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A Proof of Clausius' Theorem for Time Reversible Deterministic Microscopic A proof of Clausius’ theorem for time reversible deterministic microscopic dynamics Denis J. Evans, Stephen R. Williams, and Debra J. Searles Citation: The Journal of Chemical Physics 134, 204113 (2011); doi: 10.1063/1.3592531 View online: http://dx.doi.org/10.1063/1.3592531 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/134/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Statistical Mechanics of Infinite Gravitating Systems AIP Conf. Proc. 970, 222 (2008); 10.1063/1.2839122 Gravitational clustering: an overview AIP Conf. Proc. 970, 205 (2008); 10.1063/1.2839121 Comment on “Comment on ‘Simple reversible molecular dynamics algorithms for Nosé-Hoover chain dynamics’ ” [J. Chem. Phys.110, 3623 (1999)] J. Chem. Phys. 124, 217101 (2006); 10.1063/1.2202355 Electrostatic correlations and fluctuations for ion binding to a finite length polyelectrolyte J. Chem. 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Searles2 1Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia 2Queensland Micro- and Nanotechnology Centre and School of Biomolecular and Physical Sciences, Griffith University, Brisbane, Qld 4111, Australia (Received 17 February 2011; accepted 28 April 2011; published online 27 May 2011) In 1854 Clausius proved the famous theorem that bears his name by assuming the second “law” of thermodynamics. In the present paper we give a proof that requires no such assumption. Our proof rests on the laws of mechanics, a T-mixing property, an ergodic consistency condition, and on the axiom of causality. Our result relies on some recently derived theorems, such as the Evans-Searles and the Crooks fluctuation theorems and the recently discovered relaxation and dissipation theorems. © 2011 American Institute of Physics. [doi:10.1063/1.3592531] I. INTRODUCTION the Jarzynski equality to thermal processes. This then allows us in Sec. VI, to derive Clausius’ inequality for the heat, and Recently, we derived some generalizations1 of the to clarify the meaning of “temperature” that appears in this nonequilibrium work relations due to Crooks2 that are valid inequality. Finally, in the Sec. VII, we discuss other implica- for thermal rather than mechanical changes in time reversible, tions of these findings. In particular, we point out that unlike deterministic systems of interacting particles. Here, we extend dissipation, entropy does not appear to be a useful concept for this work to study cyclic integrals of the generalized work and deterministic nonequilibrium systems. heat, and in the process we give a proof of Clausius’ inequal- ity. This proof is derived without assuming the second law of thermodynamics. It also clarifies the meaning of the “tem- II. GENERALIZED CROOKS FLUCTUATION perature” that appears in Clausius’ inequality and shows that THEOREM (GCFT) the equality and the inequality only apply to periodic cyclic processes. Clausius’ inequality does not apply to aperiodic We consider two closed N-particle systems: 1, 2. These cycles. systems may have the same or different Hamiltonians, tem- Clausius’ inequality states that if a system responds peri- peratures, or volumes; it does not matter. Nor does the odically to a cyclic thermal process, then ensemble matter: microcanonical, canonical, or isothermal isobaric. A protocol, and the corresponding time-dependent therm dQ dynamics, is then defined that will eventually transform equi- lim lim P ≥ 0, (1) N→∞ t→∞ T librium system 1 into equilibrium system 2. The systems are λ where Qtherm is the heat transferred to the reservoir from the distinguished by introducing a parameter, , which takes on λ λ system, N is the number of particles in the system, and t is a value 1 in system 1 and 2 in system 2, and the trans- λ λ = λ the time over which the process occurs. In the past, there has formation is also parameterised through (t) with (0) 1 λ τ = λ been some question as to what temperature, T, should be used and ( ) 2. The equations of motion are therefore non- in this relation (especially for the strict inequality), and in- autonomous (i.e., they depend explicitly on absolute time). deed the question of how to define temperature, in general, Following Ref. 4, we define a generalized dimensionless τ has been an open question (see, Ref. 3). In this paper we pro- “work,” Xτ ( ), for a