## Nonequilibrium Statistcal Mechanics and Thermodynamics

*24 Jul 2023 22:17*

In equilibrium, we can use functions of states --- free energies, thermodynamic potentials --- to determine the most probable state. In fact, we can even determine the probability of arbitrary states. Out of equilibrium, it would seem that the natural generalization would be to use a functional of a sequence of states, of a trajectory, to determine the probability of trajectories. In the case of small, linear deviations from equilibrium, the Onsager-Machlup (or Onsager-Rayleigh) "action" gives us such a functional of trajectories. What works far from equilibrium? In equilibrium, one can link the thermodynamic potentials to functions which specify the rate of decay of large deviations, and this is still true out of equilibrium (see, e.g., Touchette's great review paper), but this is more of a mathematical result than a "physical" one.

Here's an argument for the ubiquity of effective actions. Markov processes
have Gibbs distributions over sequences of states, and Gibbs distributions,
just by definition, arise from an effective action. (*) Many nonequilibrium
systems can be described by Markov processes (say, deterministic trajectory
plus noise). But I'd go further and argue that *every* nonequilbrium
system can be represented as a Markov process --- that if you haven't found
one, you're not looking hard enough. (That argument's in
a separate paper.) So it
should *always* be possible to find an effective action. But this
doesn't establish that there should be a common form for these actions across
different systems, which is what e.g., Keizer and Woo (separately) claim.

Are there universal criteria for the stability of non-equilibrium steady states, or must be actually investigate entire paths? Landauer argued for the latter, convincingly to my mind, but I need to learn more here.

Approach to equilibrium doesn't interest me so much as sustained non-equilibrium situations, but like everybody else I suppose they're strongly connected. Fluctuation-dissipation results are accordingly interesting, especially ones which do not assume nearness to equilibrium. The Evans-Searles fluctuation theorem, which is well-supported by experiments (see e.g. the Carberry et al. paper) is extremely interesting.

I should try to explain some ideas about the role of smooth dynamical systems in the statistical mechanics here, but anyone who's geeky enough to be interested really ought to read Ruelle's review article rather than listen to me, and, after that, Dorfman's book.

*: Pedantically, a Gibbs distribution over a collection of variables \( X_t \), index by \( \mathcal{T} \), collectively \( X \), will take the form \( \mathbb{P}(X=x) \propto e^{-u(x)} \), and the "potential" \( u(x) = \sum_{A \in 2^{\mathcal{T}}}{u_A(x_A)} \), where \( u_A \) is a function only of the values of the random variables at the indices \( A \). (Obvious measure-theoretic complications for continuous index sets are obvious.) The sets \( A \) which actually contribute to the potential are called "cliques". In the particular case of Markov processes in discrete time, the cliques are pairs of successive indices \( \left\{ t, t+1 \right\} \), so we can think of \( u(x) \) as being basically a discrete-time effective action. For continuous-time Markov processes, see under Path Integrals and Feynman Diagrams for Classical Stochastic Processes.

See also

- Recommended, big picture:
- S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics
- J. R. Dorfman, Introduction to Chaos in Nonequilibrium Statistical Mechanics
- Dieter Forster, Hydrodynamic Fluctuations, Broken Symmetry,
and Correlation Functions [An excellent book which looks
*horrible.*Bless Donald Knuth for delivering us from type-writen equations!] - Josef Honerkamp, Stochastic Dynamical Systems
- Joel Keizer, Statistical Thermodynamics of Nonequilibrium Processes [Review: Molecular Fluctuations for Fun and Profit]
- David Ruelle, "Smooth Dynamics and New Theoretical Ideas in
Nonequilibrium Statistical Mechanics," Journal of Statistical
Physics
**95**(1999): 393--468 = chao-dyn/9812032 - Geoffrey Sewell, Quantum Mechanics and Its Emergent Macrophysics [Including nonequilibrium quantum statistical mechanics]
- Eric Smith, "Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions", arxiv:1102.3938
- Hugo Touchette, "The Large Deviations Approach to Statistical
Mechanics", Physics Reports
**478**(2009): 1--69, arxiv:0804.0327

- Recommended, historical:
- Mark Kac, Probability in Physical Sciences and Related Topics
- Lars Onsager, "Reciprocal relations in irreversible processes",
Physical Review
**37**(1931): 405--426 (part I) and**38**(1931): 2265--2279 (part II) - Lars Onsager and S. Machlup, "Fluctuations and Irreversible
Processes", Physical Review
**91**(1953): 1505--1512

