater than or less than the backward hopping rate throughout the entire chain, the three standard steady-state phases are preserved. Our analysis reveals the sensitivity of the current to the relative phase between the forward and backward hopping rate functions.
Mean Field Approach to the Giant Wormhole Problem
Gamba, A.; Kolokolov, I.; Martellini, M.
1992-01-01
We introduce a gaussian probability density for the space-time distribution of wormholes, thus taking effectively into account wormhole interaction. Using a mean-field approximation for the free energy, we show that giant wormholes are probabilistically suppressed in a homogenous isotropic ``large'' universe.
Mean field theory for lattice gauge systems with fermions
We extend recent mean field calculations for lattice gauge theories to include fermions. We find that the addition of a Wilson fermion leads to an almost negligible change of the weak to strong coupling transition point. The plaquette average is also only weakly affected. (author)
Social networking and individual outcomes beyond the mean field case
Y.M. Ioannides; A.R. Soetevent
2007-01-01
We study individually optimized continuous outcomes in a dynamic environment in the presence of social interactions, and where the interaction topology may be either exogenous and time varying, or endogenous. The model accommodates more general social effects than those of the mean-field type. We ad
Clustering in atomic nuclei: a mean field perspective
In this paper the physics of clustering in atomic nucleus as seen from a mean field perspective will be discussed. Special attention is paid to phenomena involving octupole deformation like the α structure of 20Ne or the emission of heavy clusters. The stabilizing role of spin for cluster-like highly deformed states is also discussed in the case of 36 Ar
Robust mean field games for coupled Markov jump linear systems
Moon, Jun; Başar, Tamer
2016-07-01
We consider robust stochastic large population games for coupled Markov jump linear systems (MJLSs). The N agents' individual MJLSs are governed by different infinitesimal generators, and are affected not only by the control input but also by an individual disturbance (or adversarial) input. The mean field term, representing the average behaviour of N agents, is included in the individual worst-case cost function to capture coupling effects among agents. To circumvent the computational complexity and analyse the worst-case effect of the disturbance, we use robust mean field game theory to design low-complexity robust decentralised controllers and to characterise the associated worst-case disturbance. We show that with the individual robust decentralised controller and the corresponding worst-case disturbance, which constitute a saddle-point solution to a generic stochastic differential game for MJLSs, the actual mean field behaviour can be approximated by a deterministic function which is a fixed-point solution to the constructed mean field system. We further show that the closed-loop system is uniformly stable independent of N, and an approximate optimality can be obtained in the sense of ε-Nash equilibrium, where ε can be taken to be arbitrarily close to zero as N becomes sufficiently large. A numerical example is included to illustrate the results.
Variational Wigner-Kirkwood approach to relativistic mean field theory
Estal, Manuel del; Centelles Aixalà, Mario; Viñas Gausí, Xavier
1997-01-01
The recently developed variational Wigner-Kirkwood approach is extended to the relativistic mean field theory for finite nuclei. A numerical application to the calculation of the surface energy coefficient in semi-infinite nuclear matter is presented. The new method is contrasted with the standard density functional theory and the fully quantal approach.
Chaotic time series prediction using mean-field theory for support vector machine
Cui Wan-Zhao; Zhu Chang-Chun; Bao Wen-Xing; Liu Jun-Hua
2005-01-01
This paper presents a novel method for predicting chaotic time series which is based on the support vector machines approach, and it uses the mean-field theory for developing an easy and efficient learning procedure for the support vector machine. The proposed method approximates the distribution of the support vector machine parameters to a Gaussian process and uses the mean-field theory to estimate these parameters easily, and select the weights of the mixture of kernels used in the support vector machine estimation more accurately and faster than traditional quadratic programming-based algorithms. Finally, relationships between the embedding dimension and the predicting performance of this method are discussed, and the Mackey-Glass equation is applied to test this method. The stimulations show that the mean-field theory for support vector machine can predict chaotic time series accurately, and even if the embedding dimension is unknown, the predicted results are still satisfactory. This result implies that the mean-field theory for support vector machine is a good tool for studying chaotic time series.
Mean-field cosmological dynamos in Riemannian space with isotropic diffusion
de Andrade, L Garcia
2009-01-01
Mean-field cosmological dynamos in Riemannian space with isotropic diffusion}} Previous attempts for building a cosmic dynamo including preheating in inflationary universes [Bassett et al Phys Rev (2001)] has not included mean field or turbulent dynamos. In this paper a mean field dynamo in cosmic scales on a Riemannian spatial cosmological section background, is set up. When magnetic fields and flow velocities are parallel propagated along the Riemannian space dynamo action is obtained. Turbulent diffusivity ${\\beta}$ is coupled with the Ricci magnetic curvature, as in Marklund and Clarkson [MNRAS (2005)], GR-MHD dynamo equation. Mean electric field possesses an extra term where Ricci tensor couples with magnetic vector potential in Ohm's law. In Goedel universe induces a mean field dynamo growth rate ${\\gamma}=2{\\omega}^{2}{\\beta}$. In this frame kinetic helicity vanishes. In radiation era this yields ${\\gamma}\\approx{2{\\beta}{\\times}10^{-12}s^{-1}}$. In non-comoving the magnetic field is expressed as $B\\ap...
Charge and parity projected relativistic mean field model with pion for finite nuclei
We construct a new relativistic mean field model by explicitly introducing a π-meson mean field with charge number and parity projection. We call this model the charge and parity projected relativistic mean field (CPPRMF) model. We take the chiral σ model Lagrangian for the construction of finite nuclei. We apply this framework first for the 4He nucleus as a pilot case and study the role of the π-meson field on the structure of nuclei. We demonstrate that it is essential to solve the mean field equation with the variation introduced after the projection in order to take the pionic correlations into account explicitly. We study the ground-state properties of 4He by varying several parameters, such as the σ-meson mass and the ω-meson coupling constant. We are able to construct a good ground state for 4He. A depression appears in the central region of the density distribution, and the second maximum and the position of the dip in the form factor of 4He are naturally obtained in the CPPRMF model
Damping of Collective Nuclear Motion and Thermodynamic Properties of Nuclei beyond Mean Field
Luo, Hong-Gang; Cassing, W.; Wang, Shun-Jin
1999-01-01
The dynamical description of correlated nuclear motion is based on a set of coupled equations of motion for the one-body density matrix $\\rho (11';t)$ and the two-body correlation function $c_2(12,1'2';t)$, which is obtained from the density-matrix hierarchy beyond conventional mean-field approaches by truncating 3-body correlations. The resulting equations nonperturbatively describe particle-particle collisions (short-range correlations) as well as particle-hole interactions (long-range corr...
Expansion Around the Mean-Field Solution of the Bak-Sneppen Model
Marsili, M. [Institut de Physique Theorique, Universite de Fribourg Perolles, Fribourg, CH-1700 (Switzerland); De Los Rios, P. [Max-Planck-Institut fuer Physik Komplexer Systeme, Noethnitzer Str.38, D-01187 Dresden (Germany); Maslov, S. [Department of Physics, Brookhaven National Laboratory, Upton, New York 11973 (United States)
1998-02-01
We study a recently proposed equation for the avalanche distribution in the Bak-Sneppen model. We demonstrate that this equation indirectly relates {tau} , the exponent for the power law distribution of avalanche sizes, to D , the fractal dimension of an avalanche cluster. We compute this relation numerically and approximate it analytically up to the second order of expansion around the mean-field exponents. Our results are consistent with Monte Carlo simulations of the Bak-Sneppen model in one and two dimensions. {copyright} {ital 1998} {ital The American Physical Society}
Merging Belief Propagation and the Mean Field Approximation: A Free Energy Approach
Riegler, Erwin; Kirkelund, Gunvor Elisabeth; Manchón, Carles Navarro;
2013-01-01
We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al. We show that the message passing fixed-point equations obtained with this combination...... correspond to stationary points of a constrained region-based free energy approximation. Moreover, we present a convergent implementation of these message passing fixed-point equations provided that the underlying factor graph fulfills certain technical conditions. In addition, we show how to include hard...
