A Proof of Clausius' Theorem for Time Reversible Deterministic Microscopic

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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

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A proof of Clausius’ theorem for time reversible deterministic microscopic dynamics Denis J. Evans,1 Stephen R. Williams,1,a) and Debra J. 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’ inequality. 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

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. It is very important to note that the time τ is the time (or vice versa). We can calculate the generalized work af- at which the parametric change in λ is complete. This means ter the parameter change has ceased but during the period that at time τ this system is not necessarily at equilibrium: in which the system relaxes, as specified by the relaxation theorem,5 to the new equilibrium. From Eq. (2), we see that f (, 0) = feq,1(), but f (,τ) = feq,2(). The generalized work is defined with respect to two different equilibrium dis- exp[Xτ+s () − Xτ ()] tributions and the end points of finite time phase space tra- f , ((0))d(0)Z(λ ) f , ((τ))d(τ)Z(λ ) jectories: (s): 0≤ s ≤ τ. The recently proved relaxation = eq 1 1 eq 2 2 τ + τ + λ λ theorem5 says that if the system is T-mixing and if the initial feq,2( ( s))d ( s)Z( 2) feq,1( (0))d (0)Z( 1) equilibrium distribution is an even function of the momenta, f , ((τ))d(τ) = eq 2 , ∀s > 0. (7) then lim f (, t) = feq,2(). τ + τ + t→∞ feq,2( ( s))d ( s) ∀ ∈ In order for Xτ ( ) to be well defined, (0) D1, If we look at the last line of Eq. (7), we recognize that it is then (τ) ∈ D and both f , ((0)) = 0 and f , ((τ)) 2 eq 1 eq 2 simply the integrated dissipation function, , , ((τ)) de- = eq 2 s 0. This is known as the ergodic consistency for the fined in the Evans-Searles fluctuation theorem,6 for equilib- generalized work. rium system 2, evaluated at a phase (τ), and integrated for a ∂ τ /∂ We identify ( ) (0) as the Jacobian determinant time s. It is important to note that both the numerator and the and note that denominator of Eq. (7) involve forward time integrations from ∂(τ) d(τ) system 2 equilibrium [i.e., there is no forward and reverse as = . (3) ∂(0) d(0) in Eq. (2)]. Therefore,

The GCFT considers the probability, peq, f (Xt = B ± dB), [Xτ+ () − Xτ ()] s of observing values of Xt = B ± dB for forward trajec- τ feq,2((τ)) tories starting from the initial equilibrium distribution 1, ≡ Xs ((τ)) = ln − ds ((τ + s)) feq,2((τ + s)) 0 f1(, 0) = feq,1(), and the probability, peq,r (Xt =−B = τ = , ∀ ∈ , ∀ ≤ , ∓ dB), of observing Xt =−B ± dB for reverse trajec- eq,2,s ( ( )) 0 (0) D2 s 0 (8) f , tories but starting from the equilibrium given by eq 2( ), where is the phase space expansion factor: system 2. ∂ Consider two equilibrium ensembles from which initial ≡ · . (9) trajectories can be selected with known equilibrium distribu- ∂ tions: feq,1() and feq,2(). If initially we select phases from The last line in Eq. (8) is identically zero because the dissipa- feq,1(), employ a particular protocol (f) and correspond- tion function eq () for all equilibrium systems is identically

zero,5 and we know from the relaxation theorem that the sys- In deriving this relation the fact that ex ≥ 1 + x, ∀x ∈ R is tem does eventually relax to the unique, ergodic, dissipation- used.7 Taking the logarithms of both sides and then multiply- less equilibrium state of system 2. ing both sides by –1, we get λ ≥ Z( 1) . III. GENERALIZED JARZYNSKI EQUALITY Xτ ln (12) Z(λ2) The generalized Jarzynski equality (GJE), can be thought This is clearly the analogue of the second law inequality6 of as the analogue of the nonequilibrium partition identity6 for ensembles of systems with changing free energy. Some (NPI) evaluated for the generalized work. We say “analogue” authors refer to work inequalities like Eq. (12) as Clausius’ because the introduction of forward and reverse paths in the inequality,8 however, we reserve that term for cyclic inequal- definition of the generalized work is quite different from the ities of the heat, since as Planck remarked,9 “this is the form use of forward only paths, for the NPI. of the second law first enunciated by Clausius.” As is