- Recommended, close-ups:
- D. M. Carberry, J. C. Reid, G. M. Wang, E. M. Sevick, Debra
J. Searles and Denis J. Evans, "Fluctuations and Irreversibility: An
Experimental Demonstration of a Second-Law-Like Theorem Using a Colloidal
Particle Held in an Optical Trap", Physical Review Letters
**92**(2004): 140601 [An*extremely*good paper, giving a very nice explanation of the fluctuation theorem of Evans and Searles, followed by the neatest imaginable experimental demonstration of its validity.] - S. C. Chapman, G. Rowlands and Nick W. Watkins, "The Origin of Universal Fluctuations in Correlated Systems: Explicit Calculation for an Intermittent Turbulent Cascade," cond-mat/0302624
- W. De Roeck, Christian Maes and Karel Netocny, "H-Theorems from
Autonomous Equations", Journal of
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**123**(2006): 571--584, cond-mat/0508089 ["If for a Hamiltonian dynamics for many particles, at all times the present macrostate determines the future macrostate, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a Markov property for the macroscopic evolution."] - S. F. Edwards, "New Kinds of Entropy", Journal of
Statistical Physics
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**53**(1988): 233--248 ["The relative occupation of competing states of local stability is not determined solely by the characteristics of the locally favored states, but depends on the noise along the whole path connecting the competing states. This is not new, but the sophistication of most modern treatments has obscured the simplicity of this central point, and here it is argued for in simple physical terms."] - Michael Mackey, Time's Arrow: The Origin of Thermodynamic
Behavior [This is a very valuable short introduction to the
ergodic theory of Markov operators, which is
highly relevant to the origins of irreversibility, etc., but I don't think his
approach works, because he focuses on the
*relative*entropy (Kullback-Leibler divergence from the invariant distribution), rather than the Boltzmann entropy or even the Gibbs entropy.] - Mark Millonas (ed.), Fluctuations and Order: The New Synthesis [Despite the subtitle, no synthesis is in evidence. However, many of the individual papers are very interesting.]
- Eric Smith, "Thermodynamic dual structure of linear-dissipative
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**72**(2005): 036130 - Hyung-June Woo, "Statistics of nonequilibrium trajectories and
pattern
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**64**(2003): 627--633 - R. K. P. Zia, L. B. Shaw, B. Schmittmann and R. J. Astalos, "Contrasts Between Equilibrium and Non-Equilibrium Steady States: Computer Aided Discoveries in Simple Lattice Gases," cond-mat/9906376

- Modesty forbids me to recommend:
- CRS and Cristopher Moore, "What Is a Macrostate? Subjective Measurements and Objective Dynamics," cond-mat/03003625

- To read:
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- Anthony Aguirre, Sean M. Carroll, Matthew C. Johnson, "Out of equilibrium: understanding cosmological evolution to lower-entropy states", arxiv:1108.0417
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**88**(2013): 032146 - D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and
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**98**(2007): 150601 - Francis J. Alexander and Gregory L. Eyink, "Rayleigh-Ritz
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**78**(1997): 1--4 - Bidhan Chandra Bag
- "Nonequilibrium stochastic processes: Time dependence of entropy flux and entropy production," cond-mat/0205500
- "Upper bound for the time derivative of entropy for nonequilibrium stochastic processes," cond-mat/0201434
- BCB, Suman Kumar Banik, and Deb Shankar Ray, "The noise properties of stochastic processes and entropy production," cond-mat/0104524

- M. M. Bandi, W. I. Goldburg, J. R. Cressman Jr, "Measurement of entropy production rate in compressible turbulence", nlin.CD/0607036
- Julien Barre', Freddy Bouchet, Thierry Dauxois, Stefano Ruffo, "Out-of-equilibrium states as statistical equilibria of an effective dynamics," cond-mat/0204407
- Daniel A. Beard and Hong Qian, "Relationship between Thermodynamic Driving Force and One-Way Fluxes in Reversible Chemical Reactions", q-bio.SC/0607020
- Christian Beck
- "Superstatistics in hydrodynamic turbulence," physics/0303061
- "Superstatistics: Theory and Applications," cond-mat/0303288

