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Nonequilibrium chemical potentials of steady-state lattice gas models in contact: A large-deviations approach

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Pith's one-line read This paper establishes that two driven systems in slow particle exchange acquire well-defined contact chemical potentials, and that those potentials—not the bulk alone—determine the steady-state densities.

desk verdict A clean large-deviations criterion for when contact chemical potentials exist in driven lattice gases, with honest but unquantified finite-epsilon caveats in the numerics. read the letter →

arxiv 1909.00432 v2 pith:FBBRZQG5 submitted 2019-09-01 cond-mat.stat-mech

Jules Guioth , Eric Bertin This is my paper · ORCID
classification cond-mat.stat-mech MSC 82C2260F10
keywords nonequilibriumchemicalpotentiallargedeviationslatticegasslowexchangelimitadditivitycontactdynamicszerothlawdrivensystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish when two driven, out-of-equilibrium systems that exchange particles through a weak contact can be assigned chemical potentials at all, and what those potentials predict. The answer is conditional and partly negative: if the contact is slow enough and the exchange dynamics factorizes between the two systems, the large-deviations function of the densities becomes additive, so a contact chemical potential can be attached to each system and equalizes at the steady state. That potential, however, is not a function of the bulk state alone; it carries an imprint of the contact dynamics, so no equation of state exists generically. If the contact rule does not factorize, no chemical potential can be defined, even though the most probable densities still exist. A sympathetic reader would care because this pins down when the familiar equilibrium notion of a chemical potential survives in driven steady states, and why earlier numerical violations of the zeroth law appeared.

What carries the argument

The engine of the argument is the large-deviations (Hamilton–Jacobi) equation for the joint density distribution in the slow-exchange limit, together with two structural conditions imposed on it. Macroscopic detailed balance, $I'(\rho_A|\bar\rho)=\ln[\phi(\rho_A,-1)/\phi(\rho_A,1)]$, lets the ratio of the forward and backward coarse-grained exchange rates determine the derivative of the large-deviations function. Factorization, $\phi(\rho_A,\Delta N_A)=\nu_0\phi_A(\rho_A,\Delta N_A)\phi_B(\rho_B,\Delta N_B)$, splits that derivative into a difference of two single-system terms, yielding the contact chemical potential $\mu^{\rm cont}_k(\rho_k)=\ln[\phi_k(\rho_k,-1)/\phi_k(\rho_k,1)]$. The factorization is inherited from a microscopic contact rule of the form $T_c(C'_A,C'_B|C_A,C_B)=\nu_0\theta_A(C_A,C'_A)\theta_B(C_B,C'_B)$, which holds for exponential and high-barrier Arrhenius rules but fails for heat-bath and Metropolis rules.

What would settle it

Simulate one pair of bulk driven systems with the same parameters but two different factorized contact rules and compare the measured steady-state densities over a range of overall densities; the paper predicts the two curves differ, with the difference set by the contact chemical potentials. If the densities coincide, the claimed contact-dependence fails. Conversely, if a non-factorized heat-bath contact rule still yields an additive joint density large-deviations function $I(\rho_A,\rho_B)=\gamma_A I_A(\rho_A)+\gamma_B I_B(\rho_B)$, the sufficiency claim for factorization would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that additivity of the density large-deviations function, $I(\rho_A,\rho_B)=\gamma_A I_A(\rho_A)+\gamma_B I_B(\rho_B)$, is the precise condition under which a nonequilibrium chemical potential can be defined for two driven lattice gases in slow contact. The paper shows that two sufficient conditions make additivity hold: macroscopic detailed balance at contact, $I'(\rho_A)=\ln[\phi(\rho_A,-1)/\phi(\rho_A,1)]$, and factorization of the coarse-grained exchange rate, $\phi(\rho_A,\Delta N_A)=\nu_0\,\phi_A(\rho_A,\Delta N_A)\phi_B(\rho_B,\Delta N_B)$. When both hold, each system carries a contact chemical potential $\mu^{\rm cont}_k(\rho_k)=\ln[\phi_k(\rho_k,-1)/\phi_k(\rho_k,1)]$, and the steady-state densities satisfy $\mu^{\rm cont}_A(\rho_A^*)=\mu^{\rm cont}_B(\rho_B^*)$. Because $\mu^{\rm cont}_k$ depends on the contact factors $\phi_k$, it is not a bulk equation of state: different factorized contact rules move the predicted densities, as confirmed in the exactly solvable model and in KLS simulations, while a non-factorized rule destroys additivity and with it the chemical-potential description.