the case for NPI, the GJE can be derived trivially In actual systems the right hand side will turn out to from the corresponding fluctuation theorem, Eq. (6) giving be a dimensionless free energy difference. For example, if λ systems 1, 2 are canonical and at the same temperature − = Z( 2), exp[ Xτ ( )] eq,1 (10) and have the same number of particles and volume, as we Z(λ1) will see later, ln[Z1/Z2] = βA21 = β(A2 − A1) and X τ ... = β ds W s , W where the brackets eq,1 denote an equilibrium ensemble 0 ( ) where denotes the work (i.e., the change average over the initial equilibrium distribution. We also note caused by the internal energy minus the change caused by that with the change of variables the domain of integration the heat) and Ai is the Helmholtz free energy of system i. may change. The minimum average work inequality implies in this case ≥ The GJE is very widely applicable. It relates the en- W21 A21. The minimum work is expended if the path semble average of the exponential of nonequilibrium path is reversible or quasi-static, in the case where that work is, in integrals to equilibrium thermodynamic free energy dif- fact, the difference in the Helmholtz free energies divided by ferences. The validity of Eq. (10) requires that ∀(0) kB T . In the quasi-static case there are no fluctuations in the ∈ D1, feq,1((0)) = 0, and feq,2((τ)) = 0, which implies value of the generalized work. ∀(τ) ∈ D2, feq,2((τ)) = 0. We call this the ergodic con- If the parametric protocol takes us around a closed cycle sistency condition for the GJE. that is defined in terms of the parameter λ(t), we see that since / = Equations (6) and (10) are very general, and they even ap- by definition Z1 Z2 1, ply to stochastic dynamics (see Ref. 4). Obviously, the paths do not need to be quasi-static paths as in traditional thermo- ds X˙ (s) = dX ≥0. (13) dynamics. These equations are independent of the particular protocol, provided the appropriate ergodic consistency holds. The ensemble average of the cyclic integral of the generalized In fact, it is possible to average over the initial ensemble and work is non-negative. Because the dynamics is microscop- a set of protocols, since the final answer is protocol or path ically reversible, the cyclic integral can only be zero if the independent. cycle is thermodynamically reversible—what we term quasi- Although the GJE and NPI can be proved from their ap- static. A pathway is traversed quasi-statically if the average propriate fluctuation relation, the reverse is not true because work for a forward path is equal and opposite to the average the corresponding fluctuation relations contain more informa- work for the reversed path. tion than either the NPI or the GJE. The cyclic integral of the generalized work for a quasi- As we mentioned earlier, like the CFT, the application of static cycle is zero. In fact, this is the definition of a quasi- the GJE relies on the observation of improbable fluctuations. static process. The fact that the cyclic integral of the general- = In order to yield reliable estimates of free energy differences, ized work is zero, dX 0, also implies that one must sample the trajectories that are the conjugate anti- f trajectories of the most probable trajectories. This means that dX = independent of path, (14) these formulae are of limited use for computing free energy qs i differences in the thermodynamic, or large system, limit. where the subscript qs denotes the fact that the integral is for a quasi-static or thermodynamically reversible pathway. The proof of Eq. (14) is obvious. Construct a reversible cycle i IV. MINIMUM AVERAGE GENERALIZED WORK → f, f → i. The cyclic integral must be zero, so if we vary We now derive a further simple corollary of the GJE. the path for the return leg f → i, we must always get the From Eq. (10), we see that same value for the integrated reversible work, independent of the precise path. Z(λ2) = −Xτ Finally, we can see that if the integral of the generalized λ exp[ ] eq,1 Z( 1) work for paths is independent of the pathway, that integral = exp[− Xτ eq,1] exp[−Xτ + Xτ eq,1] must be a state function (i.e., a function only of the initial and final states of the system). In fact, this is why the seemingly ≥ exp[− Xτ , ] 1 − Xτ + Xτ , , eq 1 eq 1 eq 1 abstract generalized work defined in Eq. (2) is so important. = exp[− Xτ eq,1]. (11) The generalized work for a thermodynamically reversible