- Eric Bertin, Kirsten Martens, Olivier Dauchot, and Michel Droz,
"Intensive thermodynamic parameters in nonequilibrium systems",
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**75**(2007): 031120 - Eric Bertin, Olivier Dauchot, Michel Droz, "Definition and
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**96**(2006): 120601, cond-mat/0512116 - L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim
- "Current Fluctations in Stochastic Lattice Gases",
Physical Review
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**94**(2005): 030601 = cond-mat/0407161 - "Towards a Nonequilibrium Thermodynamics: A Self-Contained Macroscopic Description of Driven Diffusive Systems", Journal of Statistical Physics
**135**(2009): 857--872 - "Lagrangian phase transitions in nonequilibrium thermodynamic systems", arxiv:1005.1489
- "Large deviation approach to non equilibrium processes in stochastic lattice gases", arxiv:math/0602557

- "Current Fluctations in Stochastic Lattice Gases",
Physical Review
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- Richard A. Blythe, "An introduction to phase transitions in stochastic dynamical systems", cond-mat/0511627
- Doriano Brogioli, "Marginally Stable Chemical Systems as Precursors
of
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**105**(2010): 058102 - Stephen G. Brush, The Kind of Motion We Call Heat: Statistical Physics and Irreversible Processes
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- Giovanni Bussi, Alessandro Laio and Michele Parrinello,
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**96**(2006): 090601 - C. Bustamante, J. Liphardt, and F. Ritort, "The Nonequilibrium
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**58**(2005): 43--48, cond-mat/0511629 - Pasquale Calabrese and Andrea Gambassi, "On the definition of a unique effective temperature for non-equilibrium critical systems", cond-mat/0406289
- T. Carlsson, L. Sjogren, E. Mamontov, and K. Psiuk-Maksymowicz,
"Irreducible memory function and slow dynamics in disordered systems",
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E
**75**(2007): 031109 - M. E. Cates and M. R. Evans (eds.), Soft and Fragile Matter: Nonequilibrium Dynamics, Metastability and Flow
- Philippe Chomaz, Francesca Gulminelli and Olivier Juillet, "Generalized Gibbs ensembles for time dependent processes", cond-mat/0412475
- E. G. D. Cohen, "Properties of nonequilibrium steady states: a path integral approach", Journal of Statistical Mechanics (2008): P07014
- Leonardo Crochik and Tania Tome, "Entropy production in the
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**72**(2005): 057103 - Daan Crommelin, "Estimation of Space-Dependent Diffusions and Potential Landscapes from Non-equilibrium Data", Journal of Statistical Physics
**149**(2012): 220--233 - Gavin E. Crooks, "Measuring Thermodynamic Length",
Physical Review
Letters
**99**(2007): 100602, arxiv:0706.0559 ["Thermodynamic length is a metric distance between equilibrium thermodynamic states. Among other interesting properties, this metric asymptotically bounds the dissipation induced by a finite time transformation of a thermodynamic system. It is also connected to the Jensen-Shannon divergence, Fisher information, and Rao's entropy differential metric."] - Gregory Bulnes Cuetara, Massimiliano Esposito, Alberto Imparato, "Exact fluctuation theorem without ensemble quantities", arxiv:1402.1873
- B. Derrida, "Non equilibrium steady states: fluctuations and large deviations of the density and of the current", cond-mat/0703762
- B. Derrida, Joel L. Lebowitz and Eugene R. Speer, "Exact Large Deviation Functional for the Density Profile in a Stationary Nonequilibrium Open System," cond-mat/0105110
- Deepak Dhar, "Pico-canonical ensembles: A theoretical description of metastable states," cond-mat/0205011 math-ph/0304043
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**13**(2003): 845--889 - Vlad Elgart and Alex Kamenev, "Rare Events Statistics in Reaction--Diffusion Systems", cond-mat/0404241 [i.e., large deviations]
- Denis J. Evans and Gary Morriss, Statistical Mechanics of Nonequilibrium Liquids
- R. M. L. Evans
- "Detailed balance has a counterpart in non-equilibrium steady states", cond-mat/0408614
- "Rules for transition rates in nonequilibrium steady states", cond-mat/0402527

