Large deviation approach to non equilibrium processes in stochastic lattice gases

A. De Sole; C. Landim; D. Gabrielli; G. Jona-Lasinio; L. Bertini

REVIEW

Large deviation approach to non equilibrium processes in stochastic lattice gases

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim

Probability · Submitted 24 Feb 2006 · 25 references

Pith's verdict: Review of the authors' own MFT program: the SSEP free-energy derivation checks out and is the real content, but the general Onsager–Machlup principle rides on a quoted identity and Section 7's transition is asserted rather than proved.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDmath/0602557 (version 1)

Pith IDOO5DQTK6, Pith's permanent record of this paper and its review

CitationA. De Sole, C. Landim, D. Gabrielli, G. Jona-Lasinio, L. Bertini. "Large deviation approach to non equilibrium processes in stochastic lattice gases." arXiv:math/0602557 (2006). https://pith.science/paper/OO5DQTK6

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This is a review of the authors' own program, and it is upfront about that. The one piece that stands alone is Section 5.4: they derive the DLS free energy for the boundary-driven SSEP from the dynamical large-deviations route, via a heat-equation transformation. That computation is correct and worth having in one place. If you are teaching MFT or need a worked example, this is a useful reference.

The paper does several things well. It spells out the physical motivation for using quasi-potentials in non-equilibrium steady states, it states the rigorous LDPs with enough context, and it gives a fair account of the Bodineau–Giacomin–Derrida contributions alongside the authors' own. As a review, it is readable and generally honest about what is rigorous and what is heuristic.

Now the soft spots, in descending order of importance. The central identity (5.6) is quoted from [1,2] with no argument, and the whole extended Onsager–Machlup principle in Section 5.3 is built on it. The authors themselves note that infinite-dimensional Hamilton–Jacobi theory is not established, so the reader is essentially being asked to accept that the general framework works on faith. The SSEP derivation would survive a failure of (5.6), which is why I still trust that part, but the 'general principle' advertised in the abstract has weaker support than the abstract suggests. Section 7 is another issue: Lemmas 7.2 and 7.3 are stated without proof, and the traveling-wave phase transition is justified by 'a computation shows' plus citations to [4,5]. That is a citation, not a derivation. In a review, that is borderline acceptable if the results are clearly marked as announced elsewhere, but here it reads as part of the review's own claim. Finally, reference [19] is listed as '1983 in preparation'—that is either a joke or a mistake, and it should be fixed.

None of these are load-bearing for the main worked computation, so the paper is not fundamentally misleading. But the balance is shifted: the concrete part is strong, the general part is mostly assertion.

Who is this for? A graduate student or researcher wanting an entry point to the authors' program, or someone who wants to see how the SSEP computation goes without digging through four papers. It is not for someone looking for new theorems.

My recommendation: send it to peer review, but with a referee who knows the area and will require the status of (5.6), Lemmas 7.2/7.3, and the traveling-wave claim to be made explicit. As a review, it deserves a serious reading, not a desk reject.

Referee report

Significance

The paper is valuable as a coherent statement of a program that has been influential in nonequilibrium statistical mechanics. Its strengths are the self-contained derivation of the SSEP quasi-potential, the independent check against the DLS free energy, and the clear articulation of the variational structure linking density and current fluctuations. It is also honest about technical gaps in the general theory, noting in Section 5.2 that infinite-dimensional Hamilton–Jacobi theory is not well established and that multiple solutions can occur. If the general adjoint-fluctuation principle is correct, it gives a broadly applicable extension of Onsager–Machlup; however, the manuscript does not itself supply the proof of the key structural identity (5.6), and several statements in Section 7 are asserted rather than demonstrated.

Major objections (3)

