Note on the derivation of the Boltzmann equation from the Liouville equation

Cohen, E. G. D.; Berlin, T. H.

doi:10.1016/0031-8914(60)90061-6

Cited by 64 publications

(19 citation statements)

References 2 publications

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“…If instead the evolution of the system leading up to the equilibrium state at t = 0 had been considered, then all the inequalities associated with the second law would have been obtained, but with their signs reversed. This emphasizes the importance of boundary conditions (in time), and touches on the deep connection between irreversibility and causality [77,78,79].…”

Section: The Need To Modelmentioning

confidence: 99%

Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale

Jarzynski

2011

Annu. Rev. Condens. Matter Phys.

999

1,133

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The reason we never observe violations of the second law of thermodynamics is in part a matter of statistics: When ∼1023 degrees of freedom are involved, the odds are overwhelmingly stacked against the possibility of seeing significant deviations away from the mean behavior. As we turn our attention to smaller systems, however, statistical fluctuations become more prominent. In recent years it has become apparent that the fluctuations of systems far from thermal equilibrium are not mere background noise, but satisfy strong, useful, and unexpected properties. In particular, a proper accounting of fluctuations allows us to rewrite familiar inequalities of macroscopic thermodynamics as equalities. This review describes some of this progress, and argues that it has refined our understanding of irreversibility and the second law.

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Section: The Need To Modelmentioning

confidence: 99%

Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale

Jarzynski

2011

Annu. Rev. Condens. Matter Phys.

999

1,133

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“…[45] have the "wrong" signs, and the Boltzmann-like equation of Ref. [46] obeys an anti-H theorem. The situation is similar here: anti-causal ensembles are associated with negative average values of dissipated work; the second law of thermodynamics, by contrast, asserts that irreversible processes are accompanied by positive average dissipated work.…”

Section: Discussionmentioning

confidence: 99%

“…In the context of linear response theory, Evans and Searles [45] have shown that anti-causal ensembles give rise to Green-Kubo "anti-transport" coefficients. In an earlier theoretical study of dilute gases, Cohen and Berlin [46] derived an anti-causal version of the Boltzmann equation, by applying the assumption of molecular chaos to future rather than past pair distribution functions. In both papers the anti-causal behavior is associated with violations of the second law of thermodynamics: the Green-Kubo coefficients of Ref.…”

Section: Discussionmentioning

confidence: 99%

Rare events and the convergence of exponentially averaged work values

2006

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Equilibrium free energy differences are given by exponential averages of nonequilibrium work values; such averages, however, often converge poorly, as they are dominated by rare realizations. I show that there is a simple and intuitively appealing description of these rare but dominant realizations. This description is expressed as a duality between "forward" and "reverse" processes, and provides both heuristic insights and quantitative estimates regarding the number of realizations needed for convergence of the exponential average. Analogous results apply to the equilibrium perturbation method of estimating free energy differences. The pedagogical example of a piston and gas [R.C. Lua and A.Y. Grosberg, J. Phys. Chem. B 109, 6805 (2005)] is used to illustrate the general discussion.The nonequilibrium work theorem,relates the work performed on a system during a nonequilibrium process, to the free energy difference between two equilibrium states of that system. The angular brackets denote an average over an ensemble of realizations (repetitions) of a thermodynamic process, during which a system evolves in time as a control parameter λ is varied from an initial value A to a final value B. W is the external work performed on the system during one realization; ∆F = F B − F A is the free energy difference between two equilibrium states of the system, corresponding to λ = A and B; and β is the inverse temperature of a heat reservoir with which the system is equilibrated prior to the start of each realization of the process. A sample of derivations of Eq. 1 can be found in Refs. [1,2,3,4,5,6,7,8,9, 10]; pedagogical and review treatments are given in Refs. [11,12,13,14,15,16]; for experimental tests of this and closely related results, see Refs. [17,18,19,20]; finally, Refs. [21,22,23,24,25] discuss quantal versions of the theorem. In principle, Eq.1 implies that ∆F can be estimated using nonequilibrim experiments or numerical simulations. If we repeat the thermodynamic process N times, and observe work valueswhere the approximation becomes an equality in the limit of infinitely many realizations, N → ∞. In practice, the average of e −βW is often dominated by very rare realizations, leading to poor convergence with N . The aim of this paper is develop an understanding of these rare but important realizations. I will argue that there is a simple description of these dominant realizations, which leads to both quantitative estimates and useful heuristic insights regarding the number of realizations needed for convergence of the average of e −βW . The organization of this paper is as follows. In Section I the central result is summarized, then illustrated using a simple example. Section II contains a derivation of this result. Section III discusses the number of realizations needed for convergence of the exponential average. Section IV focuses on the free energy perturbation method (a limiting case of Eq.1), and Section V concludes with a brief discussion.