- Massimo Falcioni, Luigi Palatella, Simone Pigolotti, Lamberto Rondoni and Angelo Vulpiani, "Boltzmann entropy and chaos in a large assembly of weakly interacting systems", nlin.CD/0507038
- Gregory Falkovich and Alexander Fouxon, "Entropy production away from equilibrium", nlin.CD/0312033 ["we express the entropy production via a two-point correlation function... the long-time limit gives the sum of the Lyapunov exponents"]
- Roger Filliger and Max-Olivier Hongler, "Relative entropy and
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**38**(2005): 1247--1255 - Silvio Franz, "How glasses explore configuration space," cond-mat/0212091
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- Giovanni Gallavotti
- "Entropy creation in nonequilibrium thermodynamics: a review", cond-mat/0312657
- "Stationary nonequilibrium statistical mechanics", cond-mat/0510027
- "Fluctuation relation, fluctuation theorem, thermostats and entropy creation in non equilibrium statistical Physics", cond-mat/0612061

- J. Galvao Ramos, Aurea R. Vasconcellos and Roberto Luzzi, "Nonlinear Higher-Order Thermo-Hydrodynamics II: Illustrative Examples", cond-mat/0412231
- Pierre Gaspard, The Statistical Mechanics of Irreversible Phenomena
- Piotr Garbaczewski
- "Information Entropy Balance and Local Momentum Conservation Laws in Nonequilibrium Random Dynamics," cond-mat/0301044
- "Shannon versus Kullback-Leibler Entropies in Nonequilibrium Random Motion", cond-mat/0504115

- Nicolas B. Garnier and Daniel K. Wojcik, "Spatiotemporal Chaos: The
Microscopic Perspective", Physical Review
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**96**(2006): 114101 - Pierre Gaspard, "Time-Reversed Dynamical Entropy and
Irreversibility in Markovian Random Processes", Journal of Statistical
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**117**(2004): 599--615 - T. Gilbert, J. R. Dorfman and P. Gaspard, "Entropy Production,
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**85**(2000): 1606--1609 - S. Goldstein and J. L. Lebowitz, "On the (Boltzmann) Entropy of Nonequilibrium Systems," cond-mat/0304251
- S. Goldsten, J. L. Lebowitz and Y. Sinai, "Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems," math-ph/9804016
- Giacomo Gradenigo, Alessandro Sarracino, Andrea Puglisi, Hugo Touchette, "Fluctuation relations without uniform large deviations", Journal of Physics A
**46**(2013): 335002, arxiv:1303.2851 - T. Hanney and R. B. Stinchcombe, "Real-space renormalisation group approach to driven diffusive systems", cond-mat/0606515
- R. J. Harris, A. Rákos, G. M. Schuetz, "Breakdown of Gallavotti-Cohen symmetry for stochastic dynamics", cond-mat/0512159
- Hisao Hayakawa and Michio Otsuki, "Nonequilibrium identities and response theory for dissipative particles", Physical Review E
**88**(2013): 032117 - Kumiko Hayashi and Hiroaki Takagi, "Fluctuation Thoerem applied
to Dictyostelium discoideum system", Journal of the Physical
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**10**(2007): 105001, arxiv:0710.0523 - Kumiko Hayashi, Hiroshi Ueno, Ryota Iino, and Hiroyuki Noji,
"Fluctuation Theorem Applied to F1-ATPase", Physical
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**104**(2010): 218103 - Malte Henkel, "Ageing, dynamical scaling and its extensions in many-particle systems without detailed balance", cond-mat/0609672
- Haye Hinrichsen, "Critical Phenomena in Nonequilibrium Systems," cond-mat/0001070
- Steven Huntsman, "Effective statistical physics of Anosov systems", arxiv:1009.2127
- Pablo I. Hurtado, Carlos Pérez-Espigares, Jesús J. del Pozo, and Pedro L. Garrido, "Symmetries in fluctuations far from equilibrium",
Proceedings of the National Academy
of Sciences (USA)
**108**(2011): 7704--7709 - A. Imparato and L. Peliti
- "Work probability distribution in systems driven out of equilibrium", cond-mat/0507080
- "The distribution function of entropy flow in stochastic systems", cond-mat/0611078