pathway is always a history independent function of the The extended, time-dependent internal energy is 2 2 thermodynamic states of the system at the end points of the HE (,α,t) = H0() + (3(Nth − 1)/2)kB T (t)α τα and the path. extended phase space of the system is = (,α). The Liouville equation states6, 10 df/dt =−f , and using Eq. (15) it is easy to show that kB T = therm V. NONEQUILIBRIUM WORK RELATIONS FOR kB T (∂/∂)˙ +(∂/∂α)˙α =−3(Nth−1)kB T α =−Q˙ , therm THERMAL PROCESSES where Q˙ is the rate of decrease in HE due to the ther- We wish to consider a realistic model of a system that is mostat or equivalently the rate of increase of energy by the driven away from equilibrium by a reservoir whose tempera- imaginary external thermostat. From the relaxation theorem, ture is changing. For this case the simple parametric change the unique equilibrium distribution function for this system at a fixed temperature T is then5 in the Hamiltonian or the external field usually employed in √ the derivation of the JE or the CFT, is not applicable and care τα 3(N − 1)/(2π) f (, T,α) = th is needed in developing the physical mechanisms, see Ref. 1. eq Z(T ) Here, we could address this issue by considering a system × −β , ,α δ , of interest, containing some very slowly relaxing constituents, exp( HE ( T )) (pth) (16) such as soft matter or pitch, in contact with a rapidly relaxing where Z(T ) is the canonical partition function and λ(t) reservoir. The reservoir may be formed from a copper block or ≡ T (t). another highly thermally conductive material. Changing the We now consider applying the GCFT, Eq. (6), when a temperature of the reservoir (say with a thermostatically con- thermal rather than a mechanical process occurs. Consider a trolled heat exchanger) then drives the system of interest out thermostated system of N particles whose target kinetic tem- of equilibrium. The change in the temperature is slow enough perature is changed from T1 to T2 over a period 0 < t <τ.We that the reservoir may be treated to high accuracy, as under- do not change the Hamiltonian during this process. For sim- going a quasi-static temperature change. The slowly relaxing plicity we consider a canonical ensemble for the two equi- system of interest is far from equilibrium. We employ gener- librium states, Eq. (16), and use the equations of motion, alized versions of the CFT and the JE to describe this system. Eq. (15). The temperature dependence of the reservoir is Importantly, the quantities that appear in the theory are phys- achieved by making the Nosé-Hoover target temperature T (t) ically measurable variables. in Eq. (15) a time dependent variable. Another mechanism for achieving the required result From Eqs. (15) and (16), we see that the generalized di- would be (following Planck9) to have a set of large equilib- mensionless work is rium thermostats that can be thermally coupled to the sys- Xτ ( ;0,τ) = β H ( (τ)) tem of interest in a protocol sequence. If these thermostats are 2 E large they can be regarded as being in thermal equilibrium. If τ −β + β ˙ therm , they are sufficiently remote from the system of interest there 1 HE ( (0)) dt (t)Q ( (t)) 0 is no way the system of interest can “know” the precise math- (17) ematical details of how heat is ultimately taken from or added to the system of interest. where β(t) = 1/(kB T (t)) is the inverse, time-dependent tar- For convenience from a theoretical perspective, we get temperature. (Note: as noted above dE = dW − dQtherm choose the Nosé-Hoover thermostating mechanism,10 and the = dW + dQsoi because, following Planck, our change in the equations of motion, including the thermostat multiplier, are heat refers to the thermal reservoir (therm) rather than to the then system of interest.) Now if we take the derivative of the extended Hamiltonian while the temperature is changing, but pi q˙ i = , with no other external agent acting on the system, we obtain m = − α + γ , d p˙ i Fi (q) Si ( ( )pi th) H ( (t)) =−Q˙ therm( (t)) dt E N · / i=1 Si pi pi m 1 α˙ = − 1 , (15) 3 2 2 − τ 2 + (N − 1)k T˙ (t)α (t)τα . (18) 3(Nth 1)kB T (t) α 2 therm B We then obtain