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“…In the case of the Bogoliubov theory (see Appendix A), it is the "initial condition" (1.3,) that introduces the time arrow by defining the "past" (Cohen and Berlin 1960). Equation (11.4) by itself still has no time arrow, as is easily seen from the symmetry of (11.4) in the two directions of time.…”

Section: (4) the Landau Dampingmentioning

confidence: 92%

“…Equatioil (1.2) states that the correlation effect vanishes as one traces the system bacltward in time T >> rO/u, where ro is the range of internzolecular interaction and ZL is the average velocity of the molecules. In other theories of irreversible processes, other assumptions are introduced (Cohen 1961). It has been sl~o\v~l that in the classical theory a less specific Ansatz in the form of a relation between the probabilities of the states of the gas a t two instants ol time will lead to a definite time arrow and hence to a theory of irreversible processes.…”

Section: Ti-ie Nature O F T H E T H E O R Y O F Irreversible Procementioning

confidence: 99%

Kinetic Equation Describing Irreversible Processes in Ionized Gases

Wu¹,

Rosenberg²

1962

Can. J. Phys.

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The present \vorl; deals with the formulation of the theory of irreversible processes in ionized gases on the basis of the theory of Bogoli~tbov for neutral gases. In a n estension of the \vork of Guernsey and Uhlenbeck, the kinetic equation is obtained for a spatially inhomogeneous system in an approximation to the "first order" ill the interaction strength (e') ancl to all orders in the correlation erect (e2/v, a being the specific volume). The equation, unlibe the usual Vlasov equation, is not invariaut wit11 respect to time-reversal and shows a genuine "dainping" in the sense of an irreversible approach to equilibrium. 111 the present paper, brief discussions have also been given of the nature of a theory of irreversible processes, the so-called "divergence difficulty", due to the long-range Coulomb forces, the I,a~~clau clampil~g-, the recent general theory of irrcvcrsible processes of Prigoginc and Balescu, and the theory of ionized gases of Balcscu. INTRODUCTIONUp to about 1946, the Boltzmann equation was the basis of the theory of the irreversible processes in a gas towards ther~nodylianlic equilibrium. I t was ltnowil very early that the theory iilvolves the so-called Stosszahlansatz which is not derivable from the dynainical equations of nlotiolis but must be understood on statistical considerations in the lnacroscopic descriptioii of the gas. I t is the Stosszal~lansatz that gives the Boltzmann equation the "irreversible character" in time. Tlie success of the equatioil in dealing with all the transport phenomena in gases, especially fro111 the of Chapman and of Ensltog, is well li110wn. I t was soon realized, however, by many physicists that the theory iilvolving the explicit asssunlptioli of bi~iary collisions between molecules cannot be easily generalized to apply to gases a t higher densities in which higher order collisioils become significant. About 1946, a few iildependent attempts were made by Bogoliubov, Born, and Green, I

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Note on the derivation of the Boltzmann equation from the Liouville equation

Cited by 64 publications

References 2 publications

Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale

Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale

Rare events and the convergence of exponentially averaged work values

Kinetic Equation Describing Irreversible Processes in Ionized Gases

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