Nuclei, hypernuclei, and neutron stars in a relativistic mean-field model
An essential aim of this thesis consisted in the obtainment of an optimal description of finite also strangeness carrying nuclei in the framework of a relativistic mean-field model. For this the model parameters were fitted to experimental nuclear and hypernuclear data. By the so optimized parametrizations the - among others - equations of state of neutron matter were extrapolated and by solving of the Oppenheimer-Volkoff equation neutron star properties calculated. In this connection also the possible existence of a quark phase in the interior of neutron stars was considered. (orig.)
Mean-Field Dynamical Semigroups on C*-ALGEBRAS
Duffield, N. G.; Werner, R. F.
We study a notion of the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra { A} with itself. In analogy with the theory of semigroups on Banach spaces we give abstract conditions for the existence of these limits. These conditions are verified in the case of semigroups whose generators are determined by the successive resymmetrizations of a fixed operator, as well as generators which can be approximated by generators of this type. This includes the time evolutions of the mean-field versions of quantum lattice systems. In these cases the limiting dynamical semigroup is given by a continuous flow on the state space of { A}. For a class of such flows we show stability by constructing a Liapunov function. We also give examples where the limiting evolution is given by a diffusion, rather than a flow on the state space of { A}.
A Local Mean Field Analysis of Security Investments in Networks
Lelarge, Marc
2008-01-01
Getting agents in the Internet, and in networks in general, to invest in and deploy security features and protocols is a challenge, in particular because of economic reasons arising from the presence of network externalities. Our goal in this paper is to carefully model and quantify the impact of such externalities on the investment in, and deployment of, security features and protocols in a network. Specifically, we study a network of interconnected agents, which are subject to epidemic risks such as those caused by propagating viruses and worms, and which can decide whether or not to invest some amount to self-protect and deploy security solutions. We make three contributions in the paper. First, we introduce a general model which combines an epidemic propagation model with an economic model for agents which captures network effects and externalities. Second, borrowing ideas and techniques used in statistical physics, we introduce a Local Mean Field (LMF) model, which extends the standard mean-field approxi...
Nuclear collective vibrations in extended mean-field theory
The extended mean-field theory, which includes both the incoherent dissipation mechanism due to nucleon-nucleon collisions and the coherent dissipation mechanism due to coupling to low-lying surface vibrations, is briefly reviewed. Expressions of the strength functions for the collective excitations are presented in the small amplitude limit of this approach. This fully microscopic theory is applied by employing effective Skyrme forces to various giant resonance excitations at zero and finite temperature. The theory is able to describe the gross properties of giant resonance excitations, the fragmentation of the strength distributions as well as their fine structure. At finite temperature, the success and limitations of this extended mean-field description are discussed. (authors)
Progress in nuclear structure beyond the mean-field approximation
Although self-consistent mean-field methods, or implementations of the density functional theory for atomic nuclei, are becoming increasingly accurate, some observables are not well reproduced by those models. In particular, the fragmentation and the decay properties of both single-particle and vibrational states cannot be accounted for. Models based on the introduction of further correlations or, in other words, that go beyond the mean-field approximation, have often been discussed in the past. We have recently developed a consistent model based on the use of a Skyrme-type force without the intervention of any other ad hoc parameter. A few typical results are discussed, after we have mentioned briefly the essential features of the model. Moreover, we discuss the necessity of fitting a new force within this context, the difficulties arising because of divergences that need to be renormalized, and our roadmap for curing these divergences
Characterizing the mean-field dynamo in turbulent accretion disks
Gressel, Oliver
2015-01-01
The formation and evolution of a wide class of astrophysical objects is governed by turbulent, magnetized accretion disks. Understanding their secular dynamics is of primary importance. Apart from enabling mass accretion via the transport of angular momentum, the turbulence affects the long-term evolution of the embedded magnetic flux, which in turn regulates the efficiency of the transport. In this paper, we take a comprehensive next step towards an effective mean-field model for turbulent astrophysical disks by systematically studying the key properties of magnetorotational turbulence in vertically-stratified, isothermal shearing boxes. This allows us to infer emergent properties of the ensuing chaotic flow as a function of the shear parameter as well as the amount of net-vertical flux. Using the test-field method, we furthermore characterize the mean-field dynamo coefficients that describe the long-term evolution of large-scale fields. We simultaneously infer the vertical shape and the spectral scale depen...
Self-organized criticality in nonconservative mean-field sandpiles
Juanico, Dranreb Earl
2007-01-01
A mean-field sandpile model that exhibits self-organized criticality (SOC) despite violation of the grain-transfer conservation law during avalanches is proposed. The sandpile consists of $N$ agents and possesses background activity with intensity $\\eta\\in[0,1]$. Background activity is characterized by transitions between two stable agent states. Analysis employing theories of branching processes and fixed points reveals a transition from sub-critical to SOC phase that is determined by $\\eta ...
Energy Dependent Isospin Asymmetry in Mean-Field Dynamics
Gaitanos, T
2011-01-01
The Lagrangian density of Relativistic Mean-Field (RMF) theory with non-linear derivative (NLD) interactions is applied to isospin asymmetric nuclear matter. We study the symmetry energy and the density and energy dependences of nucleon selfenergies. At high baryon densities a soft symmetry energy is obtained. The energy dependence of the isovector selfenergy suppresses the Lane-type optical potential with increasing energy and predicts a $\\rho$-meson induced mass splitting between protons and neutrons in isospin asymmetric matter.
Finite-size scaling for mean-field percolation
Gaveau, B. (Univ. P.M. Curie, Paris (France)); Schulman, L.S. (Univ. P.M. Curie, Paris (France) Clarkson Univ. Patsdam, NY (United States) Columbia Univ., New York (United States))
1993-02-01
By studying transfer matrix eigenvalues, correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime. Near criticality, finite-size scaling behavior is derived and an analytic technique is provided for obtaining the finite-size scaling function. Our methods involve the generating function, matched asymptotic expansions, and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability.
RPA correlations and nuclear densities in relativistic mean field approach
Van Giai, N. [Institut de Physique Nucleaire, CNRS, UMR 8608, F-91406 Orsay (France)]|[Universite Paris-Sud, F-91406 Orsay (France); Liang, H.Z. [Institut de Physique Nucleaire, CNRS, UMR 8608, F-91406 Orsay (France)]|[Universite Paris-Sud, F-91406 Orsay (France)]|[School of Physics, Peking University, 100871 Beijing (China); Meng, J. [School of Physics, Peking University, 100871 Beijing (China)
2007-02-15
The relativistic mean field approach (RMF) is well known for describing accurately binding energies and nucleon distributions in atomic nuclei throughout the nuclear chart. The random phase approximation (RPA) built on top of the RMF is also a good framework for the study of nuclear excitations. Here, we examine the consequences of long range correlations brought about by the RPA on the neutron and proton densities as given by the RMF approach. (authors)
Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study
Haule, Kristjan; Kotliar, Gabriel
2007-01-01
We use cluster Dynamical Mean Field Theory to study the simplest models of correlated electrons, the Hubbard model and the t-J model. We use a plaquette embedded in a medium as a reference frame to compute and interpret the physical properties of these models. We study various observables such as electronic lifetimes, one electron spectra, optical conductivities, superconducting stiffness, and the spin response in both the normal and the superconducting state in terms of correlation functions...