where τα is an arbitrary Nosé-Hoover time constant. The ˙ value of T (t) is the target temperature of the thermostat and d β =−β T (t) + ˙ , = , [ (t)HE ( (t))] (t) H0( (t)) Q( (t)) Si 0 1 is a switch that controls which particles are coupled dt T (t) N = to the Nosé-Hoover thermostat: i=1 Si Nth. In our model (19) the particles that are coupled to the thermostat can be taken to be remote from the system of interest. This ensures that the and combining Eqs. (19) and (17), the generalized “power” particles in the system of interest are ignorant of the precise for a change in the target temperature with time is details of this unphysical thermostat. These thermostated par- ˙ = β˙ . γ X( (t)) (t)H0( (t)) (20) ticles are also subject to a fluctuating force th that is chosen to ensure that the total momentum of the thermostated parti- Note that the right hand side of Eq. (20) only depends N ≡ = cles i=1 Si pi pth 0 is identically zero. upon physical variables and not the unphysical thermostat

multiplier α or the extended Hamiltonian. Equation (10) then thermostat. At any instant the numerical value of the target becomes temperature is, in fact, the equilibrium thermodynamic tem- τ Z perature that the entire system would relax to, if at that same − β˙ = 2 exp dt (t)H0( (t)) moment, this parameter was fixed at its current value. We Z1 0 1 know that this is so from the relaxation theorem for T-mixing = −β + β . exp[ 2 A2 1 A1] systems. We will often use the phrase that the temperature ap- (21) pearing in Eqs. (1) and (23) is at any instant of time, the equilibrium thermodynamic temperature of the underlying equi- Since for quasi-static processes the target kinetic temperature librium state. of the Nosé-Hoover thermostat is, in fact, equal to the equilib- If the cycle is traversed rapidly so that the system of in- rium thermodynamic temperature, one can see that this equa- terest is not in thermodynamic equilibrium, the actual thermo- tion is consistent with thermodynamics because in the quasi- dynamic temperature of the system of interest is of course not static limit, equilibrium thermodynamics gives us the relation defined. If the thermostat is composed of a large Hamiltonian re- τ τ d gion coupled to the system of interest and a remote Nosé- dt β˙(t)U(t) = dt T˙ (t) [β(t)A(t)] dT Hoover thermostated region, we can argue that the precise 0 0 τ d details of the thermostat cannot possibly be “known” to the = dt [β(t)A(t)] system of interest and are therefore unimportant. dt 0 If the thermostat is comparable in size to the system = β2 A2 − β1 A1. (22) of interest and if the cycle is traversed quickly, both the For this case we see that the dimensionless work is the rate of system of interest and the thermostat will be away from change of dimensionless free energy. equilibrium. At any point in the cycle there is a profound difference between the nonequilibrium state generated by Gaussian processes and Nosé-Hoover processes. However, for VI. CLAUSIUS’ INEQUALITY AND THE both types of thermostat, Eqs. (1) and (23) take the same form. THERMODYNAMIC TEMPERATURE For Gaussian thermostats the change in the kinetic temper- We now turn our attention away from work to heat. ature of the thermostat is instantaneous, whereas for Nosé- ∼ τ As before we consider a periodic protocol. However for Hoover thermostats there is a variable phase lag α in the heat (and unlike work), we can only deduce useful re- Eq. (15). (The value of this feedback time constant is arbi- sults if the system responds periodically to the cyclic pro- trary.) The only “temperature” any of these systems have in tocol. We note that if we periodically cycle a given proto- common is the equilibrium thermodynamic temperature that col, not all systems will respond periodically. The necessary they will relax to if at any point in the cycle the parametric and sufficient conditions for the system to respond periodi- change is halted. At any point in the cycle the precise na- cally are not known. Clausius’ inequality only applies if, in ture of the nonequilibrium state (e.g., the instantaneous av- the long time limit (t →∞), the average system response is erage pressure or energy) is highly dependent on the phase τ periodic. lag α, or whether the thermostat is Gaussian or