Load-bearing premise

The whole construction rests on a strict separation of time scales: particle exchanges across the contact are so rare that, conditioned on the current densities, the two systems' microstates are statistically independent, each in its own isolated steady state.

Editorial extensions

If this is right

  • When macroscopic detailed balance and factorization both hold, the steady-state densities of two systems in contact are fixed by equality of contact chemical potentials, $\mu^{\rm cont}_A(\rho_A^*)=\mu^{\rm cont}_B(\rho_B^*)$, and this equality predicts the densities measured in simulations.
  • The contact chemical potential is not a bulk property: it depends on which factorized microscopic rule realizes the contact, so no equation of state exists generically; only a contact fine-tuned with the drive restores one.
  • The zeroth law of thermodynamics holds only within classes of systems that share a factorized contact rule, effectively including half of the contact in each system; transitivity fails otherwise.
  • $\mu^{\rm cont}_k$ can be expressed as the equilibrium chemical potential $\mu^{\rm eq}_k$ or the isolated-system potential $\mu^{\rm iso}_k$ plus an excess term measuring the nonequilibrium modification of the stationary distribution and/or extra work at the contact.
  • For non-factorized contact rules, the large-deviations function is non-additive and no chemical potential can be defined, even though macroscopic detailed balance holds for single-particle exchange and the most probable densities remain well defined.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension would be to use a driven system with a known contact chemical potential as a probe of another driven system, with the caveat that the measured value will depend on the probe's own contact rule, so comparisons require identical contact dynamics.
  • One could test the slow-exchange boundary directly by measuring the joint density large-deviations function at increasing exchange rates; the paper's framework predicts that additivity breaks generically away from the slow limit, so observing additivity persist at finite rates would mark the regime where a chemical-potential description still works.
  • The same additivity criterion could be applied to other conserved quantities, such as energy or volume, yielding analogous contact temperature or contact pressure whose contact dependence would mirror the chemical-potential result.
  • Since macroscopic detailed balance alone is insufficient, a multi-particle exchange rule that factorizes in the bulk but not at the contact would provide a sharp numerical test of the factorization condition's role.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops a large-deviations framework for two driven lattice gases exchanging particles through a slow contact. In the limit where contact jumps are rare compared to bulk relaxation, the coarse-grained density dynamics is governed by a Hamilton-Jacobi equation. The authors show that if the macroscopic transition rates at contact satisfy a macroscopic detailed-balance relation and factorize into system-dependent factors, then the stationary large-deviations function is additive and one can attach contact chemical potentials μ_cont_A, μ_cont_B that equalize in the steady state. They argue that these potentials generally depend on the contact dynamics and therefore do not obey an equation of state, and they relate them to equilibrium or isolated-system chemical potentials through excess terms. The framework is applied to an exactly solvable driven lattice gas and to the KLS model using numerical simulations.

Significance. If the central claims hold, the paper clarifies the microscopic conditions under which a nonequilibrium chemical potential can be defined via additivity of the density large-deviations function, and it sharpens earlier phenomenological discussions by Pradhan, Seifert, Sasa, Tasaki, and Chatterjee et al. The Hamilton-Jacobi/additivity derivation in Secs. III–IV is internally consistent, and the central equalization prediction is parameter-free. The exactly solvable model provides a nontrivial analytic test, and the KLS comparison is a genuine numerical check rather than a fit. The negative corollary that the nonequilibrium chemical potential is contact-dependent and lacks an equation of state is clearly formulated. The paper is also careful to label Eq. (44) as a postulate, which limits the generality of the μ_cont–μ_iso relation but does not affect the core additivity criterion.