Advanced Mean Field Theory of Restricted Boltzmann Machine
Huang, Haiping; Toyoizumi, Taro
2015-01-01
Learning in restricted Boltzmann machine is typically hard due to the computation of gradients of log-likelihood function. To describe the network state statistics of the restricted Boltzmann machine, we develop an advanced mean field theory based on the Bethe approximation. Our theory provides an efficient message passing based method that evaluates not only the partition function (free energy) but also its gradients without requiring statistical sampling. The results are compared with those...
Local Mean Field Dynamics of Ising Spin Glasses
Hung, Tran; Li, Mai; Cieplak, Marek
1995-01-01
The Glauber dynamics of three-dimensional Ising spin glasses are studied numerically in the local mean field approximation. The aging effect is observed in both field cooled and zero field cooled regimes but the remanent magnetization never reaches true equilibrium. The dynamic susceptibility behaves like in experiments. A double peak structure in the real part of the susceptibility plotted as a function of temperature may appear for non-symmetric distributions of the exchange couplings. This...
Merging Belief Propagation and the Mean Field Approximation
Riegler, Erwin; Kirkelund, Gunvor Elisabeth; Manchón, Carles Navarro; Fleury, Bernard Henri
We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al., which allows to use the same objective function (Kullback-Leibler divergence) as a....... Our results can be applied, for example, to algorithms that perform joint channel estimation and decoding in iterative receivers. This is demonstrated in a simple example....
Rotating nuclei at extreme conditions: Cranked relativistic mean field description
Afanasjev, A V
1999-01-01
The cranked relativistic mean field (CRMF) theory is applied for the description of superdeformed (SD) rotational bands observed in sup 1 sup 5 sup 3 Ho. The question of the structure of the so-called SD band in sup 1 sup 5 sup 4 Er is also addressed and a brief overview of applications of CRMF theory to the description of rotating nuclei at extreme conditions is presented.
Introduction to mean-field theory of spin glass models
Janiš, Václav
Jülich : Forschungszentrums Jülich, 2015 - (Pavarini, E.; Koch, E.; Coleman, P.), s. 8.1-8.28 ISBN 978-3-95806-074-6 Institutional support: RVO:68378271 Keywords : mean-field theory * spin-glass models * ergodicity breaking * real replicas * asymptotic expansions Subject RIV: BM - Solid Matter Physics ; Magnetism http://www.cond-mat.de/events/correl15/manuscripts/janis.pdf
Disorder Chaos in the Spherical Mean-Field Model
Chen, Wei-Kuo; Hsieh, Hsi-Wei; Hwang, Chii-Ruey; Sheu, Yuan-Chung
2015-07-01
We study the problem of disorder chaos in the spherical mean-field model. It concerns the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction states that if the disorders in the Hamiltonians are slightly decoupled, then the overlap will be concentrated near a constant value. Following Guerra's replica symmetry breaking scheme, we establish this at the levels of the free energy and the Gibbs measure.
Time-dependent mean-field games in the superquadratic case
Gomes, Diogo A.
2016-04-06
We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as key techniques used in the subquadratic case, which was studied in a previous publication of the authors, do not extend to the superquadratic setting. The main objective of the present paper is to address these difficulties. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton−Jacobi equation are obtained here through a novel set of techniques. These explore the parabolic nature of the problem through the nonlinear adjoint method. Well-posedness is proven by combining Lipschitz regularity for the Hamilton−Jacobi equation with polynomial estimates for solutions of the Fokker−Planck equation. Existence of classical solutions is then established under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to current understanding of superquadratic Hamilton−Jacobi equations.
Mean-Field Description of Plastic Flow in Amorphous Solids
Lin, Jie; Wyart, Matthieu
2016-01-01
Failure and flow of amorphous materials are central to various phenomena including earthquakes and landslides. There is accumulating evidence that the yielding transition between a flowing and an arrested phase is a critical phenomenon, but the associated exponents are not understood, even at a mean-field level where the validity of popular models is debated. Here, we solve a mean-field model that captures the broad distribution of the mechanical noise generated by plasticity, whose behavior is related to biased Lévy flights near an absorbing boundary. We compute the exponent θ characterizing the density of shear transformation P (x )˜xθ, where x is the stress increment beyond which they yield. We find that after an isotropic thermal quench, θ =1 /2 . However, θ depends continuously on the applied shear stress; this dependence is not monotonic, and its value at the yield stress is not universal. The model rationalizes previously unexplained observations and captures reasonably well the value of exponents in three dimensions. Values of exponents in four dimensions are accurately predicted. These results support the fact that it is the true mean-field model that applies in large dimensions, and they raise fundamental questions about the nature of the yielding transition.
Mean-field effects on matter and antimatter elliptic flow
We report our recent work on mean-field potential effects on the elliptic flows of matters and antimatters in heavy ion collisions leading to the production of a baryon-rich matter. Within the framework of a multiphase transport (AMPT) model that includes both initial partonic and final hadronic interactions, we have found that including mean-field potentials in the hadronic phase leads to a splitting of the elliptic flows of particles and their antiparticles, providing thus a plausible explanation of the different elliptic flows between p and anti-p, K+ and K-, and π+ and π- observed by the STAR Collaboration in the Beam Energy Scan (BES) program at the Relativistic Heavy Ion Collider (RHIC). Using a partonic transport model based on the Nambu-Jona-Lasinio (NJL) model, we have also studied the effect of scalar and vector mean fields on the elliptic flows of quarks and antiquarks in these collisions. Converting quarks and antiquarks at hadronization to hadrons via the quark coalescence model, we have found that the elliptic flow differences between particles and antiparticles also depend on the strength of the quark vector coupling in baryon-rich quark-gluon plasma, providing thus the possibility of extracting information on the latter's properties from the BES program at RHIC. (authors)
Relativistic heavy ion collisions with realistic non-equilibrium mean fields
Fuchs, C; Wolter, H H
1996-01-01
We study the influence of non-equilibrium phase space effects on the dynamics of heavy ion reactions within the relativistic BUU approach. We use realistic Dirac-Brueckner-Hartree-Fock (DBHF) mean fields determined for two-Fermi-ellipsoid configurations, i.e. for colliding nuclear matter, in a local phase space configuration approximation (LCA). We compare to DBHF mean fields in the local density approximation (LDA) and to the non-linear Walecka model. The results are further compared to flow data of the reaction Au on Au at 400 MeV per nucleon measured by the FOPI collaboration. We find that the DBHF fields reproduce the experiment if the configuration dependence is taken into account. This has also implications on the determination of the equation of state from heavy ion collisions.