- Claude Itzykson and Jean-Michel Drouffe, Statistical Field Theory (2 vols.)
- M. V. Ivanchenko, O. I. Kanakov, V. D. Shalfeev and S. Flach,
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- Alex Kamenev, Field Theory of Non-Equilibrium Systems
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- Kyogo Kawaguchi and Yohei Nakayama, "Fluctuation theorem for hidden entropy production", Physical Review E
**88**(2013): 022147 - R. Kawai, J. M. R. Parrondo, C. Van den Broeck, "Dissipation: The phase-space perspective", cond-mat/0701397
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**100**(2008): 030601 - Teruhisa S. Komatsu, Naoko Nakagawa, Shin-Ichi Sasa and Hal Tasaki,
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**2007**: P03002 - Francois Leyvraz, Hernan Larralde, and David P. Sanders, "A Definition of Metastability for Markov Processes with Detailed Balance", cond-mat/0509754
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**87**(2013): 042108, arxiv:1302.1013 - James F. Lutsko, "Chapman-Enskog expansion about nonequilibrium states: the sheared granular fluid", cond-mat/0510749
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- Christian Maes
- "Entropy Production in Driven Spatially Extended Systems," cond-mat/0101064
- "Elements of Nonequilibrium Statistical Mechanics" [PDF]
- "Statistical Mechanics of Entropy Production: Gibbsian hypothesis and local fluctuations," cond-mat/0106464

- Christian Maes and Karel Netocny
- "Time-Reversal and Entropy," cond-mat/0202501
- "Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states", arixv:0705.2344

- C. Maes, K. Netocny, B. Shergelashvili, "A selection of nonequilibrium issues", math-ph/0701047 [Lecture notes, 55 pp.]
- Christian Maes, Karel Netocny, Bram Wynants
- "On and beyond entropy production: the case of Markov jump processes", arxiv:0709.4327
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**42**(2009): 365002, arxiv:0905.4897 - "Monotonic Return to Steady Nonequilibrium",
Physical Review Letters
**107**(2011): 010601

- Christian Maes, Frank Redig and Michel Verschuere
- "From Global to Local Fluctuation Theorems," cond-mat/0106639
- "No current without heat," cond-mat/0111281

- Christian Maes, Hal Tasaki, "Second law of thermodynamics for macroscopic mechanics coupled to thermodynamic degrees of freedom", cond-mat/0511419
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- "Local Fluctuation Theorem for Large Systems", Physical Review Letters
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**99**(2007): 130602 - Michele Pavon, "Stochastic control and nonequilibrium thermodynamical systems", Applied Mathematics and Optimiztion
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= Physical
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- Hong Qian and Timothy C. Reluga, "Nonequilibrium Thermodynamics and
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Letters
**94**(2005): 028101 - Saar Rahav and Christopher Jarzynski, "Fluctuation relations and coarse-graining", Journal of Statistical Mechanics (2007): P09012, arxiv:0708.2437
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- Edgar Roldan, Juan M.R. Parrondo, "Estimating dissipation from single stationary trajectories", arxiv:1004.2831
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- Lamberto Rondoni, Carlos Mejia-Monasterio, "Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms", arxiv:0709.1976
- David Ruelle
- "Extending the definition of entropy to nonequilibrium steady states," cond-mat/0303156
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- Shin-ichi Sasa and Teruhisa S. Komats
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**60**(1999): 159--164 = cond-mat/9901258 - "The Fluctuation Theorem and Green-Kubo Relations," cond-mat/9902021
- "Ensemble Dependence of the Transient Fluctuation Theorem," cond-mat/9906002

- "Fluctuation Theorem for Stochastic Systems,"
Physical Review E
- Udo Seifert
- "Entropy production along a stochastic trajectory and
an integral fluctuation theorem", cond-mat/0503686 = Physical Review
Letters
**95**(2005): 040602 - "Stochastic thermodynamics: principles
and perspectives", European Physical Journal B
**64**(2008): 423--431, arxiv:0710.1187

- "Entropy production along a stochastic trajectory and
an integral fluctuation theorem", cond-mat/0503686 = Physical Review
Letters
- E.M. Sevick, R. Prabhakar, Stephen R. Williams, Debra J. Searles, "Fluctuation Theorems", arxiv:0709.3888
- Geoffrey Sewell [Note to self: carefully compare these to papers by
Woo]
- "On Connections between the Quantum and Hydrodynamical Pictures of Matter", arxiv:0710.1239
- "Quantum macrostatistical picture of nonequilibrium steady states", math-ph/0403017
- "Quantum Macrostatistical Theory of Nonequilibrium Steady States", math-ph/0509069
- "Quantum Theory of Irreversibility: Open Systems and Continuum Mechanics", pp. 7--30 in E. Benatti and R. Floreanini (eds.): Lecture Notes in Physics vol. 622 (Springer-Verlag, 2003)
- "Macrostatistics and Fluctuating Hydrodynamics", arxiv:1206.2750