Nosé-Hoover If we now substitute Eq. (17) into Eq. (13) and apply it to like. 9 a periodic cycle after any cyclic transients have decayed, we In Planck’s discussion of Clausius’ inequality, at any in- can deduce that stant in the cycle, T is the equilibrium thermodynamic temperature of the particular large equilibrium reservoir with which lim ds X˙ (t + s) =lim ds β(t + s)Q˙ therm(t + s) →∞ →∞ the system of interest is currently in contact. t P t P Clausius’ thermodynamic inequality equation (1),isof dQtherm course only exact in the thermodynamic limit, and in small = lim ≥ 0, (23) →∞ t P kB T systems it can occasionally be violated as in Eq. (23). The probability ratio that for a finite system the work in- where we use the notation: P ds to denote the cyclic inte- tegral takes on a value A compared to −A can be com- gral of a periodic function. We note that the cycle is driven puted from a time dependent version of the fluctuation by a periodic parametric change. In this equation t is the theorem. time when you start the cyclic integral. Because the cycle Equations (1) and (23) show that on average we cannot β is periodic in Eq. (17), the change in HE around the cy- construct a perpetual motion machine of the second kind. A cle is identically zero. Thus for periodic cycles, Eq. (13) perpetual motion machine of the second kind would require reduces to an integral of the heat divided by the target therm/ < that P dQ T 0, so that ambient heat from the reser- temperature. voir is converted into useful work. Thus, the proof of Eqs. (1) In more usual notation Eq. (23) implies that in the large and (23) constitutes a direct mechanical proof of Clausius’ →∞ system limit, N , where fluctuations are negligible, we statement of the second “law” of thermodynamics. obtain the very well-known Clausius’ inequality,9 given in therm If the cycle is reversible we can apply Eq. (1) to the Eq. (1), that is, lim lim dQ ≥ 0. N→∞ t→∞ P T forward cycle and to the reversed cycle that must have In these equations, Eqs. (1) and (23), the time depen- the same value for magnitude of the integral but opposite dent temperature is the target temperature of the Nosé-Hoover sign. The only possible value for both integrals is therefore

zero: For this case, if and only if the system settles into a peri- dQtherm odic work cycle, Eq. (23) becomes lim = 0. (24) N→∞ qs T lim ds X˙ (t + s) = lim β ds Q˙ therm(t + s) →∞ →∞ The subscript qs, denotes a quasi-static cycle. We note that t P t P a quasi-static cycle cannot have any transients and is always ⇒ lim dQtherm ≥0, (28) periodic. →∞ t P Applying the same arguments as we did for the quasi- static cyclic integral of the generalized work shows that the where T is the fixed equilibrium thermodynamic tempera- quasi-static integral from an initial to a final state, ture of the reservoir. This shows that for periodic work processes carried out in contact with either a large thermal reser- f dQtherm lim ≡ Stherm − Stherm (25) voir at a fixed equilibrium thermodynamic temperature or a →∞ f i N qs i T small nonequilibrium thermal reservoir with a fixed under-

is, in fact, a state function, denoted by the symbol Stherm. lying equilibrium temperature, the cyclic integral of dQ is

The state function Stherm, defined in Eq. (25) is known positive on average, and for quasi-static processes, qs dQ as the change in the equilibrium entropy of the thermostating is a state function. reservoir. If we combine Eqs. (10), (17), and (21), we see that in VII. CONCLUSIONS the thermodynamic limit where fluctuations are negligible, f therm We have proved Clausius’ thermodynamic inequal- dQ A − H , A − H , lim =− f E f + i E i ity from time reversible microscopic equations of motion. →∞ N qs i T T f Ti Clausius of course proved his theorem assuming the second =−Ssoi + Ssoi, (26) “law” of thermodynamics. Our proof requires no such as- f i sumption. Our proof rests on the laws of mechanics, on the where in the second term we have used our previously proven5 axiom of causality, ergodic consistency, and the assumption relation between the partition function, the Helmholtz free en- that the system is T-mixing. This is quite a different log- ergy, and the equilibrium thermodynamic temperature. Now ical position from that used by Clausius in 