Mean-field effects may mimic number squeezing in Bose-Einstein condensates in optical lattices
This paper presents a mean-field numerical analysis, using the full three-dimensional time-dependent Gross-Pitaevskii equation (GPE), of an experiment carried out by Orzel et al. [Science 291, 2386 (2001)] intended to show number squeezing in a gaseous Bose-Einstein condensate in an optical lattice. The motivation for the present work is to elucidate the role of mean-field effects in understanding the experimental results of this work and those of related experiments. We show that the nonadiabatic loading of atoms into optical lattices reproduces many of the main results of the Orzel et al. experiment, including both loss of interference patterns as laser intensity is increased and their regeneration when intensities are lowered. The nonadiabaticity found in the GPE simulations manifests itself primarily in a coupling between the transverse and longitudinal dynamics
Ion-metal and ion-atom collisions instant replays and mean-field theories
In this paper, we describe the results of our general long-term programmatic goal of investigating the strengths and weaknesses of time-dependent mean-field theories for collisions. We have made some progress in: (a) obtaining a better formulation of the theory, which has the exact full Schroedinger equation as one limit and permits appropriate classical treatment of heavy particles correctly coupled to the quantally treated electrons; (b) restructuring our numerical treatment to make it fully three-dimensional, improve accuracy and decrease cycle time, so that larger problems more in keeping with the mean-field concept can be treated; and (c) incorporating the electrons in the conduction band of a metal into our quantal treatment, making possible the description of collisions of atoms and ions with solids. Numerical results for protons tranversing a thin metallic foil, among other examples, are presented and discussed
Short-range correlations in an extended time-dependent mean-field theory
A generalization is performed of the time-dependent mean-field theory by an explicit inclusion of strong short-range correlations on a level of microscopic reversibility relating them to realistic nucleon-nucleon forces. Invoking a least action principle for correlated trial wave functions, equations of motion for the correlation functions and the single-particle model wave function are derived in lowest order of the FAHT cluster expansion. Higher order effects as well as long-range correlations are consider only to the extent to which they contribute to the mean field via a readjusted phenomenological effective two-body interaction. The corresponding correlated stationary problem is investigated and appropriate initial conditions to describe a heavy ion reaction are proposed. The singleparticle density matrix is evaluated
Mean-field description of collapsing and exploding Bose-Einstein condensates
We perform numerical simulations based on the time-dependent mean-field Gross-Pitaevskii equation to understand some aspects of a recent experiment by Donley et al. [Nature (London) 412, 295 (2001)] on the dynamics of collapsing and exploding Bose-Einstein condensates of 85Rb atoms. These authors manipulated the atomic interaction by an external magnetic field via a Feshbach resonance, thus changing the repulsive condensate into an attractive one, and vice versa. In the actual experiment they suddenly changed the scattering length of atomic interaction from a positive to a large negative value on a preformed condensate in an axially symmetric trap. Consequently, the condensate collapsed and ejected atoms via explosion. We find that the present mean-field analysis can explain some aspects of the dynamics of the collapsing and exploding Bose-Einstein condensates
Modeling and computation of mean field equilibria in producers' game with emission permits trading
Zhang, Shuhua; Wang, Xinyu; Shanain, Aleksandr
2016-08-01
In this paper, we present a mean field game to model the production behaviors of a very large number of producers, whose carbon emissions are regulated by government. Especially, an emission permits trading scheme is considered in our model, in which each enterprise can trade its own permits flexibly. By means of the mean field equilibrium, we obtain a Hamilton-Jacobi-Bellman (HJB) equation coupled with a Kolmogorov equation, which are satisfied by the adjoint state and the density of producers (agents), respectively. Then, we propose a so-called fitted finite volume method to solve the HJB equation and the Kolmogorov equation. The efficiency and the usefulness of this method are illustrated by the numerical experiments. Under different conditions, the equilibrium states as well as the effects of the emission permits price are examined, which demonstrates that the emission permits trading scheme influences the producers' behaviors, that is, more populations would like to choose a lower rather than a higher emission level when the emission permits are expensive.
Damping of collective nuclear motion and thermodynamic properties of nuclei beyond mean field
The dynamical description of correlated nuclear motion is based on a set of coupled equations of motion for the one-body density matrix ρ(11';t) and the two-body correlation function c2(12, 1'2';t), which is obtained from the density-matrix hierarchy beyond conventional mean-field approaches by truncating three-body correlations. The resulting equations non-perturbatively describe particle-particle collisions (short-range correlations) as well as particle-hole interactions (long-range correlations). Within a basis of time-dependent Hartree-Fock states these equations of motion are solved for collective vibrations of 40Ca at several finite thermal excitation energies corresponding to temperatures T = 0 - 6 MeV. Transport coefficients for friction and diffusion are extracted from the explicit solutions in comparison to the solutions of the associated TDHF, VUU, Vlasov or damped quantum oscillator equations of motion. We find that the actual magnitude of the transport coefficients is strongly influenced by particle-hole correlations at low temperature which generate large fluctuations in the nuclear shape degrees of freedom. Thermodynamically, the specific heat and the entropy of the system as a function of temperature does not differ much from the mean-field limit except for a bump in the specific heat around T ≅ 4 MeV which we attribute to the melting of shell effects in the correlated system
Generalized quantum mean-field systems and their application to ultracold atoms
Trimborn-Witthaut, Friederike Annemarie
2011-11-18
the subspace of Bose-symmetric states and discuss their representation by quantum phase space distributions in terms of generalized coherent states. In particular, this allows for an explicit calculation of the evolution equations and bounds for the ground state energy. In the second part of this thesis we analyse the dynamics of ultracold atoms in optical lattices described by the Bose-Hubbard Hamiltonian, which provide an important example of the generalized quantum mean-field systems treated in the first part. In the mean-field limit the dynamics is described by the (discrete) Gross-Pitaevskii equation. We give a detailed analysis of the interplay between dissipation and strong interactions in different dynamical settings, where we especially focus on the relation between the mean-field description and the full many-particle dynamics given by a master equation. (orig.)
Fictive impurity approach to dynamical mean field theory
A new extension of the dynamical mean-field theory was investigated in the regime of large Coulomb repulsion. A number of physical quantities such as single-particle density of states, spin-spin correlation, internal energy and Neel temperature, were computed for a two-dimensional Hubbard model at half-filling. The numerical data were compared to our analytical results as well as to the results computed using the dynamical cluster approximation. In the second part of this work we consider a two-plane Hubbard model. The transport properties of the bilayer were investigated and the phase diagram was obtained. (orig.)
A new Mean Field Approach for Exotic Nuclei
We present a new phenomenological mean field approach aiming at the calculation of properties of exotic nuclei. This approach combines the microscopic description of the spin-orbit properties in terms of particle densities, but also vector spin-orbit densities, inspired by results obtained within the Skyrme Hartree-Fock formalism, thus including the contribution of the tensor force. At the same time, the new approach preserves the simplicity of the phenomenological Woods-Saxon calculations and, more importantly, the robustness of the latter towards extrapolations in terms of increasing number of particles and/or isospin. (author)
Fictive impurity approach to dynamical mean field theory
Fuhrmann, A.
2006-10-15
A new extension of the dynamical mean-field theory was investigated in the regime of large Coulomb repulsion. A number of physical quantities such as single-particle density of states, spin-spin correlation, internal energy and Neel temperature, were computed for a two-dimensional Hubbard model at half-filling. The numerical data were compared to our analytical results as well as to the results computed using the dynamical cluster approximation. In the second part of this work we consider a two-plane Hubbard model. The transport properties of the bilayer were investigated and the phase diagram was obtained. (orig.)