[Section 5.3, Eq. (5.6)] The identity D(ρ)+D*(ρ)=∇(χ(ρ)∇δV/δρ) is the only bridge between the Hamilton–Jacobi construction of Section 5.2 and the interpretation of (5.4) as the adjoint hydrodynamic equation. It is quoted from references [1,2] with no argument reproduced in this review. Since the extended Onsager–Machlup principle of Section 5.3 is advertised as a general principle, please include at least a derivation sketch or a precise theorem with hypotheses in an appendix, or explicitly present this identity as an assumption quoted from the literature. In its current form a reader cannot check the step, and a failure of (5.6) would invalidate the claim that the most probable creation path is the time-reverse of the adjoint relaxation.
[Section 5.2, Lemma 5.2 and Theorem 5.3] The equality V=W in Theorem 5.3 depends on the convergence of the adjoint solution ψ_t in (5.4) to the stationary profile ρ̄ as T↑∞. This convergence is established in the text only for the boundary-driven SSEP, through the heat-equation transformation (5.11)–(5.12). For the other models mentioned later (WASEP, zero range, KMP), the hypothesis is not verified in this manuscript. Please state explicitly for which models the convergence hypothesis is known and where it is proved, or restrict the general claim accordingly.
[Section 7, Lemmas 7.2 and 7.3, Eq. (7.4)] Lemmas 7.2 and 7.3 are stated as facts with citations to [4,5,6], and the KMP dynamical phase transition (7.4) is asserted via 'a computation shows' plus citations to [4,5]. Since the abstract and introduction explicitly advertise the traveling-wave phase transition as one of the main physical conclusions, the variational computation leading to Φ_m(q)<U_m(q) for large q should be presented at least in outline, or the statement should be explicitly labeled as a result quoted from [4,5] with its exact hypotheses. This is a review article, so citation is legitimate, but the current text mixes proof and quotation without clearly marking which parts are demonstrated here.

Minor objections (4)

[Section 5.4] The sentence 'A computation shows that the derivative of the functional W...' omits the algebra verifying (5.7) against (5.8); please expand this step, since it fixes the functional form of W and is central to the SSEP computation.
[Lemma 5.2] In the proof of Lemma 5.2, 'an appropriate interpolation between ψ_{-1} and ρ̄' is not specified; please define the interpolation explicitly and show that its added cost vanishes as T→∞.
[Section 7, Eq. (7.1)] In the definition of A_{T,q}, the path W should be required to be absolutely continuous in time so that \\dot W_t exists for almost every t; please state this regularity condition explicitly.
[References] Reference [19] is listed as '1983 in preparation' and should be updated or removed.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

For non-reversible lattice gases, the most likely way a fluctuation appears is the time reversal of the adjoint relaxation path.

The reading

This review paper argues that a single large-deviation principle governs fluctuations in non-equilibrium stochastic lattice gases: the probability of any density or current trajectory has an exponential rate given by a dynamical cost functional. Minimizing that functional defines a quasi-potential, and the paper's central claim is that this quasi-potential is the non-equilibrium free energy of the stationary state. For reversible systems the optimal fluctuation path is the time reverse of the ordinary relaxation path, reproducing the Onsager-Machlup picture; for non-reversible systems the same statement holds with relaxation replaced by relaxation under the adjoint dynamics. In the boundary-driven symmetric simple exclusion process the quasi-potential is written explicitly through a nonlinear boundary value problem, while in the KMP heat-flow model the current rate function is lowered by traveling-wave paths for large currents, signalling a dynamical phase transition.

What carries the argument

The central object is the rate functional $I_{[0,T]}(\pi|\gamma)$ for density trajectories, together with its current version $I_{[0,T]}(W|\gamma)$; the quasi-potential $V(\gamma)=\inf_T \inf_{\pi_T=\gamma} I_{[0,T]}(\pi|\bar\rho)$ is obtained by minimizing such functionals. The argument then runs through the Hamilton-Jacobi equation for the quasi-potential, the structural identity $D(\rho)+D^*(\rho)=\nabla(\chi(\rho)\nabla\delta V/\delta\rho)$ connecting the forward hydrodynamic operator $D$, the adjoint operator $D^*$, the mobility $\chi$, and the quasi-potential, and the adjoint hydrodynamic equation $\partial_t\rho = -D(\rho)+\nabla(\chi(\rho)\nabla\delta V/\delta\rho)$. Running this adjoint equation backward in time produces the optimal fluctuation paths. For the simple exclusion process, a nonlocal transformation converts the adjoint equation into the heat equation, which is how the required relaxation to the stationary profile is established.

Extended reading notes

Core claim

The paper's central claim is that a single dynamical variational problem organizes both relaxation and fluctuation in non-equilibrium lattice gases. The quasi-potential $V(\gamma)$, defined as the infimum of the large-deviation action over all trajectories joining the stationary profile $\bar\rho$ to the profile $\gamma$, is shown to equal the free energy $F$ appearing in the Boltzmann-Einstein formula for the stationary measure. In the boundary-driven symmetric simple exclusion process this yields the explicit formula $F(\gamma)=\int_0^1 du\{\gamma(u)\log[\gamma(u)/F(u)] + (1-\gamma(u))\log[(1-\gamma(u))/(1-F(u))] + \log[F'(u)/(\beta-\alpha)]\}$, where $F$ is the increasing solution of $F''=(\gamma-F)(F')^2/[F(1-F)]$ with $F(0)=\alpha$, $F(1)=\beta$. The same machinery applied to the time-averaged current shows that in the KMP model the optimal path for a large current is a traveling wave, giving $\Phi_m(q)<U_m(q)$; this is the paper's claimed dynamical phase transition.

