Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

Approach to Equilibrium of A Large Fermion System

Kyoji Nishikawa

Published Jan 1, 1960• 8 citations
Publisher

Abstract

Using Feynman diagrams, the dissipative hehaviour of a large quantum system of many fermions is discussed. On the basis of linked cluster expansion of the time-dependent perturbation theory, the master equation is derived for the time sufficiently long compared to the effective collision time and for a slowly varying initial distribution in the momentum space. Van Hove's result for weak coupling limit and the equation for low density limit are obtained as special cases.

DOI
10.1143/jpsj.15.78

Related papers

Linear response theory revisited. II. The master equation approach

K. M. van Vliet

We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.

Journal of Mathematical Physics•Dec 1, 1979•81 citations

Dynamics of nuclear fluid. V. Extended time-dependent Hartree-Fock approximation illuminates the approach to thermal equilibrium

Cheuk-Yin Wong, Henry H. K. Tang

In the time-dependent Hartree-Fock approximation, the fermions are assumed to interact only through the mean field and the collisions between particles are neglected. We formulate an extended time-dependent Hartree-Fock approximation which incorporates particle collisions due to the residual interaction, with the usual time-dependent Hartree-Fock approximation as the collisionless limit. It is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of time-dependent occupation numbers. The final set of coupled equations consists of a modified time-dependent Hartree-Fock equation and a master equation for the occupation numbers. These results are physically transparent and describe properly the physics of the collision process. They may also be simple enough to be of practical use to study heavy-ion collisions or the dynamics of other fermion systems. Furthermore, as the configuration-space analog of the quantum Boltzmann equation, many important results concerning statistical dynamics are obtained. Concepts such as entropy, temperature, and local and thermal equilibrium can be quantitatively introduced. The well-known $H$ theorem that entropy never decreases can be readily recovered. With the collision term explicitly exhibited, the macroscopic equations (equations of continuity, momentum flux, and energy) and their associated conservation theorems can also be derived. Analytic solutions for the master equation for simple cases lead to new "level crossing" formulas having characteristics distinctly different from the Landau-Zener level-crossing formula and illuminate the salient features as to how a nonequilibrium fermion system approaches thermal equilibrium.NUCLEAR REACTIONS Extension of time-dependent Hartree-Fock approximation. Collisions between particles. Master equation for occupation probabilities. Entropy, temperature, and thermal equilibrium. $H$ theorem. Analytic solution of the master equation. New types of level crossing formula.

Physical Review C•Oct 1, 1979•62 citations

Theory of Strongly Coupled Many-Fermion Systems. I. Convergence of Linked-Cluster Expansions

Leon N. Cooper

A strongly coupled system---the limiting case of a highly degenerate many-fermion system for which the variation of the kinetic energy is neglected, and the interaction restricted to a region of momentum space neighboring the Fermi surface---has been analyzed in a manner not dependent upon assumptions about the convergence of power series expansions or on partial summations of infinite series. The vacuum expectation value of the resolvent operator, ${〈\frac{1}{(H\ensuremath{-}z)}〉}_{0}$, is expressed as the Laplace transform of the exponential of a function linearly dependent on the volume of the system. It is shown that the linked-cluster expansion of the vacuum expectation value of the resolvent operator has a zero radius of convergence as a power series in the coupling constant. The most serious physical consequence of this is that a nontrivial interaction never results in a "normal" system.

Physical Review•May 15, 1961•5 citations

Out-of-equilibrium quantum dynamics of fermionic gases in the presence of localized particle loss

Francesco Tarantelli, Ettore Vicari

We address the effects of dissipative defects giving rise to a localized particle loss, in one-dimensional noninteracting lattice fermionic gases confined within a region of size $\ensuremath{\ell}$. We consider homogeneous systems within hard walls and inhomogeneous systems where the particles are trapped by space-dependent external potentials, such as harmonic traps. We model the dissipative particle-decay mechanism by Lindblad master equations governing the time evolution of the density matrix. The resulting quantum dynamics is analyzed in protocols starting from the ground state of the Hamiltonian for ${N}_{0}$ particles, then evolving under the effect of one dissipative particle-loss defect, for example, at the center of the system. We study the interplay between time, size $\ensuremath{\ell}$, and the number ${N}_{0}$ of initial particles, considering two different situations: (i) fixed number ${N}_{0}$ of initial particles; and (ii) fixed ratio ${N}_{0}/\ensuremath{\ell}$, corresponding to the thermodynamic limit of the initial equilibrium state. We show that the quantum evolutions of the particle number and density develop various intermediate and asymptotic dynamic regimes, and nontrivial large-time states when the dissipative mechanism acts at the center of the system.

arXiv:2112.05180•Physical Review A•Apr 21, 2022•10 citations

MICROSCOPIC COEFFICIENTS FOR THE QUANTUM MASTER EQUATION OF A FERMI SYSTEM

Eliade N. Stefanescu, Aurel Sǎndulescu

In a previous paper, we derived a master equation for fermions, of Lindblad's form, with coefficients depending on microscopic quantities. In this paper, we study the properties of the dissipative coefficients taking into account the explicit expressions of: (a) the matrix elements of the dissipative potential, evaluated from the condition that, essentially, this potential induces transitions among the system eigenstates without significantly modifying these states, (b) the densities of the environment states according to the Thomas–Fermi model, and (c) the occupation probabilities of these states taken as a Fermi–Dirac distribution. The matrix of these coefficients correctly describes the system dynamics: (a) for a normal, Fermi–Dirac distribution of the environment population, the decays dominate the excitation processes; (b) for an inverted (exotic) distribution of this population, specific to a clustering state, the excitation processes are dominant.

