Zeroth law of thermodynamics for nonequilibrium steady states in contact

Transcription

1 Zeroth law of thermodynamics for nonequilibrium steady states in contact Sayani Chatterjee 1, Punyabrata Pradhan 1 and P. K. Mohanty 2,3 1 Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Kolkata , India 2 TCMP Division, Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata , India 3 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany We ask what happens when two systems having a nonequilibrium steady state are kept in contact and allowed to exchange a quantity, say mass, which is conserved in the combined system. Will the systems eventually evolve to a new stationary state where certain intensive thermodynamic variable, like equilibrium chemical potential, equalizes following zeroth law of thermodynamics and, if so, under what conditions is it possible? We argue that the zeroth law would hold, provided both systems have short-ranged spatial correlations and they interact weakly to exchange mass with rates satisfying a balance condition - reminiscent of detailed balance in equilibrium. This proposition is proved for driven systems in general in the limit of small exchange rates (i.e., weak interaction) and is demonstrated in various conserved-mass transport processes having nonzero spatial correlations. PACS numbers: Ln, y, a Introduction. - Zeroth law is the cornerstone of equilibrium thermodynamics. It states that, if two systems are separately in equilibrium with a third one, they are also in equilibrium with each other [1]. An immediate consequence of the zeroth law is the existence of state functions - a set of intensive thermodynamic variables (ITV) which equalize for two systems in contact. For example, if two systems are kept in contact and allowed to exchange a conserved quantity, say mass, they eventually achieve equilibrium where chemical potential becomes uniform throughout the combined systems. The striking feature of this thermodynamic structure is that all equilibrium systems form equivalence classes where each class is specified by a particular ITV. Then a system, an element of a particular class, is related to any other system in the class by a property that they have the same value of the ITV. We ask whether a similar thermodynamic characterization is possible for systems having a nonequilibrium steady state (NESS). Can equalization of an ITV, governing equilibration between two steady-state systems in contact, be used to construct such equivalence classes? The answer is nontrivial; in fact, it is not even clear if such a formulation is at all possible [2 12]. In this paper, we find an affirmative answer to this question, which can lead to an extraordinary thermodynamic structure where a vast class of systems having a NESS, equilibrium systems of course included, form equivalence classes. There have been extensive studies in the past to find a suitable statistical mechanical framework for systems having a NESS [2, 3, 5, 9 16]. Though the studies have not yet converged to a universal picture, it has been realized that suitably chosen mass exchange rates at the contact could possibly lead to proper formulation of steadystate thermodynamics [10, 11, 16 19]. An appropriate contact dynamics is crucial because, without it, properties of mass fluctuations in a system would be different, depending on whether the system is in contact (grand canonical) or not in contact (canonical) with other system; in other words, without an appropriate contact dynamics, canonical and grand canonical ensembles would not be equivalent [17 20]. The situation is analogous to that in equilibrium where ensemble equivalence, and the consequent thermodynamic structure, is ensured by contact dynamics with mass exchange rates satisfying detailed balance w.r.t. the Boltzmann distribution. However in nonequilibrium, in the absence of a priori knowledge of microscopic steady-state structure, the intriguing questions, (a) whether there indeed exist a class of exchange rates leading to a well-defined thermodynamic structure