- T. Speck and U. Seifert, "The Jarzynski relation, fluctuation theorems, and stochastic thermodynamics for non-Markovian processes", Journal of Statistical Mechanics (2007) L09002, arxiv:0709.2236
- Jaeyoung Sung, "Validity condition of the Jarzynski relation for a classical mechanical system", cond-mat/0506214
- Tooru Taniguchi, E. G. D. Cohen, "Onsager-Machlup theory for
nonequilibrium steady states and fluctuation
theorems", cond-mat/0605548
=? Journal
of Statistical Physics
**130**(2007): 633--667 - Hal Tasaki
- "From Quantum Dynamics to the Second Law of Thermodynamics," cond-mat/0005128
- "The second law of Thermodynamics as a theorem in quantum mechanics," cond-mat/0011321

- Uwe C. Tauber, "Field Theory Approaches to Nonequilibrium Dynamics", cond-mat/0511743
- C. Tietz, S. Schuler, T. Speck, U. Seifert, and J. Wrachtrup,
"Measurement of Stochastic Entropy Production", Physical Review
Letters
**97**(2006): 050602 = cond-mat/0607407 - Alexei V. Tkachenko, "Generalized Entropy Approach to Far-from-Equilibrium Statistical Mechanics," cond-mat/0005198
- H. Touchette and E. G. D. Cohen, "A novel fluctuation relation for
a Lévy
particle", cond-mat/0703254
=? Physical Review
E
**76**(2007) 020101 - H. Touchette, M. Costeniuc, R.S. Ellis, and B. Turkington, "Metastability within the generalized canonical ensemble", cond-mat/0509802
- Hugo Touchette, Rosemary J. Harris, "Large deviation approach to nonequilibrium systems", arxiv:1110.5216
- E. H. Trepagnier, C. Jarzynski, F. Ritort, G. E. Crooks, C. J.
Bustamante and J. Liphardt, "Experimental test of Hatano and Sasa's
nonequilibrium steady-state equality", Proceedings of the
National Academy of Sciences USA
**101**(2004): 15033--15037 - Bruce Turkington, "An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics", Journal of Statistical Physics
**152**(2013): 569--597, arxiv:1207.2692 - Ramses van Zon, H. van Beijeren and J. R. Dorfman, "Kinetic Theory of Dynamical Systems," chao-dyn/9906040
- Aurea R. Vasconcellos, J. Galvao Ramos and Roberto Luzzi, "Nonlinear Higher-Order Thermo-Hydrodynamics: Generalized Approach in a Nonequilibrium Ensemble Formalism", cond-mat/0412227
- G. M. Wang, J. C. Reid, D. M. Carberry, D. R. M. Williams,
E. M. Sevick, and Denis J. Evans, "Experimental study of the fluctuation
theorem in a nonequilibrium steady state", PRE
**71**(2005): 046142 - Stephen R. Williams, Debra J. Searles, Denis J. Evans, "Numerical study of the Steady State Fluctuation Relations Far from Equilibrium", cond-mat/0601328
- Hyung-June Woo, "Variational
formulation of nonequilibrium thermodynamics for hydrodynamic pattern
formations," Physical Review E
**66**(2002) 066104 - Jeroen Wouters, Valerio Lucarini, "Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach",
Journal of Statistical Physics
**151**(2013): 850--860, arxiv:1208.3080 - Bram Wynants, "Structures of nonequilibrium fluctuations: dissipation and activity", arxiv:1011.4210
- V. I. Yukalov, "Principle of Pattern Selection for Nonequilibrium Phenomena," cond-mat/0110107
- Francesco Zamponi, "Is it possible to experimentally verify the fluctuation relation? A review of theoretical motivations and numerical evidence", cond-mat/0612019
- Juan Zanella and Esteban Calzetta, "Renormalization group and
nonequilibrium action in stochastic field theory,"
Physical Review
E
**66**(2002): 036134 - H. D. Zeh, Physical Basis of the Direction of Time
- R. K. P. Zia, B. Schmittmann, "A possible classification of nonequilibrium steady states", cond-mat/0605301
- D. N. Zubarev et al., Statistical Mechanics of Nonequilibrium Processes
- Robert Zwanzig, Nonequilibrium Statistical Mechanics

- To write someday, when I'd understand it:
- "Variational Principles in Nonequilibrium Statistical Mechanics"