major comments (4)
  1. [Secs. V D and V E, Eq. (11)] The slow-exchange factorization P(CA,CB|ρA,ρB)=PA(CA|ρA)PB(CB|ρB) is the load-bearing premise for the coarse-grained rates (10) and all subsequent results, and the authors correctly note in Sec. VII that additivity is generically broken beyond this limit. The numerical validations in Secs. V D and V E, however, use only ε=0.01 and provide no ε-dependence or extrapolation toward ε=0; the KLS comparison is purely numerical and no data or code are supplied. Since the theory does not quantify corrections in ε, agreement at a single finite value does not by itself establish that the simulations realize the ε→0 limit. Please add an ε-dependence study or at least an estimate of the leading correction, or explicitly present the comparisons as illustrative rather than as direct validation of the limit.
  2. [Sec. IV C 3, Eq. (44)] The authors state that they 'postulate, without proof' the representation PVk,k(Ck|ρk)=Zneq^{-1}e^{-βHk+Υneq}. The subsequent relation μcont_k=μiso_k+ln(φ_{k,ΔΥneq}/φ_k), Eq. (46), is therefore conditional on an unproved ansatz. The central additivity result does not depend on Eq. (44), but the paper should either present Eq. (46) explicitly as a conjecture or provide evidence for the assumed form of the stationary distribution.
  3. [Sec. V C 2] The text says that for the Kawasaki rule the coarse-grained rates do not factorize, 'so that the large-deviations function is not additive, implying that a chemical potential cannot be defined.' This inference is not valid as stated, because Sec. IV A 1 explicitly notes that factorization is a sufficient but not necessary condition. One needs to show that the ratio φ(ρA,−1)/φ(ρA,+1) in the Kawasaki example cannot be written as a difference of single-density functions, or the conclusion should be qualified as restricted to the class of factorized contacts.
  4. [Secs. V C–V E, Eqs. (63), (69), (72)] The printed formulas are inconsistent with the definitions. Eq. (30) defines μcont_k as the logarithm of a ratio, but Eqs. (69) and (72) present μcont_k as a bare ratio, omitting the logarithm. In Eq. (63), the numerator should follow from Eq. (B8) with a factor μ(n+1) in the exponent rather than μn. These inconsistencies make the theoretical curves in Figs. 2, 3, and 5 irreproducible from the text; please correct the formulas and state explicitly which quantity was actually computed in the simulations.
minor comments (5)
  1. [Fig. 2 caption] The caption says 'Densities ρA (red) and ρB (blue) versus time,' but the horizontal axis is labeled f_A; please correct the caption or the axis label.
  2. [Figs. 3–6] The KLS simulation results are reported without error bars or statistical uncertainty estimates; please add standard errors or state that fluctuations are smaller than the symbol size.
  3. [Reproducibility] No data or code availability statement is included; given that the KLS validation is an important part of the paper, a reproducibility statement would be helpful.
  4. [Sec. III C] The statement that macroscopic detailed balance is 'always verified' for single-particle exchange would be clearer if the two-term cancellation in Eq. (20) were shown explicitly for ΔNA=±1.
  5. [References] Ref. [52] is cited as 'to be published'; if the perturbative solution discussed in Sec. III C is needed, please either include the relevant calculation or remove the forward reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: additivity and contact chemical potentials are derived from stated factorization/detailed-balance assumptions, and simulations are parameter-free consistency tests.

full rationale

The paper's central derivation is self-contained rather than circular. The slow-exchange factorization (Eq. 11) and the microscopic factorization of contact rates (Eq. 32) are explicitly stated physical assumptions; macroscopic detailed balance (Eq. 21) is either a consequence of the Hamilton-Jacobi equation for single-particle exchange or an explicitly identified condition for multi-particle exchange. The additivity of the large-deviations function (Eq. 26) and the equalization of contact chemical potentials (Eq. 31) are mathematical consequences of these assumptions, not inputs. The chemical potential at contact is defined by Eq. (30) as a ratio of coarse-grained rates, but the predictive content lies in computing these rates from the isolated stationary distributions and then comparing the resulting density prediction with simulations of two coupled systems; no fitted parameters enter. The KLS and exactly solvable model comparisons are genuine parameter-free tests that could have failed if the factorization assumption were violated. The self-citation to Ref. [66] supplies an independent exact steady-state distribution for the solvable model; that result does not assume or contain the contact-chemical-potential claim, so it is real evidence rather than load-bearing circularity. Eq. (44) is explicitly labeled a postulate for relating contact and isolated chemical potentials, and is not disguised as a derivation. No equation is shown to be equivalent to its own input by construction, and no fitted quantity is renamed as a prediction. The unquantified finite-exchange-rate corrections noted in Sec. VII are a correctness/approximation concern, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation introduces no fitted constants; all comparisons are parameter-free given the model dynamics. The main unproved inputs are the slow-exchange factorization, the postulated form of the stationary distribution in Eq. (44), the orthogonality of the contact to the drives, and the LDP/convexity assumptions. The exact-solvable-model steady state is quoted from the authors' prior paper [66].