Relativistic Mean Field Study of the Z = 117 Isotopic Chain
The properties of the Z = 117 isotopic chain are studied within the framework of the axially deformed relativistic mean field theory (RMFT) in the blocked BCS approximation. The ground-state properties, such as binging energies, deformations as well as the possible α decay energies and lifetimes are calculated with the parameter set of NL-Z2 and compared with results from the finite range droplet model. The analysis by RMFT shows that the isotopes in the range of mass number A = 291 ∼ 300 exhibit higher stability, which suggests that they may be promising nuclei to be hopefully synthesized in the lab among the nuclei Z = 117. (nuclear physics)
Relativistic mean field study of the Z=117 isotopic chain
The properties of the Z=117 isotopic chain are studied within the framework of the axially deformed relativistic mean field theory (RMFT) in the blocked BCS approximation. The ground-state properties, such as binding energies, deformations as well as the possible α decay energies and lifetimes are calculated with the parameter set of NL-Z2 and compared with results from the finite range droplet model. The analysis by RMFT shows that the isotopes in the range of mass number A=291-300 exhibit higher stability, which suggests that they may be promising nuclei to be hopefully synthesized in the lab among the nuclei Z-117. (authors)
A Mean Field Game Approach to Scheduling in Cellular Systems
Manjrekar, Mayank; Ramaswamy, Vinod; Shakkottai, Srinivas
2013-01-01
We study auction-theoretic scheduling in cellular networks using the idea of mean field equilibrium (MFE). Here, agents model their opponents through a distribution over their action spaces and play the best response. The system is at an MFE if this action is itself a sample drawn from the assumed distribution. In our setting, the agents are smart phone apps that generate service requests, experience waiting costs, and bid for service from base stations. We show that if we conduct a second-pr...
Relativistic mean field study of clustering in nuclei
Clustering phenomenon in exotic, light, heavy and superheavy nuclei is studied within the relativistic mean field (RMF) approach. Numerical calculations are done by using the axially deformed harmonic oscillator basis. The calculated nucleon density distributions and deformation parameters are analyzed to look for the cluster configurations. In case of light nuclei, the calculations explain many of the well established cluster structures in both the ground and intrinsic excited states. In the heavy and superheavy nuclei, interesting results are obtained and the results indicate new possibilities of exotic clusters at the centre of superheavy nuclei. (author)
Benchmarking mean-field approximations to level densities
Alhassid, Y.; Bertsch, G. F.; Gilbreth, C. N.; Nakada, H.
2016-04-01
We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus 162Dy, representing a heavy deformed nucleus, and 148Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are found to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point approximation to extract the number-projected densities is not a significant source of error compared to other errors inherent to the mean-field theory. We also present an alternative formulation of the saddle-point approximation that makes direct use of an approximate particle-number projection and avoids computing the usual three-dimensional Jacobian of the saddle-point integration. We find that the pairing condensate is less amenable to approximate particle-number projection methods because of the explicit violation of particle-number conservation in the pairing condensate. Nevertheless, the Hartree-Fock-Bogoliubov theory is accurate to less than one unit of entropy for 148Sm at the neutron threshold energy, which is above the pairing phase transition. This result provides support for the commonly used "back-shift" approximation, treating pairing as only affecting the excitation energy scale. When the ground state is strongly deformed, the Hartree-Fock entropy is significantly
A mechanical approach to mean field spin models
Genovese, Giuseppe; Barra, Adriano
2008-01-01
Inspired by the bridge pioneered by Guerra among statistical mechanics on lattice and analytical mechanics on 1+1 continuous Euclidean space-time, we built a self-consistent method to solve for the thermodynamics of mean-field models defined on lattice, whose order parameters self average. We show the whole procedure by analyzing in full details the simplest test case, namely the Curie-Weiss model. Further we report some applications also to models whose order parameters do not self-average, ...
A mechanical approach to mean field spin models
Genovese, Giuseppe
2008-01-01
Inspired by the bridge pioneered by Guerra among statistical mechanics on lattice and analytical mechanics on 1+1 continuous Euclidean space-time, we built a self-consistent method to solve for the thermodynamics of mean-field models defined on lattice, whose order parameters self average. We show the whole procedure by analyzing in full details the simplest test case, namely the Curie-Weiss model. Further we report some applications also to models whose order parameters do not self-average, by using the Sherrington-Kirkpatrick spin glass as a guide.
Relativistic mean field study of clustering in light nuclei
The clustering phenomenon in light, stable and exotic nuclei is studied within the relativistic mean field (RMF) approach. Numerical calculations are done by using the axially deformed harmonic oscillator basis. The calculated nucleon density distributions and deformation parameters are analyzed to look for the cluster configurations. The calculations explain many of the well-established cluster structures in both the ground and intrinsic excited states. Comparisons of our results with other model calculations and the available experimental information suggest that the RMF theory is well suited for studying clustering in light nuclei. A few discrepancies and their possible sources are also discussed
Nuclear Density-Dependent Effective Coupling Constants in the Mean-Field Theory
Lee, J H; Lee, S J; Lee, Jae Hwang; Lee, Young Jae; Lee, Suk-Joon
1996-01-01
It is shown that the equation of state of nuclear matter can be determined within the mean-field theory of $\\sigma \\omega$ model provided only that the nucleon effective mass curve is given. We use a family of the possible nucleon effective mass curves that reproduce the empirical saturation point in the calculation of the nuclear binding energy curves in order to obtain density-dependent effective coupling constants. The resulting density-dependent coupling constants may be used to study a possible equation of state of nuclear system at high density or neutron matter. Within the constraints used in this paper to $M^*$ of nuclear matter at saturation point and zero density, neutron matter of large incompressibility is strongly bound at high density while soft neutron matter is weakly bound at low density. The study also exhibits the importance of surface vibration modes in the study of nuclear equation of state.
We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green close-quote s functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article. copyright 1996 The American Physical Society
HBT pion interferometry with phenomenological mean field interaction
Hattori, Koichi
2010-01-01
In order to extract the information of the hadron production dynamics in ultra-relativistic heavy ion collisions, the space-time structure of the hadron source has been measured using Hanbury Brown and Twiss interferometry. We study the distortion of the source images due to the effect of a final state interaction. We describe the interactions, taking place while penetrating through the cloud formed by evaporating particles, in terms of an one-body mean field potential localized in the vicinity of the source region. Adopting the semi-classical method, the modification of the propagation of an emitted particle is examined. In analogy to the optical model applied to the nuclear reactions, our phenomenological model has an imaginary part of the potential, which describes the absorption in the cloud. In this work, we focus on the pion interferometry and the mean field interaction obtained using a phenomenological $\\pi\\pi$ forward scattering amplitude in the elastic channels. The p-wave scattering with rho meson r...
The application of mean field theory to image motion estimation.