trajectory of duration , originating vide a proof of Eq. (1) which also clarifies exactly what this from the phase point (0), temperature refers to. peq,1(d(0))Z(λ1) In Secs. II and III, we present generalized forms of the exp[Xτ ((0))] = lim d(0)→0 p , (d(τ))Z(λ ) Crooks fluctuation theorem and the Jarzynski equality (JE), eq 2 2 λ respectively. Section IV presents the minimum average gener- feq,1( (0))d (0)Z( 1) ≡ , ∀(0) ∈ D1, alized work relationship, a corollary of the Jarzynski equality, feq,2((τ))d(τ)Z(λ2) and discusses its application to cyclic integrals of the gener- (2) alized work. These three relationships provide the formal re- sults required for the derivation of Clausius’ inequality, later where Z(λi ) is the partition function for the system i = 1, 2, in the paper. In Section V, we introduce a framework whereby D1 is the accessible phase space domain for system 1 (e.g., we can apply the generalized Crooks fluctuation theorem and coordinates in a fixed special range (−L, +L) and momenta range (−∞, +∞)), and feq,i () is the equilibrium phase a)Author to whom correspondence should be addressed. Electronic mail: space distribution for system i. Without loss of generality we [email protected]. assume that both equilibrium distributions functions are even 0021-9606/2011/134(20)/204113/7/$30.00134, 204113-1 © 2011 American Institute of Physics Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.118 On: Thu, 01 Sep 2016 03:57:30 204113-2 Evans, Williams, and Searles J. Chem. Phys. 134, 204113 (2011) functions of the momentum. This simply means that we are ing time-dependent dynamics, defined by a parameter λ f (s) not moving relative to either system. In Eq. (2), d(0) is an with λ f (0) = λ1 and λ f (τ) = λ2, then the probability that the infinitesimal phase space volume centred on (0). phase variable defined in Eq. (2) takes on the value B ± dB is If the system is not canonical the partition function Z is given by i just the normalization factor for the equilibrium distribution = ± = . function and feq,i () = exp[Fi ()]/Zi , where Fi ()issome peq,1( Xτ, f B dB) d feq,1( ) (4) = ± real single valued phase function. Xτ, f B dB Although the physical significance of the generalized If initially we select phases from feq,2() with a par- work, X, might seem obscure at this point, we will show that ticular protocol (r) which is the time-reverse of (f), λr (s) for particular choices of dynamics and ensemble, it is related = λ f (τ − s), and corresponding time-dependent dynamics, to important thermodynamic properties and when it is evalu- so λr (0) = λ2 and λr (τ) = λ1, then the probability that ated along quasi-static paths it is, in fact, a path independent the phase variable defined in Eq. (2) takes on the state function. value −B is given by peq,2(Xτ, f =−B ∓ dB) = d feq, (). Before proceeding further with the analysis, it is useful Xτ, f =−B∓dB 2 to consider precisely what the generalized work is dependent We note that a trajectory starting at point , and evolv- upon. First, it is a function of the equilibrium states 1 and 2. ing forward in time with the forward protocol to the point This occurs via the equilibrium distributions appearing in (τ) will be related by a time reversal mapping to a trajectory Eq. (2) and also the partition functions for those states—see starting at M T (τ) and evolving with the time-reverse proto- τ Eq. (2). Second, it is a function of the endpoints of the pos- col. If S f/r is the time evolution operator with forward/reverse sibly nonequilibrium phase space trajectory that takes phase protocol, (0) to (τ). As we will see later it is also a function of how τ τ M T S M T S (0) = (0). (5) much heat is gained or lost from the system over the dura- r f tion of that trajectory. This heat loss determines the ratio of Also, if Xτ, f ((0)) = B, then Xτ, f ((0)) =−B.Aswas phase space volumes d(0)/d(τ). Last, it is a function of shown in Ref. 1, we can therefore write the duration of the trajectories τ. = ± λ peq,1( Xτ, f B dB) = Z( 2). The probability of observing ensemble members within exp[B] (6) peq,2(Xτ,r =−B ∓ dB) Z(λ1) the infinitesimal phase volume d, centred on the phase vec- We note that at time τ, the system that is evolving from tor , in the initial equilibrium distribution function, feq,i () feq,1() would not have relaxed to the distribution feq,2() is peq,i (d) = feq,i ()d.
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