for steady-state systems in general and (b) how the rates could be determined, still remain unsettled. In this paper, we show that the equilibrium thermodynamics can be extended to systems having a nonequilibrium steady state, where zeroth law is obeyed and equilibration between two systems (labeled by = 1, 2) in contact can be characterized by equalization of an intensive thermodynamic variable, provided spatial correlations in individual systems are short-ranged and mass exchange occurs weakly across the contact with the exchange rates satisfying u 12 (ε) u 21 (ε) = e F, (1) a reminiscent of detailed balance condition in equilibrium. Here u (ε) is the rate with which a mass of size ε is chipped off from system to, and F is the change in a nonequilibrium free energy of the contact regions. Note that Eq. 1 requires a free energy function to exit, which, we argue, is the case in systems having short-ranged spatial correlations. We show that this free energy can in principle be obtained from a fluctuationresponse relation. The size of the contact regions, oth-

2 2 erwise arbitrary, should be much larger than correlation lengths, therefore making the contact regions effectively independent of the rest of the systems. Proof. - Let us consider two systems = 1,2 of size V, having mass variables m {m i 0} defined at the sites i V. Each of the systems, while not in contact with each other (canonical ensemble), has a nonequilibrium steady state distribution P (m ) = ω (m ) W (M,V ) δ ( M i V m i ) (2) where ω (m ) is the steady-state weight of a microscopic configuration m and W (M,V ) = dm ω (m )δ(m i V m i ), is the partition sum ( dm implies integral over all m i with i V ). The delta function ensures conservation of mass M = i V m i or mass density ρ = M /V of individual systems. On the other hand, when the systems 1 and 2 are in contact, mass exchange at the contact region breaks conservation of M 1,2 whereas the total mass M = M 1 + M 2 remains conserved. We refer this situation as a grand canonical ensemble. To have a consistent thermodynamic structure, it is necessary that individual systems have a well defined canonical free-energy functions F 1,2 for systems 1 or 2 which should not change upon contact, i.e., free energy of the combined system F = F 1 + F 2 is additive. This additivity principle, once satisfied, has the following immediate consequences: (i) Equalization of an intensive thermodynamic variable (ii) a fluctuationresponse relation and (iii) zeroth law; all of them follows from standard statistical mechanics [1]. Let us consider a situation where mass exchange takes place between two systems through contact regions (see Fig. 1.(a)) each with volume v (taken same for both systems for simplicity) which is much larger than finite spatial correlation length ξ but otherwise arbitrary. As subsystems much larger than the correlation lengths would be statistically independent in the thermodynamic limit, the steady state subsystem mass distribution can be written as product of weights which depends only on mass of the subsystems [3]. Thus, when ξ 1,2 v V 1,2, we could view each individual system composed of two subsystems - contact region (of size v and mass M) c and the rest (of size V v and mass M b = M M) c - whose steady-state weights are factorized, i.e., product of two coarse-grained weights, as reflected in the partition sum W (M,V ) dmw c (M M)W c (M), c (3) thus leading to the existence of a canonical free energy F lnw which is additive, i.e., F (M,V ) = F (M c,v)+f (M b,v v). Total free energy F (M) = inf M c [F (M c )+F (M M c )] is obtained thorough minimization w.r.t. M c. We now demand that the canonical description where M 1 and M 2 are individually conserved, must be equivalent to the grand canonical ensemble where only total mass M = M 1 + M 2 is conserved. That is, the microscopic weights of the combined system must be product of the individual microscopic weights P(m 1,m 2 ) = ω 1(m 1 )ω 2 (m 2 ) δ (M M 1 M 2 ),(4) W(M) with W(M) = dm 1 W 1 (M 1,V 1 )W 2 (M M 1,V 2 ), so that