Zhang, J; Hanauer, G G
1995-01-01
Previously, Markov random field (MRF) model-based techniques have been proposed for image motion estimation. Since motion estimation is usually an ill-posed problem, various constraints are needed to obtain a unique and stable solution. The main advantage of the MRF approach is its capacity to incorporate such constraints, for instance, motion continuity within an object and motion discontinuity at the boundaries between objects. In the MRF approach, motion estimation is often formulated as an optimization problem, and two frequently used optimization methods are simulated annealing (SA) and iterative-conditional mode (ICM). Although the SA is theoretically optimal in the sense of finding the global optimum, it usually takes many iterations to converge. The ICM, on the other hand, converges quickly, but its results are often unsatisfactory due to its "hard decision" nature. Previously, the authors have applied the mean field theory to image segmentation and image restoration problems. It provides results nearly as good as SA but with much faster convergence. The present paper shows how the mean field theory can be applied to MRF model-based motion estimation. This approach is demonstrated on both synthetic and real-world images, where it produced good motion estimates. PMID:18289956
Non-local correlations within dynamical mean field theory
Li, Gang
2009-03-15
The contributions from the non-local fluctuations to the dynamical mean field theory (DMFT) were studied using the recently proposed dual fermion approach. Straight forward cluster extensions of DMFT need the solution of a small cluster, where all the short-range correlations are fully taken into account. All the correlations beyond the cluster scope are treated in the mean-field level. In the dual fermion method, only a single impurity problem needs to be solved. Both the short and long-range correlations could be considered on equal footing in this method. The weak-coupling nature of the dual fermion ensures the validity of the finite order diagram expansion. The one and two particle Green's functions calculated from the dual fermion approach agree well with the Quantum Monte Carlo solutions, and the computation time is considerably less than with the latter method. The access of the long-range order allows us to investigate the collective behavior of the electron system, e.g. spin wave excitations. (orig.)
Benchmarking mean-field approximations to level densities
Alhassid, Y; Gilbreth, C N; Nakada, H
2015-01-01
We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point appr...
Non-local correlations within dynamical mean field theory
The contributions from the non-local fluctuations to the dynamical mean field theory (DMFT) were studied using the recently proposed dual fermion approach. Straight forward cluster extensions of DMFT need the solution of a small cluster, where all the short-range correlations are fully taken into account. All the correlations beyond the cluster scope are treated in the mean-field level. In the dual fermion method, only a single impurity problem needs to be solved. Both the short and long-range correlations could be considered on equal footing in this method. The weak-coupling nature of the dual fermion ensures the validity of the finite order diagram expansion. The one and two particle Green's functions calculated from the dual fermion approach agree well with the Quantum Monte Carlo solutions, and the computation time is considerably less than with the latter method. The access of the long-range order allows us to investigate the collective behavior of the electron system, e.g. spin wave excitations. (orig.)
Active matter beyond mean-field: Ring-kinetic theory for self-propelled particles
Chou, Yen-Liang; Ihle, Thomas
2014-01-01
A ring-kinetic theory for Vicsek-style models of self-propelled agents is derived from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle pre-collisional correlations and cluster formation which both seem important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small density expansion of the collision operator and de...
What can we learn from recent non-relativistic mean field calculations ?
In the present contribution, we discuss the relevance of fully self-consistent HF plus RPA (or HF-BCS plus QRPA) calculations based on Skyrme effective forces, and their impact on our present knowledge of basic properties of the nuclear equation of state, like the incompressibilty and the symmetry energy. Finally, we address the problem whether correlations beyond mean field can alter the picture obtained at the (Q)RPA level. Throghout the paper, the comparison with the results obtained using Gogny forces, or RMF Lagrangians, will be emphasized
Mean field mutation dynamics and the continuous Luria-Delbr\\"uck distribution
Kashdan, Eugene
2011-01-01
The Luria-Delbr\\"uck mutation model has a long history and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using some mathematical tools from nonlinear statistical physics. Starting from the classical formulations we derive the corresponding differential models and show that under a suitable mean field scaling they correspond to generalized Fokker-Planck equations for the mutants distribution whose solutions are given by the corresponding Luria-Delbr\\"uck distribution. Numerical results confirming the theoretical analysis are also presented.
Accretion disks and dynamos: toward a unified mean field theory
Conversion of gravitational energy into radiation near stars and compact objects in accretion disks and the origin of large-scale magnetic fields in astrophysical rotators have often been distinct topics of active research in astrophysics. In semi-analytic work on both problems it has been useful to presume large-scale symmetries, which necessarily results in mean field theories; magnetohydrodynamic turbulence makes the underlying systems locally asymmetric and highly nonlinear. Synergy between theory and simulations should aim for the development of practical, semi-analytic mean field models that capture the essential physics and can be used for observational modeling. Mean field dynamo (MFD) theory and alpha-viscosity accretion disk theory have exemplified such ongoing pursuits. Twenty-first century MFD theory has more nonlinear predictive power compared to 20th century MFD theory, whereas alpha-viscosity accretion theory is still in a 20th century state. In fact, insights from MFD theory are applicable to accretion theory and the two are really artificially separated pieces of what should ultimately be a single coupled theory. I discuss pieces of progress that provide clues toward a unified theory. A key concept is that large-scale magnetic fields can be sustained via local or global magnetic helicity fluxes or via relaxation of small-scale magnetic fluctuations, without appealing to the traditional kinetic helicity driver of 20th century textbooks. These concepts may help explain the formation of large-scale fields that supply non-local angular momentum transport via coronae and jets in a unified theory of accretion and dynamos. In diagnosing the role of helicities and helicity fluxes in disk simulations, it is important to study each disk hemisphere separately to avoid being potentially misled by the cancelation that occurs as a result of reflection asymmetry. The fraction of helical field energy in disks is expected to be small compared to the total field in
Relativistic Mean-Field Models and Nuclear Matter Constraints
Dutra, M; Carlson, B V; Delfino, A; Menezes, D P; Avancini, S S; Stone, J R; Providência, C; Typel, S
2013-01-01
This work presents a preliminary study of 147 relativistic mean-field (RMF) hadronic models used in the literature, regarding their behavior in the nuclear matter regime. We analyze here different kinds of such models, namely: (i) linear models, (ii) nonlinear \\sigma^3+\\sigma^4 models, (iii) \\sigma^3+\\sigma^4+\\omega^4 models, (iv) models containing mixing terms in the fields \\sigma and \\omega, (v) density dependent models, and (vi) point-coupling ones. In the finite range models, the attractive (repulsive) interaction is described in the Lagrangian density by the \\sigma (\\omega) field. The isospin dependence of the interaction is modeled by the \\rho meson field. We submit these sets of RMF models to eleven macroscopic (experimental and empirical) constraints, used in a recent study in which 240 Skyrme parametrizations were analyzed. Such constraints cover a wide range of properties related to symmetric nuclear matter (SNM), pure neutron matter (PNM), and both SNM and PNM.
Nonhelical mean-field dynamos in a sheared turbulence
Rogachevskii, I
2008-01-01
Mechanisms of nonhelical large-scale dynamos (shear-current dynamo and effect of homogeneous kinetic helicity fluctuations with zero mean) in a homogeneous turbulence with large-scale shear are discussed. We have found that the shear-current dynamo can act even in random flows with small Reynolds numbers. However, in this case mean-field dynamo requires small magnetic Prandtl numbers (i.e., ${\\rm Pm} < {\\rm Pm}^{\\rm cr}<1$). The threshold in the magnetic Prandtl number, ${\\rm Pm}^{\\rm cr} = 0.24$, is determined using second order correlation approximation (or first-order smoothing approximation) for a background random flow with a scale-dependent viscous correlation time $\\tau_c=(\
Double binding energy differences: Mean-field or pairing effect?