assumptions (6)
  • domain assumption Slow-exchange time-scale separation and conditioned factorization: P(CA,CB|ρA,ρB)=PA(CA|ρA)PB(CB|ρB).
    Introduced in Sec. III A, Eq. (11), as the small-ϵ limit of the coarse-grained dynamics. It is the precondition for replacing coupled dynamics by contact rates averaged over isolated steady states.
  • domain assumption Contact is orthogonal to the driving forces and its microscopic rates do not depend on fA, fB.
    Stated in Sec. II B and used throughout; it makes Tc drive-independent and compatible with ordinary local detailed balance at equilibrium.
  • ad hoc to paper The stationary distribution of an isolated driven system has the form PVk,k(Ck|ρk)=Z_neq^{-1} e^{-βHk+Υneq} (Eq. 44).
    Explicitly introduced without proof in Sec. IV C 3 to derive μcont=μiso+η. This is the clearest unproved input in the paper.
  • domain assumption The large-deviations function I exists, is convex, and has a unique minimum in the absence of phase transitions.
    Invoked in Sec. III B following standard large-deviations practice; needed to identify the steady state through I'(ρ*)=0.
  • domain assumption The exact stationary distribution of the solvable model, Eq. (55), is taken from the authors' earlier work [66].
    Quoted rather than rederived in this paper; it underpins all exact-solvable-model chemical potential formulas.
  • domain assumption For the additive cases, microscopic contact rates factorize as Tc(C'A,C'B|CA,CB)=ν0 θA(C'A,CA) θB(C'B,CB) (Eq. 32).
    A sufficient condition for additivity; verified for the natural, Sasa-Tasaki, and exponential rules, and deliberately violated by the Kawasaki example.

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Pith. "Pith review of Nonequilibrium chemical potentials of steady-state lattice gas models in contact: A large-deviations approach." pith.science (2026). https://pith.science/paper/FBBRZQG5

@misc{pith2026190900432,
  author       = {Pith},
  title        = {Pith review of: Nonequilibrium chemical potentials of steady-state lattice gas models in contact: A large-deviations approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBBRZQG5}},
  note         = {Machine review of arXiv:1909.00432}
}
read the original abstract

We introduce a general framework to describe the stationary state of two driven systems exchanging particles or mass through a contact, in a slow exchange limit. The definition of chemical potentials for the systems in contact requires that the large-deviations function describing the repartition of mass between the two systems is additive, in the sense of being a sum of contributions from each system. We show that this additivity property is satisfied on condition that a macroscopic detailed balance condition holds at contact, and that the coarse-grained contact dynamics satisfies a factorization property. However, the nonequilibrium chemical potentials of the systems in contact keep track of the contact dynamics, and thus do not obey an equation of state. These nonequilibrium chemical potentials can be related either to the equilibrium chemical potential, or to the nonequilibrium chemical potential of the isolated systems. Results are applied both to an exactly solvable driven lattice gas model, and to the Katz-Lebowitz-Spohn model using a numerical procedure to evaluate the chemical potential.

Figures

Figures reproduced from arXiv: 1909.00432 by the authors.

Figure 1
Figure 1. FIG. 1: Plots of the chemical potentials associated with [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Numerical simulations of two lattice models [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Average density profile along the [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: FIG. 3: Numerical simulations of two KLS models A [PITH_FULL_IMAGE:figures/full_fig_p018_3.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Average density profile along the [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Numerical simulations of two KLS models [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

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Reference graph

Works this paper leans on

101 extracted references · 73 canonical work pages

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    This implies that the ratio 8 ϕ(ρA,−∆NA)/ϕ(ρA, ∆NA) should take a factorized form with respect to systems A and B

    Factorization condition of the contact dynamics When macroscopic detailed balance (21) holds, the additivity property of the large-deviations function I(ρA|¯ρ) should be directly related to the coarse- grained transition rates ϕ. This implies that the ratio 8 ϕ(ρA,−∆NA)/ϕ(ρA, ∆NA) should take a factorized form with respect to systems A and B. A sufficient c...

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    As seen in Eq

    Microscopic transition rates: Factorization condition We now relate the factorization assumption (28) of the coarse-grained transition rates to the properties of the microscopic transition rates Tc. As seen in Eq. (10), the transition rates ϕ(ρA, ∆NA) are averages of the micro- scopic transition rates over the product of stationary dis- tributions of the ...

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    Sasa-Tasaki dynamics The first one, discussed in [2, Appendix B], will be called the Sasa-Tasaki rule.[80] This rule is claimed to model a high energy barrier separating systems A and B. If the energy barrier is high, the transition rate takes an Arrhenius expression: Tc(C′ A,C′ B|CA,CB) =ϵ { e−β∆HA if ∆NA =−1 e−β∆HB if ∆NA = +1 , (34) ∆Hk = Hk(C′ k)−Hk(Ck...