the joint mass distribution is also factorized P(M 1,M 2 ) = W 1(M 1 )W 2 (M 2 ) δ (M M 1 M 2 ), (5) W(M) and thus additivity principle is ensured. Consequently, total free energy of the combined systems F(M) lnw = inf M1 [F 1 (M 1 ) + F 2 (M M 1 )] is additive and a chemical potential µ (ρ ) = F / M equalizes. Now we show how, in the weak interaction limit, balance condition in Eq. 1 ensures additivity principle in Eq. 5 - the main result of this paper. Let mass exchange occur at the contact with rate u (ε) where a mass ε is transferred from system to. The rate may depend on the mass values at the contact regions (not explicitly shown in u ). Mass conservation in the individual systems is then broken in this process (M M ε and M M + ε), generating a mass flow. To attain stationarity, average mass current J 12 (ε) generated by all possible microscopic exchanges u 12 where the chipped off mass ε flows from system 1 to 2 must be balanced by the reverse current J 21 (ε). Since total mass M = M 1 + M 2 is conserved, current balance condition J 12 (ε) = J 21 (ε) can be written, using only one of the mass variables, say M 1, as P(M 1, M M 1)U 12(M 1, ε) = P(M 1 ε, M M 1 +ε)u 21(M M 1 +ε, ε), where U (x,y) is an effective rate with which mass y is transferred from system, having mass x, to. The current balance, along with Eq. 5, gives U 12 (M 1,ε) U 21 (M M 1 + ε,ε) = e F (6) where F = 2 =1 (F final F initial ) difference in free energy of the combined system. To obtain a condition on the actual microscopic exchange rate u 12 (ε), we use the expression of current J (ε) as the average rate u = dm 1 dm 2 P(m 1,m 2 )u and write U 12 (M 1,ε) = J 12 (ε)/p(m 1,M 2 ), using Eq. 5, as [ 2 ] U 12 (M 1,ε) = dm1dm c 2 c W (M)e c µmc u 12 (ε), ε 0 =1 where Z = dm c W (M c ) e µmc. Finally, using U12 and a similar expression obtained for U 21 and then Eq. 6, we get the desired balance condition (i.e., Eq. 1) u 12 = W 1(M1 c ε) u 21 W 1 (M1) c W 2 (M c 2 + ε) W 2 (M c 2) Z = e F c = e F. (7)

3 3 In the last step, we used the contact free energy F c = lnw (M) c and equate F c = 2 =1 F c = F as F = 2 =1 (F c + F), b F b bulk free energy, and F b = 0 (i.e., changes occur only at the contact regions). To know what class of rates satisfy Eq. 7, we still need to know canonical coarse-grained weight W (M) c which can be calculated from knowing the mass fluctuation in a subvolume v using a fluctuation-response relation [21] 2 f (ρ ) ρ 2 = 1 ψ (ρ ) (8) where free energy density f (ρ ) = [lnw (M c )]/v and ψ (ρ ) = σ 2 v/v with variance of subsystem mass σ 2 v = (M c ) 2 v 2 ρ 2 in system. We emphasize here that, even when the detailed microscopic weight ω is not known, W can still be obtained, either analytically or numerically, from the subsystem mass fluctuations; this makes our formulation work both in theory and in practice, as illustrated later. Eq. 7 does not specify the contact dynamics (CD) uniquely; two simple choices are CD I : u = u 0 p(ε)w(m c (c) ε)/w(m c ), c (9) CD II : u = u 0 p(ε)min{1,e F }, (10) where constant u 0 1 implies the limit of weak interaction and p(ε) is a probability that mass ε is chosen for exchange. The resemblance between the rate in Eq. 10 and the familiar Metropolis rate is striking. In equilibrium, Eq. 7 reduces to the condition of detailed balance, albeit on a coarse-grained level. The balance condition in Eq. 7 is necessary, but not sufficient to ensure that the steady state has required product form as in Eq. 4; it is only in the weak interaction limit when the balance condition becomes sufficient as, in this limit, the individula systems have sufficient time to relax to the distribution in Eq. 4, leaving no inhomogeneities at the contact region. This completes the proof. Illustrations. - First we consider nonequilibrium processes where steady-state structures are exactly known. The simplest among them are zero range processes (ZRP) having a factorized steady state (FSS) [22]. We take two such systems = 1,2 where their steady-state weights ω (m ) = i V h (m i ) are simply product of factors