Load-bearing premise

The argument stands on the identity $D(\rho)+D^*(\rho)=\nabla(\chi(\rho)\nabla\delta V/\delta\rho)$, cited from earlier work rather than proved here, and on the assumption that the adjoint hydrodynamic flow converges to the stationary profile; that convergence is established only for the simple exclusion process, through its heat-equation transformation.

Editorial extensions

If this is right

For the boundary-driven symmetric simple exclusion process, the static large-deviation rate of the empirical density is exactly the quasi-potential, so density fluctuations in the stationary state are controlled by the nonlinear boundary value problem (3.3).
The most probable path that creates a density fluctuation is the time reversal of the path by which the adjoint process relaxes; in reversible systems this recovers the Onsager-Machlup statement that creation and relaxation are mutual time reversals.
The time-averaged current obeys a large-deviation principle with a convex rate function; when the optimal strategy is a time-independent density profile the rate equals the cost $U(q)$ of maintaining the current at a fixed profile, and when it is not, time-dependent minimizers lower the rate.
In the KMP heat-flow model, the inequality $\Phi_m(q)<U_m(q)$ for large currents means there is a dynamical phase transition: the system breaks time-translation symmetry by choosing a traveling-wave density profile.
The dynamic variational problem is the source of the static theory: the rate function of the stationary empirical density is obtained from the quasi-potential, so thermodynamics emerges from the analysis of dynamical fluctuations.

Reading between the lines

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

If the identity behind the adjoint-relaxation principle holds generally, the non-equilibrium free energy of a model could be measured by watching the time-reversed process relax, bypassing the difficult task of computing the stationary measure.
The traveling-wave mechanism is likely a generic feature of models with strictly convex mobility: similar dynamical phase transitions should appear in boundary-driven heat or charge transport once the time-averaged current is conditioned to be large.
The variational formulation naturally extends to higher dimensions, where the analogue of the one-dimensional traveling wave is a divergence-free traveling current pattern; testing whether such patterns beat stationary profiles would be a direct check of the scenario.
For finite $N$ and $T$, the sharpness of the inequality $\Phi_m(q)<U_m(q)$ in the KMP model is an open quantitative question; numerical measurement of the rate function at fixed current could identify the critical current where the optimal path changes from constant to traveling wave.

Original abstract

We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,873 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems.	contribution 0.95	0.49	0.20	asserted 0.96
s1	Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems.	background 0.87	0.11	0.06	asserted 0.92
s2	In this way, some principles of wide validity have been obtained leading to interesting physical consequences.	interpretation 0.68	0.32	0.39	asserted 0.98

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.80.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (25)

[19]

Galves A., Kipnis C., Spohn H., 1983 in preparation

View on Pith 1983
[1]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Fluctuations in stationary non equilibrium states of irreversible processes. Phys. Rev. Lett. 87, 040601 (2001)

View on Pith 2001
[2]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Macroscopic ﬂuctuation theory for stationary non equilibrium state. J. Statist. Phys. 107, 635–675 (2002)

View on Pith 2002
[3]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Large deviations for the boundary driven simple exclusion process. Math. Phys. Anal. Geom. 6, 231–267 (2003)

View on Pith 2003
[4]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Current ﬂuctuations in stochastic lattice gases. Phys. Rev. Lett. 94, 030601 (2005)

View on Pith 2005
[5]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Non equilibrium current ﬂuctuations in stochastic lattice gases. Preprint 2005, arXiv: cond-mat/0506664

View on Pith Pith review arXiv 2005
[6]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Large deviations of the empirical current in interacting particle systems. Preprint 2005, arXiv: math.PR/0512394

View on Pith arXiv 2005
[7]

Bertini L., Gabrielli D., Lebowitz J.L., Large deviations for a stochastic model of heat ﬂow. J. Statist. Phys. 121, 843–885 (2005)

View on Pith 2005
[8]

Bodineau T., Derrida B., Current ﬂuctuations in non-equilibrium diﬀusive systems: an addi- tivity principle. Phys. Rev. Lett. 92, 180601 (2004)

View on Pith 2004
[9]

Bodineau T., Derrida B., Distribution of current in non-equilibrium diﬀusive syste ms and phase transitions. Preprint 2005, arXiv: cond-mat/0506540

View on Pith Pith review arXiv 2005
[10]