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    It reads as Tc(C′ A,C′ B|CA,CB) =ϵe−β 2 ∆HAe−β 2 ∆HB, (35) where one has used the same notations as for the Sasa- Tasaki dynamics

    Exponential rule Another classic rule for which the factorization condi- tion holds is when τ(x) =ex/2. It reads as Tc(C′ A,C′ B|CA,CB) =ϵe−β 2 ∆HAe−β 2 ∆HB, (35) where one has used the same notations as for the Sasa- Tasaki dynamics. This exponential rule could be relevant in the case when interactions between A and B are neg- ligible compared to the int...

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    (30)) and those of isolated systems

    General formula using detailed balance at contact We discuss here the relationships between the chemical potentials of systems in contact (see Eq. (30)) and those of isolated systems. First, one can notice that when macroscopic detailed balance condition (21) as well as the factorization condi- tion (32) hold, it is sufficient to compute quantities for ∆NA ...

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    This idea was first the one of McLennan [57, 58] who com- puted corrections due to the driving force up to first or- der

    Relation between µcont and µ eq If one takes the equilibrium state as the reference, PVk,k (Ck) can be obtained from a perturbative expan- sion with respect to the equilibrium distribution. This idea was first the one of McLennan [57, 58] who com- puted corrections due to the driving force up to first or- der. Based on this idea to compute perturbatively th...

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    Relation between µcont and µiso Rather than taking equilibrium as the reference situ- ation, one can also consider the out-of-equilibrium state on its own. Indeed, even if a general procedure to define a nonequilibrium free energy is not yet established, one can sometimes, but rarely, directly compute the nonequi- librium stationary distribution which brin...

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    Contact dependence of the excess chemical potential In each case, one sees that the chemical potential at contactµcont k (ρk) is equal to a chemical potential related to the isolated system (either the equilibrium one or the stationary nonequilibrium one) and an excess chemical potential which generically reads as ηk(ρk) = lnφk, Υk(ρk, +1) φk(ρk, +1) . (4...

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    One can nevertheless wonder what happens when microscopic transition rates at contact do depend on the driving forces or when there is an extra work performed at contact

    Driven systems with a drive-dependent contact dynamics Until now, we have considered a contact orthogonal to the nonconservative driving forces, leading to transition 12 rates at contact independent of the driving forces of sys- temsA andB and verifying detailed balance with r...

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    In particular, this situation happens in biological systems and more specifically in cells

    Equilibrium systems with an active contact Even at equilibrium, the case where the microscopic transition rates involve an extra work is quite interesting. In particular, this situation happens in biological systems and more specifically in cells. Indeed, let us consider two co...

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    The number of sites is assumed to be even and we write |Λ| = 2L with L an integer

    Definition and steady-state distribution As for the ZRP, one considers a one-dimensional lat- tice Λ of |Λ| sites. The number of sites is assumed to be even and we write |Λ| = 2L with L an integer. Each site i is occupied by ni > 0 particles that cannot exceed a maximum number ...

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    (57) The symbol ∝ means here that the transition rates are equal to the right-hand side up to a constant factor that sets the typical time scale associated with the transition

    Natural dynamics We first consider the case when the transition rate at contact is similar to the dynamics in the bulk, i.e., the transition rate depends on the final configuration: Tc(n′ iA,n′ jB|niA,njB)∝e−εA(n′ iA)e−εB(n′ jB ). (57) The symbol ∝ means here that the transition ...

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    Sasa-Tasaki rule For the Sasa-Tasaki rule which models a high energy barrier separating both systems, the probability to trans- fer a particle fromA toB (respectively fromB toA) only depends on the energy to go from theA side (respectively B side) bottom of the barrier to its ...

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    Kawasaki or heat bath rule The Kawasaki, or heat-bath, rule, is a standard choice of transition rate. It does not factorize in two terms that respectively depend on A and B: Tc(n′ iA,n′ jB|niA,njB) (59) ∝ 2 1 +e [ εA(n′ iA)−εA(niA) ] + [ εB(n′ jB )−εB(njB ) ]. Note that anothe...

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    Natural dynamics and the Sasa-Tasaki rule We start by considering the natural dynamics (57) and the Sasa-Tasaki rule (58) as the dynamics of the con- tact. Since these microscopic dynamics are factorized, the coarse-grained transition rates also take a factorized form, ϕ(ρA, ∆...

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