1854. The re- if we compare Eqs. (25) and (26), we see that in the thermo- quirement of causality is extensively discussed in Ref. 6.Ba- dynamic limit, the equilibrium entropy, the Helmholtz free sically, causality is required for the proof of all fluctuation energy, and the energy must be related by the following theorems. equation: The necessary and sufficient conditions for a system to soi be T-mixing are not presently known. However, it is extremely A / = H , / − S T / , (27) f i E f i f/i f i easy to test whether a system is T-mixing. In thermostated sys- where we have used the fact that the heat gained by the ther- tems driven by a dissipative field, if the system is T-mixing it mostat is equal and opposite to the heat gained by the system will relax to a time independent nonequilibrium steady state of interest. where the averages of suitably smooth phase functions be- Equation (25) tells us another very important piece of come time independent at long times. This is easily seen information. The integration factor for the heat in quasi- from the dissipation theorem.12 Similarly in T-mixing relax- static (i.e., reversible) processes is the time dependent ther- ing systems that possibly interact with a thermostat but which modynamic temperature. The fact that the equilibrium tem- are not driven by external dissipative fields, averages of suit- perature is the integrating factor for the heat ultimately ably smooth phase functions become time independent at long comes from the form of the canonical equilibrium distribu- times. tion function. The relaxation theorem says that this distri- Our derivation allows us to understand the mean- bution is unique5 for T-mixing systems. The consequence ing of the “temperature” appearing in Clausius’ inequality of this is that the integrating factor for the heat is also for nonequilibrium systems. Temperature can only be de- unique. fined for equilibrium systems but Clausius’ (strict) inequal- We make a final observation on Clausius’ inequality. Sup- ity refers to nonequilibrium systems. We also now know pose instead of cycling the temperature we perform a work that Clausius’ inequality is only valid for periodic cycles. cycle keeping the temperature of the thermal reservoir fixed. Planck9 also realized this. Most textbooks do not state this In this case we can make the thermal reservoir very large and requirement. in the limit where it becomes infinitely large compared to the When subject to a periodic parametric cycle, not all system of interest, it can be viewed as being in thermody- systems will settle into a periodic response. For small sys- namic equilibrium at a fixed equilibrium thermodynamic tem- tems (not covered by Clausius’ proof of his inequality), we perature. know that the inequality is only valid if we take ensemble At a great physical distance from the system of interest averages. We now also know that the integrating factor for we could fix the reservoirs temperature with either a Nosé- the heat is unique for T-mixing systems. Other conclusions Hoover or a Gaussian isokinetic thermostat—it does not mat- follow. ter. The system of interest cannot distinguish the fine details Our proof used the vehicle of the Nosé-Hoover ther- of how the heat is ultimately and remotely removed. mostat. It provided a simple means of constructing a time

dependent thermal protocol. The formal results are indepen- system is zero: dent of how near or far this Nosé-Hoover thermostated re- d Q˙ uni gion is from the system of interest. If the thermostat is moved S˙uni =−k d f uni(, t)ln[f uni(, t)] = = 0. B dt T progressively further from the system of interest, there is no D way that the system of interest can “know” the details of (31) how the thermostating is achieved. This means that the formal results are independent of the Nosé-Hoover thermostat. In our proof of the relaxation theorem for systems in contact Indeed, we could have derived the same formal results us- with a heat bath,5 we proved that the Helmholtz free energy ing a variation of a Gaussian thermostat where the kinetic en- and the partition function are related by ergy of the thermostated particles was forced to be a periodic function of time with no fluctuations away from the periodic A =−kB T ln d exp[−HE ()/kB T ] . (32) function. D If we consider the thermal equilibration of a T-mixing We also proved for equilibrium systems, the equivalence of system that has a nonuniform initial temperature distri- the target kinetic temperature employed in a Nosé-Hoover bution, we see from the relaxation theorem5 that the fi- thermostat and the thermodynamic temperature for canonical nal equilibrium state is the one with a spatially uniform systems. thermodynamic temperature distribution—as in the canoni- Now we have arrived at a completely new logical posi- cal or microcanonical distributions. The proof shows5 that tion. We have proved the zeroth, first, and second “laws” of in T-mixing systems these equilibrium distribution func- thermodynamics; the latter in the form of Clausius’ equality. tions are unique. This is equivalent to a proof of the ze- This means that logically we can now construct thermody- roth “law” of thermodynamics. Subsystems of an equi- namics without any assumptions except the laws of mechan- librium system must all be at the same thermodynamic ics, the assumption of T-mixing, ergodic consistency, and the temperature. axiom of causality. We now look again at Eq. (23). If our system is subject to ACKNOWLEDGMENTS a periodic thermal protocol and if the system settles into a periodic cycle, the ensemble averaged heat absorbed by the ther- We would like to thank the referees for their close reading therm therm mostat (dQ ) is non-negative lim P dQ /T ≥0. of an earlier draft of this manuscript. We would also like to t→∞ If the sign was reversed we would have been able to construct thank the Australian Research Council who supported part of a perpetual motion machine of the second kind. So we have this research. given a proof of the second “law” of thermodynamics, since 1S. R. Williams, D. J. Searles, and D. J. Evans, Phys. Rev. Lett. 100, 250601 Clausius’ statement of that “law” refers to the impossibility (2008). 2 of constructing such a machine. G. E. Crooks, J. Stat. Phys. 90, 1481 (1998); Phys. Rev. E 60, 2721(1999). 3G. P. Morriss and L. Rondoni, Phys. Rev. E 59, R5 (1999). There is a complementary inequality for the system of 4J. C. Reid, E. M. Sevick, and D. J. Evans, Europhys. Lett. 72, 726 (2005). interest (soi), namely, 5D. J. Evans, D. J. Searles, and S. R. Williams, “A Simple Mathematical Proof of Boltzmann’s Equal aprioriProbability Hypothesis,” in Diffusion soi Fundamentals III, edited by C. Chmelik, N. Kanellopoulos, J. Karger, and lim P dQ /T ≤0. (29) t→∞ D. Theodorou (Leipziger Universitatsverlag, Leipzig, 2009), p. 367–374; J. Stat. Mech.: Theory Exp. 2009, P07029 (2009). If we combine the system of interest and the thermostat we 6D. J. Evans and D. J. Searles, Adv. Phys. 51, 1529 (2002). see that for the combined system (uni) we find that for all 7D. Chandler, Introduction to Modern Statistical Mechanics (Oxford Uni- periodic cycles, nonequilibrium and quasi-static, versity Press, New York, 1987), p. 137. 8See, for example, p. 332 of C. Jarzynski, Annu. Rev. Condens. Matter Phys. uni 2, 329 (2011). lim P dQ /T =0. (30) 9 t→∞ M. Planck, Treatise on Thermodynamics,3rdEd.,translated from the 7th German Edition (Dover, New York, 1945), p. 99. Because the temperature appearing in Eqs. (29) and (30) is the 10D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium temperature of the underlying equilibrium state, this tempera- Liquids (Academic, London, 1990), pp. 251 and 252. 11P. Ehrenfest and T. Ehrenfest, The Conceptual Foundations of the Statistical ture is the same for the system of interest and for the thermal Approach in Mechanics, translated by M. Kac and G. E. Uhlenbeck (Dover, reservoir. New York, 1990) [P. Ehrenfest and T. Ehrenfest, “Begriffliche Grundla- At first sight Eq. (30) seems somewhat unexpected, at gen der Statistischen Auffassung in der Mechanik,” in The Encyklopadie least for nonequilibrium systems, but we have known since der Mathematischen Wissenschaften (B. G. Teubner, Leipzig, 1912), 10, 11 vol. IV:2:II, No. 6.] Gibbs’ work that the time derivative of the fine grained 12D. J. Evans, D. J. Searles, and S. R. Williams, J. Chem. Phys. 128, 014504 Gibbs entropy S˙ of a thermally isolated nonequilibrium (2008); (Erratum) 128, 249901 (2008).

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