h (m i ), function of only single-site mass variable. In this case, the individual systems exactly satisfy Eq. 3 with weights of contact region and the rest of system being W c = (f ) v and W s = (f ) V v, respectively. When mass exchange occurs through the contact regions with either of the rates u (ǫ) specified in Eqs. 9 and 10, it is easy to check that the joint distribution P(m 1,m 2 ) i V exp[ f (m i )] (i.e., the product of individual weight factors ω (m )) where f (m i ) = lnh (m i ) indeed becomes the steady-state weight of the combined system and thus additivity principle in Eq. 5 is exactly satisfied. This scenario holds even for v = 1 (i.e., when mass exchange occurs between two specific sites) and u 0 arbitrary. Note that, for FSS, Eq. 7 reduces to detailed balancing at the contact. Next we consider a general situation - keeping in contact systems having nonzero spatial correlations - where we study mass transport processes having a pair factorized steady state (PFSS) [23]. We consider two systems = 1,2, each having a PFSS of form ω (m ) = i g (m i,m i+1 ) where g a function of twosite mass variables. Provided that correlation length ξ is much smaller than v 1/d (in d dimensions), one could divide a system into ν = V /v number of almost statistically independent subsystems of volume v with subsystem masses labeled by M {M,j }. Then the joint probability distribution of the subsystem masses of systems are factorized: P({M 1, M 2 }) j V exp[ F () ({M,j })] where free energy F = j lnw (M,j ) of system is additive over the subsystems. Now let two such systems 1 and 2 be kept in contact such that mass from one specific subsystem of 1 participate in a microscopic mass-exchange dynamics with its adjacent subsystem of 2 with rates satisfying Eq. 7. In a coarse-grained level, as the subsystems could be considered as sites, the systems effectively become a set of sites with an FSS, where mass exchange occurs between two adjacent sites (here subsystems) with rates satisfying balance condition Eq. 7, and therefore the additivity principle in Eq. 5 holds exactly in the limit of v ξ 1,2 and u 0 1. To demonstrate this, we consider two one dimensional periodic lattices of L sites with continuous mass variable m i 0 at sites i = 1,2...L. The following mass conserving dynamics in the bulk leads to a PFSS [17, 23] where mass ε chosen from a distribution p b (ε) is chipped off from a site i and transferred to its right neighbor with rate u b (ε) = p b (ε) g (m i 1,m i ε) g (m i 1,m i ) g (m i ε,m i+1 ), (11) g (m i,m i+1 ) which depends on the masses at the departure site and its nearest neighbors, and on the chipped-off mass ε. For simplicity, we consider only homogeneous function g (x,y) = Γ δ g (Γx,Γy), for which ψ (ρ ) = ρ 2 /η. Then following Ref. [24], we analytically obtain W (m) = m vη 1 and chemical potential µ = η /ρ where η depends on δ. When two such systems are kept in contact, mass conservation in individual system is broken and both density ρ (t) and corresponding chemical potential µ (t) evolve until a stationarity is reached where densities adjusted so that chemical potentials equalize. We simulate using g (x,y) = (x δ +y δ +cx γ y δ γ ) and allow the two PFSS with η 1 = 2 (δ = 1, c = 0) and η 2 = 3 (δ = 2, c = 1, γ = 3/2) to exchange mass following CD I (and CD II in different

4 4 (a) System 1 System 2 µ 1 (t) υ 1/d ξ 1 ξ2 µ 2 (t) µ 1, µ (b) PFSS - PFSS (c) PFSS - PFSS (CD II) (d) MEM - MEM (e) MEM - MEM (CD II) (f) PFSS - MEM ξ 1,2 << υ 1/d FIG. 1: (Color online) Equilibration of steady states in contact: In (a), schematic representation of two systems in contact. In (b)-(e), chemical potentials µ 1(t) and µ 2(t) of systems 1 (red solid) and 2 (blue dotted) vs. rescaled time u 0t. µ 1 and µ 2, initially chosen to be different, eventually equalize. Densities (ρ 1, ρ 2) in the final steady states are respectively (3.60, 5.40) in (b), (3.57, 5.43) in (c), (5.31, 2.69) in (d), (5.32, 2.68) in (e), and (3.32, 6.68) in (f). In all cases, p(ε) = p b (ε) = exp( ε), u 0 = 0.1 and v = 10 (except