Bodineau T., Giacomin G., From dynamic to static large deviations in boundary driven exclusion particles systems. Stoch. Proc. Appl. 110, 67–81 (2004)

View on Pith 2004
[11]

De Masi A., Ferrari P., Ianiro N., Presutti E., Small deviations from local equilibrium for a process which exhibits hydrodynamical behavior. II. J. Statist. Phys. 29, 81–93 (1982)

View on Pith 1982
[12]

Derrida B., Lebowitz J.L., Speer E.R., Free energy functional for nonequilibrium systems: an exactly solvable model. Phys. Rev. Lett. 87, 150601 (2001)

View on Pith 2001
[13]

Derrida B., Lebowitz J.L., Speer E.R., Large deviation of the density proﬁle in the steady state of the open symmetric simple exclusion process. J. Statist. Phys. 107, 599–634 (2002)

View on Pith 2002
[14]

Donsker M.D., Varadhan S.R.S., Large deviations from hydrodynamic scaling limit. Comm. Pure Appl. Math. 42 243-270 (1989)

View on Pith 1989
[15]

Enaud C., Derrida B., Large deviation functional of the weakly asymmetric exclus ion process. J. Statist. Phys. 114, 537–562 (2004)

View on Pith 2004
[16]

Eyink G., Lebowitz J.L., Spohn H., Hydrodynamics of stationary nonequilibrium states for some lattice gas models. Commun. Math. Phys. 132, 253–283 (1990)

View on Pith 1990
[17]

Eyink G., Lebowitz J.L., Spohn H., Lattice gas models in contact with stochastic reservoirs: local equilibrium and relaxation to the steady state. Commun. Math. Phys. 140, 119–131 (1991)

View on Pith 1991
[18]

Freidlin M.I., W entzell A.D., Random perturbations of dynamical systems. Springer-Verlag, Berlin, New York 1998

View on Pith 1998
[20]

Graham R., T´ el T., Nonequilibrium potential for coexisting attractors. Phys. Rev. A 33 , 1322–1337 (1986)

View on Pith 1986
[21]

Kipnis C., Landim L., Scaling limits of interacting particle systems , Springer-Verlag, Berlin, New York, (1999)

View on Pith 1999
[22]

Kipnis C., Marchioro C., Presutti E., Heat ﬂow in an exactly solvable model. J. Statist. Phys. 27, 65–74 (1982)

View on Pith 1982
[23]

Kipnis C., Olla S., Varadhan S.R.S., Hydrodynamics and large deviations for simple exclusion processes. Commun. Pure Appl. Math. 42, 115–137 (1989)

View on Pith 1989
[24]

Onsager L., Machlup S., Fluctuations and irreversible processes. I Phys. Rev. 91, 1505–1512 (1953); Fluctuations and irreversible processes. II Phys. Rev. 91, 1512–1515 (1953)

View on Pith 1953
[25]

Spohn H., Long range correlations for stochastic lattice gases in a no nequilibrium steady state. J. Phys. A 16 , 4275–4291 (1983). DYNAMICAL LARGE DEVIATIONS IN STOCHASTIC LATTICE GASES 21 Lorenzo Bertini Dipartimento di Matematica, Universit `a di Roma “La Sapienza” Piazzale Aldo Moro 2, 00185 Roma, Italy e-mail: bertini@mat.uniroma1.it Alberto De Sole ...

View on Pith 1983

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper

Among other various promising attempt at discussing entropy (or a related quantity) in NESS, let us refer to the macroscopic fluctuation theory developed by Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim [ref] , exact solution of the large deviation functional by Derrida, Lebowitz, and Speer [ref] , the proposal of the additivity principle by Bodineau and Derrida [ref] , and the recent interesting proposal based on adiabatic accessibility by Lieb and Yngvason [ref] .

Exact equalities and thermodynamic relations for nonequilibrium steady states · Background and motivation · 1.00

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [19]

Galves A., Kipnis C., Spohn H., 1983 in preparation

View on Pith 1983

[2] [1]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Fluctuations in stationary non equilibrium states of irreversible processes. Phys. Rev. Lett. 87, 040601 (2001)

View on Pith 2001

[3] [2]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Macroscopic ﬂuctuation theory for stationary non equilibrium state. J. Statist. Phys. 107, 635–675 (2002)

View on Pith 2002

[4] [3]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Large deviations for the boundary driven simple exclusion process. Math. Phys. Anal. Geom. 6, 231–267 (2003)

View on Pith 2003

[5] [4]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Current ﬂuctuations in stochastic lattice gases. Phys. Rev. Lett. 94, 030601 (2005)