Qi, Chong
2012-10-01
In this Letter we present a systematic analysis on the average interaction between the last protons and neutrons in atomic nuclei, which can be extracted from the double differences of nuclear binding energies. The empirical average proton-neutron interaction Vpn thus derived from experimental data can be described in a very simple form as the interplay of the nuclear mean field and the pairing interaction. It is found that the smooth behavior as well as the local fluctuations of the Vpn in even-even nuclei with N ≠ Z are dominated by the contribution from the proton-neutron monopole interactions. A strong additional contribution from the isoscalar monopole interaction and isovector proton-neutron pairing interaction is seen in the Vpn for even-even N = Z nuclei and for the adjacent odd-A nuclei with one neutron or proton being subtracted.
Metabifurcation analysis of a mean field model of the cortex
Frascoli, Federico; Bojak, Ingo; Liley, David T J
2010-01-01
Mean field models (MFMs) of cortical tissue incorporate salient features of neural masses to model activity at the population level. One of the common aspects of MFM descriptions is the presence of a high dimensional parameter space capturing neurobiological attributes relevant to brain dynamics. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two families. After investigating and characterizing these, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli, distribution of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs ...
Shell Model and Mean-Field Description of Band Termination
Zalewski, M; Nazarewicz, W; Stoitcheva, G; Zdunczuk, H
2007-01-01
We study nuclear high-spin states undergoing the transition to the fully stretched configuration with maximum angular momentum I_max within the space of valence nucleons. To this end, we perform a systematic theoretical analysis of non-fully-stretched I_max-2 and I_max-1 f_{7/2}^n seniority isomers and d_{3/2}^{-1} f_{7/2}^{n+1} intruder states in the A~44 nuclei from the lower-fp shell. We employ two theoretical approaches: (i) the density functional theory based on the cranked self-consistent Skyrme-Hartree-Fock method, and (ii) the nuclear shell model in the full sdfp configuration space allowing for 1p-1h cross-shell excitations. We emphasize the importance of restoration of broken angular momentum symmetry inherently obscuring the mean-field treatment of high-spin states. Overall good agreement with experimental data is obtained.
Driven-dissipative Ising model: Mean-field solution
Goldstein, G.; Aron, C.; Chamon, C.
2015-11-01
We study the fate of the Ising model and its universal properties when driven by a rapid periodic drive and weakly coupled to a bath at equilibrium. The far-from-equilibrium steady-state regime is accessed by means of a Floquet mean-field approach. We show that, depending on the details of the bath, the drive can strongly renormalize the critical temperature to higher temperatures, modify the critical exponents, or even change the nature of the phase transition from second to first order after the emergence of a tricritical point. Moreover, by judiciously selecting the frequency of the field and by engineering the spectrum of the bath, one can drive a ferromagnetic Hamiltonian to an antiferromagnetically ordered phase and vice versa.
A relativistic mean field study of multi-strange system
In this paper, we study the binding energies, radii, single-particle energies, spin-orbit potential and density profile for multi-strange hypernuclei in the range of light mass to superheavy mass region within the relativistic mean field (RMF) theory. The stability of multi-strange hypernuclei as a function of introduced hyperons (Λ and Σ) is investigated. The neutron, lambda and sigma mean potentials are presented for light to superheavy hypernuclei. The inclusion of hyperons affects the nucleon, lambda and sigma spin-orbit potentials significantly. The bubble structure of nuclei and corresponding hypernuclei is studied. Nucleon and lambda halo structures are also investigated. A large class of bound multi-strange systems formed from the combination of nucleons and hyperons (n, p, Λ, Σ+ and n, p, Λ, Σ-) is suggested in the region of superheavy hypernuclei which might be stable against the strong decay. These multi-strange systems might be produced in heavy-ion reactions. (author)
Nuclear Level Density: Shell Model vs Mean Field
Sen'kov, Roman
2015-01-01
The knowledge of the nuclear level density is necessary for understanding various reactions including those in the stellar environment. Usually the combinatorics of Fermi-gas plus pairing is used for finding the level density. Recently a practical algorithm avoiding diagonalization of huge matrices was developed for calculating the density of many-body nuclear energy levels with certain quantum numbers for a full shell-model Hamiltonian. The underlying physics is that of quantum chaos and intrinsic thermalization in a closed system of interacting particles. We briefly explain this algorithm and, when possible, demonstrate the agreement of the results with those derived from exact diagonalization. The resulting level density is much smoother than that coming from the conventional mean-field combinatorics. We study the role of various components of residual interactions in the process of thermalization, stressing the influence of incoherent collision-like processes. The shell-model results for the traditionally...
Glauber Dynamics for the mean-field Potts Model
Cuff, Paul; Louidor, Oren; Lubetzky, Eyal; Peres, Yuval; Sly, Allan
2012-01-01
We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $\\beta_s(q)$ strictly lower than the critical $\\beta_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $\\beta_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $\\beta\\beta_s(q)$ the mixing time is exponentially large in $n$. Furthermore, as $\\beta \\uparrow \\beta_s$ with $n$, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of $O(n^{-2/3})$ around $\\beta_s$. These results form the first complete analysis of the critical slowdown of a dynamics with a first order phase transition.
Spectral Synthesis via Mean Field approach Independent Component Analysis
Hu, Ning; Kong, Xu
2015-01-01
In this paper, we apply a new statistical analysis technique, Mean Field approach to Bayesian Independent Component Analysis (MF-ICA), on galaxy spectral analysis. This algorithm can compress the stellar spectral library into a few Independent Components (ICs), and galaxy spectrum can be reconstructed by these ICs. Comparing to other algorithms which decompose a galaxy spectrum into a combination of several simple stellar populations, MF-ICA approach offers a large improvement in the efficiency. To check the reliability of this spectral analysis method, three different methods are used: (1) parameter-recover for simulated galaxies, (2) comparison with parameters estimated by other methods, and (3) consistency test of parameters from the Sloan Digital Sky Survey galaxies. We find that our MF-ICA method not only can fit the observed galaxy spectra efficiently, but also can recover the physical parameters of galaxies accurately. We also apply our spectral analysis method to the DEEP2 spectroscopic data, and find...
Mean field games with nonlinear mobilities in pedestrian dynamics
Burger, Martin
2014-04-01
In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results.
Quantum de Finetti theorems and mean-field theory from quantum phase space representations
Trimborn, F.; Werner, R. F.; Witthaut, D.
2016-04-01
We introduce the number-conserving quantum phase space description as a versatile tool to address fundamental aspects of quantum many-body systems. Using phase space methods we prove two alternative versions of the quantum de Finetti theorem for finite-dimensional bosonic quantum systems, which states that a reduced density matrix of a many-body quantum state can be approximated by a convex combination of product states where the error is proportional to the inverse particle number. This theorem provides a formal justification for the mean-field description of many-body quantum systems, as it shows that quantum correlations can be neglected for the calculation of few-body observables when the particle number is large. Furthermore we discuss methods to derive the exact evolution equations for quantum phase space distribution functions as well as upper and lower bounds for the ground state energy. As an important example, we consider the Bose-Hubbard model and show that the mean-field dynamics is given by a classical phase space flow equivalent to the discrete Gross-Pitaevskii equation.