in (d) and (e) where v = 1). simulations) with u 0 = 0.1, p(ε) = p b (ε) = exp( ε) and L 1 = L 2 = The contact volume v = 10 is taken much larger than ξ which is only about a few lattice sites here. Simulations in Figs. 1(b) and 1(c) demonstrate that, starting from arbitrary initial densities, the combined system reaches a stationary state where µ 1 = µ 2. The equalization of an ITV, i.e., the above mentioned chemical potential, indeed implies zeroth law which we verify next for three steady states having a PFSS: PFSS1 (δ = 1, c = 0; η 1 = 2), PFSS2 (δ = 3, c = 0; η 2 = 4) and PFSS3 (δ = 2, c = 1.0, γ = 1.5; η 3 = 3) with CD I. First, PFSS1 with density ρ and PfSS2 with density ρ are separately equilibrated with a third system PFSS3 with density ρ Then, PFSS1 with density ρ 1 and PFSS2 with density ρ 2 are brought into contact. The two resulting densities after equilibration remain almost unchanged, confirming zeroth law. The zeroth law can be similarly verified for CD II. There are numerous examples [25 29], where nonequilibrium processes with a conserved mass show shortranged spatial correlations, but the exact steady-state structures are not known. How does one find a contact dynamics which ensures Eq. 5 in these cases? We address the question in a class of widely studied mass transport processes [30 33], as another demonstration of how our formulation can be implemented in practice. In these models, we call them mass exchange models (MEM), in one dimension the continuous masses m i 0 and m i+1 0 at randomly chosen nearest neighbors i and i + 1 respectively are updated from time t to t + dt as m i (t + dt) = λ m i (t) + r(1 λ )m sum and m i+1 (t + dt) = λ m i+1 (t) + (1 r)(1 λ )m sum where, m sum = m i + m i+1 is the sum of nearest neighbor masses, r is a random number uniformly distributed in [0,1], and 0 < λ < 1 a model dependent parameter. As the spatial correlations are nonzero but very small, the subsystem weight factor in the steady states of individual systems can be obtained, even for v = 1 to a very good approximation, as W (m) = m vη 1 with η = (1 + 2λ )/(1 λ ) [21]. In panels (d) and (e) of Fig. 1, we observe equalization of chemical potentials µ 1 = η 1 /ρ 1 and µ 2 = η 2 /ρ 2 (respective ITV in this case) of systems 1 and 2 respectively for both contact dynamics I and II and for u 0 = 0.1, v = 1, L 1 = L 2 = 100 and p(ε) = p b (ε) = exp( ε). The zeroth law can be readily verified for MEM as done in the case of PFSS. There is no particular difficulty when systems having different kind of bulk dynamics are in contact; equilibration occurs as long as there is a common conserved quantity which is exchanged following Eq. 7. We demonstrate this in panel (f) of Fig. 1, taking two systems, PFSS and MEM, in contact with CD I: Chemical potentials eventually equalize and zeroth law follows. Summary. - In this paper, we demonstrate that weakly interacting nonequilibrium systems, with shortranged spatial correlations and having a conserved quantity which is exchanged upon contact between two systems, have an equilibrium-like thermodynamic structure in steady state, provided the exchange rates satisfy a balance condition (Eq. 1). The balance condition leads to zeroth law of thermodynamics and fluctuation relations analogous to the equilibrium fluctuation-dissipation theorems. In other words, for mass exchange rates satisfying the balance condition, we construct equivalence classes consisting of systems having a nonequilibrium steady state. Systems in each class are specified by value of an intensive thermodynamic variable which remains same when any two systems in the class exchange mass according to Eq. 1. In essence, our study provides a prescription to dynamically generate different equivalent nonequilibrium ensembles and thus would help in formulating a nonequilibrium thermodynamics for driven systems in general. S. C. acknowledges the financial support from the Council of Scientific and Industrial Research, India [09/575(0099)/2012-EMR-I].

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