View on Pith 2005

[6] [5]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Non equilibrium current ﬂuctuations in stochastic lattice gases. Preprint 2005, arXiv: cond-mat/0506664

View on Pith Pith review arXiv 2005

[7] [6]

Bertini L., De Sole A., Gabrielli D., Jona–Lasinio G., La ndim C., Large deviations of the empirical current in interacting particle systems. Preprint 2005, arXiv: math.PR/0512394

View on Pith arXiv 2005

[8] [7]

Bertini L., Gabrielli D., Lebowitz J.L., Large deviations for a stochastic model of heat ﬂow. J. Statist. Phys. 121, 843–885 (2005)

View on Pith 2005

[9] [8]

Bodineau T., Derrida B., Current ﬂuctuations in non-equilibrium diﬀusive systems: an addi- tivity principle. Phys. Rev. Lett. 92, 180601 (2004)

View on Pith 2004

[10] [9]

Bodineau T., Derrida B., Distribution of current in non-equilibrium diﬀusive syste ms and phase transitions. Preprint 2005, arXiv: cond-mat/0506540

View on Pith Pith review arXiv 2005

[11] [10]

Bodineau T., Giacomin G., From dynamic to static large deviations in boundary driven exclusion particles systems. Stoch. Proc. Appl. 110, 67–81 (2004)

View on Pith 2004

[12] [11]

De Masi A., Ferrari P., Ianiro N., Presutti E., Small deviations from local equilibrium for a process which exhibits hydrodynamical behavior. II. J. Statist. Phys. 29, 81–93 (1982)

View on Pith 1982

[13] [12]

Derrida B., Lebowitz J.L., Speer E.R., Free energy functional for nonequilibrium systems: an exactly solvable model. Phys. Rev. Lett. 87, 150601 (2001)

View on Pith 2001

[14] [13]

Derrida B., Lebowitz J.L., Speer E.R., Large deviation of the density proﬁle in the steady state of the open symmetric simple exclusion process. J. Statist. Phys. 107, 599–634 (2002)

View on Pith 2002

[15] [14]

Donsker M.D., Varadhan S.R.S., Large deviations from hydrodynamic scaling limit. Comm. Pure Appl. Math. 42 243-270 (1989)

View on Pith 1989

[16] [15]

Enaud C., Derrida B., Large deviation functional of the weakly asymmetric exclus ion process. J. Statist. Phys. 114, 537–562 (2004)

View on Pith 2004

[17] [16]

Eyink G., Lebowitz J.L., Spohn H., Hydrodynamics of stationary nonequilibrium states for some lattice gas models. Commun. Math. Phys. 132, 253–283 (1990)

View on Pith 1990

[18] [17]

Eyink G., Lebowitz J.L., Spohn H., Lattice gas models in contact with stochastic reservoirs: local equilibrium and relaxation to the steady state. Commun. Math. Phys. 140, 119–131 (1991)

View on Pith 1991

[19] [18]

Freidlin M.I., W entzell A.D., Random perturbations of dynamical systems. Springer-Verlag, Berlin, New York 1998

View on Pith 1998

[20] [20]

Graham R., T´ el T., Nonequilibrium potential for coexisting attractors. Phys. Rev. A 33 , 1322–1337 (1986)

View on Pith 1986

[21] [21]

Kipnis C., Landim L., Scaling limits of interacting particle systems , Springer-Verlag, Berlin, New York, (1999)

View on Pith 1999

[22] [22]

Kipnis C., Marchioro C., Presutti E., Heat ﬂow in an exactly solvable model. J. Statist. Phys. 27, 65–74 (1982)

View on Pith 1982

[23] [23]

Kipnis C., Olla S., Varadhan S.R.S., Hydrodynamics and large deviations for simple exclusion processes. Commun. Pure Appl. Math. 42, 115–137 (1989)

View on Pith 1989

[24] [24]

Onsager L., Machlup S., Fluctuations and irreversible processes. I Phys. Rev. 91, 1505–1512 (1953); Fluctuations and irreversible processes. II Phys. Rev. 91, 1512–1515 (1953)

View on Pith 1953

[25] [25]

Spohn H., Long range correlations for stochastic lattice gases in a no nequilibrium steady state. J. Phys. A 16 , 4275–4291 (1983). DYNAMICAL LARGE DEVIATIONS IN STOCHASTIC LATTICE GASES 21 Lorenzo Bertini Dipartimento di Matematica, Universit `a di Roma “La Sapienza” Piazzale Aldo Moro 2, 00185 Roma, Italy e-mail: bertini@mat.uniroma1.it Alberto De Sole ...

View on Pith 1983