International Journal of Modern Physics E•Apr 1, 2002•6 citations

Correlation Effects in Many Fermion Systems. II. Linked Clusters

Hugh P. Kelly

In a previous paper a set of coupled equations was derived for the ground-state wave function and energy of a finite system of interacting Fermions. The equations are now modified so as to be more applicable to systems in which the number of particles becomes large. The resulting equations are shown to be equivalent to those obtained from many-body perturbation theory.

Physical Review•Jun 15, 1964•105 citations
Using Feynman diagrams, the dissipative hehaviour of a large quantum system of many fermions is discussed. On the basis of linked cluster expansion of the time-dependent perturbation theory, the master equation is derived for the time sufficiently long compared to the effective collision time and for a slowly varying initial distribution in the momentum space. Van Hove's result for weak coupling limit and the equation for low density limit are obtained as special cases.
We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.
In the time-dependent Hartree-Fock approximation, the fermions are assumed to interact only through the mean field and the collisions between particles are neglected. We formulate an extended time-dependent Hartree-Fock approximation which incorporates particle collisions due to the residual interaction, with the usual time-dependent Hartree-Fock approximation as the collisionless limit. It is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of time-dependent occupation numbers. The final set of coupled equations consists of a modified time-dependent Hartree-Fock equation and a master equation for the occupation numbers. These results are physically transparent and describe properly the physics of the collision process. They may also be simple enough to be of practical use to study heavy-ion collisions or the dynamics of other fermion systems. Furthermore, as the configuration-space analog of the quantum Boltzmann equation, many important results concerning statistical dynamics are obtained. Concepts such as entropy, temperature, and local and thermal equilibrium can be quantitatively introduced. The well-known HHH theorem that entropy never decreases can be readily recovered. With the collision term explicitly exhibited, the macroscopic equations (equations of continuity, momentum flux, and energy) and their associated conservation theorems can also be derived. Analytic solutions for the master equation for simple cases lead to new "level crossing" formulas having characteristics distinctly different from the Landau-Zener level-crossing formula and illuminate the salient features as to how a nonequilibrium fermion system approaches thermal equilibrium.NUCLEAR REACTIONS Extension of time-dependent Hartree-Fock approximation. Collisions between particles. Master equation for occupation probabilities. Entropy, temperature, and thermal equilibrium. HHH theorem. Analytic solution of the master equation. New types of level crossing formula.
A strongly coupled system---the limiting case of a highly degenerate many-fermion system for which the variation of the kinetic energy is neglected, and the interaction restricted to a region of momentum space neighboring the Fermi surface---has been analyzed in a manner not dependent upon assumptions about the convergence of power series expansions or on partial summations of infinite series. The vacuum expectation value of the resolvent operator, ${〈\frac{1}{(H\ensuremath{-}z)}〉}_{0}$, is expressed as the Laplace transform of the exponential of a function linearly dependent on the volume of the system. It is shown that the linked-cluster expansion of the vacuum expectation value of the resolvent operator has a zero radius of convergence as a power series in the coupling constant. The most serious physical consequence of this is that a nontrivial interaction never results in a "normal" system.
We address the effects of dissipative defects giving rise to a localized particle loss, in one-dimensional noninteracting lattice fermionic gases confined within a region of size $\ensuremath{\ell}$. We consider homogeneous systems within hard walls and inhomogeneous systems where the particles are trapped by space-dependent external potentials, such as harmonic traps. We model the dissipative particle-decay mechanism by Lindblad master equations governing the time evolution of the density matrix. The resulting quantum dynamics is analyzed in protocols starting from the ground state of the Hamiltonian for N0{N}_{0}N0​ particles, then evolving under the effect of one dissipative particle-loss defect, for example, at the center of the system. We study the interplay between time, size $\ensuremath{\ell}$, and the number N0{N}_{0}N0​ of initial particles, considering two different situations: (i) fixed number N0{N}_{0}N0​ of initial particles; and (ii) fixed ratio ${N}_{0}/\ensuremath{\ell}$, corresponding to the thermodynamic limit of the initial equilibrium state. We show that the quantum evolutions of the particle number and density develop various intermediate and asymptotic dynamic regimes, and nontrivial large-time states when the dissipative mechanism acts at the center of the system.
In a previous paper, we derived a master equation for fermions, of Lindblad's form, with coefficients depending on microscopic quantities. In this paper, we study the properties of the dissipative coefficients taking into account the explicit expressions of: (a) the matrix elements of the dissipative potential, evaluated from the condition that, essentially, this potential induces transitions among the system eigenstates without significantly modifying these states, (b) the densities of the environment states according to the Thomas–Fermi model, and (c) the occupation probabilities of these states taken as a Fermi–Dirac distribution. The matrix of these coefficients correctly describes the system dynamics: (a) for a normal, Fermi–Dirac distribution of the environment population, the decays dominate the excitation processes; (b) for an inverted (exotic) distribution of this population, specific to a clustering state, the excitation processes are dominant.