Quantum de Finetti theorems and mean-field theory from quantum phase space representations
We introduce the number-conserving quantum phase space description as a versatile tool to address fundamental aspects of quantum many-body systems. Using phase space methods we prove two alternative versions of the quantum de Finetti theorem for finite-dimensional bosonic quantum systems, which states that a reduced density matrix of a many-body quantum state can be approximated by a convex combination of product states where the error is proportional to the inverse particle number. This theorem provides a formal justification for the mean-field description of many-body quantum systems, as it shows that quantum correlations can be neglected for the calculation of few-body observables when the particle number is large. Furthermore we discuss methods to derive the exact evolution equations for quantum phase space distribution functions as well as upper and lower bounds for the ground state energy. As an important example, we consider the Bose–Hubbard model and show that the mean-field dynamics is given by a classical phase space flow equivalent to the discrete Gross–Pitaevskii equation. (paper)
A mean-field thermodynamic description of the kinetics of overdriven interfaces
Haxhimali, Tomorr; Belof, Jonathan; Sadigh, Babak
A key aspect of an accurate description of shock-induced structural phase transitions is the rigorous computation of the dynamics of the interfaces between coexisting phases. In the wake of the shock, the system will be exposed to strong gradient fields that give rise to overdriven interfaces during the induced phase transformation. In this work we take a mean-field approach using a time-dependent Ginzburg-Landau formalism to describe the dynamics of such overdriven interfaces. We make a connection of the mean-field result to a quasi-Langevin description, the Kardar-Parisi-Zhang (KPZ) equation, of the kinetics of the interface. Further, larger coarse-grained descriptions of the phase transition such as the Kolmogorov-Johnson-Mehl-Avrami (KJMA) model, which are commonly coupled to hydrodynamic equations that describe the evolution of the temperature and pressure during the shock propagation, ignore the details of the dynamics and structure of the interfacial regions. Overlaying the KPZ description of the interface evolution to these coarse-grained methods will result in physically more accurate multiscale models for shock propagation. We will present results from our efforts in this regard. This work is performed under the auspices of the U. S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows
Peirano, Eric; Pozorski, Jacek; Minier, Jean-Pierre
2010-01-01
The purpose of this paper is to give an overview in the realm of numerical computations of polydispersed turbulent two-phase flows, using a mean-field/PDF approach. In this approach, the numerical solution is obtained by resorting to a hybrid method where the mean fluid properties are computed by solving mean-field (RANS) equations with a classical finite volume procedure whereas the local instantaneous properties of the particles are determined by solving stochastic differential equations (SDEs). The fundamentals of the general formalism are recalled and particular attention is focused on a specific theoretical issue: the treatment of the multiscale character of the dynamics of the discrete particles, that is the consistency of the system of SDEs in asymptotic cases. Then, the main lines of the particle/mesh algorithm are given and some specific problems, related to the integration of the SDEs, are discussed, for example, issues related to the specificity of the treatment of the averaging and projection oper...
Mean-field approximation of two coupled populations of excitable units
Franović, Igor; Todorović, Kristina; Vasović, Nebojša; Burić, Nikola
2013-01-01
The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of N stochastic excitable units each is performed by studying an approximate system, obtained by replacing each population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the interensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of 4N stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset, and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical interpopulation couplings.
Variational extensions of the time-dependent mean-field theory
Using the Balian-Veneroni variational principle, we propose two consistent extensions of the time-dependent mean-field theory for many-boson systems. A first approximation, devised to take into account the effect of correlations, is obtained by means of a development of the optimal density operator suggested by the maximum entropy principle around a Gaussian operator. We discuss the relevance of the evolution equations and their possible generalizations. We present an application to an one-dimensional example. In a second type of approximation, to optimize the prediction of characteristic functions of one-body observables and of transition probabilities, we select for both, the variational observable and the density matrix, the class of exponential operators of quadratic forms. We obtain coupled evolution equations of an unusual kind called 'two-point boundary value problem'. To solve them, we construct a suitable numerical algorithm. A test of the method is presented on two examples in one dimension. In a first case, we study the collision of a particle against a Gaussian barrier. The method improves significantly mean-field predictions relative to reflexion and transmission ratios. The study of the motion of a particle in a quartic well reveals the existence of several different solutions for the transition probabilities predicted by the Balian-Veneroni method
Identifying Deficiencies of Standard Accretion Disk Theory: Lessons from a Mean-Field Approach
Hubbard, Alexander
2008-01-01
Turbulent viscosity is frequently used in accretion disk theory to replace the microphysical viscosity in order to accomodate the observational need for in- stabilities in disks that lead to enhanced transport. However, simply replacing the microphysical transport coefficient by a single turbulent transport coeffi- cient hides the fact that the procedure should formally arise as part of a closure in which the hydrodynamic or magnetohydrodynamic equations are averaged, and correlations of turbulent fluctuations are replaced by transport coefficients. Here we show how a mean field approach leads quite naturally two transport coefficients, not one, that govern mass and angular momentum transport. In particular, we highlight that the conventional approach suffers from a seemingly inconsistent neglect of turbulent diffusion in the surface density equation. We constrain these new transport coefficients for specific cases of inward, outward, and zero net mass transport. In addition, we find that one of the new trans...
Neutron-rich nuclei of mass A=100-110 are of great interest for the study of nuclear structure far from stability. Previous experimental and theoretical studies suggest a complex evolution of deformation and collectivity in the isotopic chains of Zr, Mo, Ru and Pd. In order to extend information on the evolution of the collectivity towards higher spin states and more neutron-rich nuclei, lifetimes of excited states were measured in nuclei produced through a fusion-fission reaction in inverse kinematic at GANIL. Fission fragments were separated and identified in both A and Z with the high acceptance magnetic spectrometer VAMOS while the EXOGAM germanium detectors array was used for the coincident gamma-ray detection. Lifetimes of about twenty excited states were extracted using the plunger device of Cologne. This is the first RDDS measurement on fission fragments which are identified in A and Z on an event-by-event basis. The study of this mass region is completed by theoretical calculations using self consistent mean field and beyond mean field methods implemented with the Gogny force (D1S). The structure of the ground states and the excited states is described with Hartree-Fock-Bogoliubov calculations with constraints placed on the axial and triaxial deformations. Individual excitations are investigated through blocking calculations and the high spin states are studied through cranking calculations. Finally, an approximated generator coordinate method (GCM+GOA) using the 5DCH Hamiltonian is used to describe the low energy collective states and to interpret the experimental evolution of the collectivity. (author)
Chen, Xuwen
2010-01-01
In this paper, we consider the Hamiltonian evolution of N weakly interacting Bosons. Assuming triple collisions with singular potentials, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t,x,y) and derive an evolution equation for k. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work in [20,11,12] where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order O(1/$\\sqrt{N}$).
A Mean-Field Theory for Coarsening Faceted Surfaces
Norris, Scott A
2009-01-01
A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in two-phase systems [1-3], but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling the work of Smoluchowski [4] and Schumann [5] on coalescence. The model is solved by the exponential distribution, but agreement with experiment is limited by the assumption that neighboring facet lengths are uncorrelated. However, the method concisely describes the essential processes operating in the scaling state, illuminates a clear path for future refinement, and offers a framework for the investigation of faceted surfaces evolving under arbitrary dynamics. [1] I. Lifshitz, V. Slezov, Soviet Physics JETP 38 (1959) 331-339. [2] I. Lifshitz, V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35-50. [3] C. Wagner, Elektrochemie 65 (1961) 581-591. [4] M. von S...
Mean-field inference of Hawkes point processes
Bacry, Emmanuel; Gaïffas, Stéphane; Mastromatteo, Iacopo; Muzy, Jean-François
2016-04-01
We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d-dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the maximum likelihood estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.