Nonequilibrium Statistcal Mechanics and ThermodynamicsNotebooks Nonequilibrium Statistcal Mechanics and Thermodynamics

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. What works far from equilibrium? In equilibrium, one can link the thermodynamic potentials to functions which specify the rate of decay of large deviations --- is this still true out of equilibrium? (Eyink, below, says yes, but I want to be sure. Keizer, for instance, also proposes an action, but I'm not sure it's the same as Eyink's, and, if they differ, which is right?) Here's my 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 Eyink and Keizer claim. (The papers by Woo make similar claims, with special reference to hydrodynamics and spatially-extended systems; this is very exciting, and I need to re-read all of them carefully.) 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. Having just read the paper by Carberry et al., below, I am seized with the desire to read up on the Evans-Searles fluctuation theorem, which now seems incredibly cool. I should try to explain the new 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. See also Pattern Formation; Self-organization; Self-organized Critcality; Statistical Mechanics; Foundatons of Statisticcal Mechanics; Stochastic

Processes; Interacting Particle Systems; Large Deviations The Nonequilibrium Thermodynamics of Small Systems - Physics Today July 2005 advanced search

Small systems found throughout physics, chemistry, and biology manifest striking properties as a result of their tiny dimensions. Examples of such systems include magnetic domains in ferromagnets, which are typically smaller than 300 nm; quantum dots and biological molecular machines that range in size from 2 to 100 nm; and solidlike clusters that are important in the relaxation of glassy systems and whose dimensions are a few nanometers. Scientists nowadays are interested in understanding the properties of such small systems. For example, they are beginning to investigate the dynamics of the biological motors responsible for converting chemical energy into useful work in the cell (see the article by Terence Strick, Jean-François Allemand, Vincent Croquette, and David Bensimon, PHYSICS TODAY October 2001, page 46). Those motors operate away from equilibrium, dissipate energy continuously, and make transitions between steady states. Until the early 1990s, researchers had lacked experimental methods to investigate such properties of small systems as how they exchange heat and work with their environments. The development of modern techniques of microscopic manipulation has changed the experimental situation. In parallel, during the past decade, theorists have developed several results collectively known as fluctuation theorems (FTs), some of which have been experimentally tested. The much-improved experimental access to the energy fluctuations of small systems and the formulation of the principles that govern both energy exchanges and their statistical excursions are starting to shed light on the unique properties of microscopic systems. Ultimately, the knowledge physicists are gaining with their new experimental and theoretical tools may serve as the basis for a theory of the nonequilibrium thermodynamics of small systems. Molecular machines Thermodynamics describes energy exchange processes of macroscopic systems: Objects as varied as liquids, magnets, superconductors, and even black holes comply with its laws. In macroscopic systems, behavior is reproducible and fluctuations (deviations from the typically observed, average behavior) are small. It is only under some special conditions that thermal fluctuations produce readily detectable consequences in macroscopic systems. Well-known examples include the opalescence of light in a fluid at its critical point and the blue color of the sky, which is a result of light scattering. Figure 1 As a system's dimensions decrease, fluctuations away from equilibrium

begin to dominate its behavior. In particular, in a nonequilibrium small system, thermal fluctuations can lead to observable and significant deviations from the system's average behavior. Therefore, such systems are not well described by classical thermodynamics. Systems of this type abound in the laboratory, where scientists are building motors with dimensions of less than 100 nm (see figure 1a), and in the cell, where the biological function and efficiency of molecules such as the molecular motor kinesin are determined by molecular size (figure 1b). Kinesin is one of many molecular machines. In the cell, those machines use the energy of bond hydrolysis to perform useful work such as the replication, transcription, and repair of DNA and the translation of RNA. Kinesin's role is to carry subcellular cargoes along microtubules. On average, a kinesin motor takes one 8-nm step every 10–15 milliseconds. A single adenosinetriphosphate (ATP) molecule is hydrolyzed per step, and the chemical energy released is tightly coupled to movement and force generation. The kinesin is highly processive—that is, it takes many steps before detaching from its microtubule track. How efficient is the kinesin motor and how much energy does it dissipate as it moves along the track? The chemical energy released by ATP hydrolysis is about 20 kBT. (In the world of small systems, the product of Boltzmann's constant and temperature is a convenient energy unit.) The motor does about 12 kBT of work with each step. Thus, the machine's efficiency is roughly 60% and it dissipates about 650 kBT per second into its environment. Molecular machines are unlike macroscopic ones in that they can harness thermal fluctuations and rectify them using energy from chemical sources. Consider, for example, RNA polymerase, an enzyme that moves along DNA to produce a newly synthesized RNA strand—a process called transcription. Although it has not yet been proven unequivocally, evidence suggests that during transcription the polymerase moves by extracting energy from the thermal bath and uses bond hydrolysis to ensure that only "forward" fluctuations are captured. That is, the enzyme rectifies thermal fluctuations to directed motion. The amount of energy required to step from one DNA base to another, the shape of the enzyme, the structural roughness of DNA, and the information encoded in the steps (that is, the base sequence of the DNA helix) are all essential aspects that are attributable to the smallness of the system and that ultimately determine its dynamics. Figure 2 As with macroscopic systems, for small systems one can distinguish between two situations in which the systems' behavior and properties do not change with time: equilibrium states and nonequilibrium steady states. Systems in nonequilibrium steady states have net currents that flow across them, but their properties do not display observable time dependence. Examples of nonequilibrium

steady-state systems include an object in contact with two thermal sources at different temperatures, for which the current is a heat flux; a resistor with electric current flowing across it; and the kinesin–microtubule system, for which kinesin motion is the current. Such systems require a constant input of energy to maintain their steady state. Because the systems constantly dissipate net energy, they operate away from equilibrium. Most biological systems, including molecular machines and even whole cells, are nonequilibrium steady states. In a nonsteady-state system, the most general case, one or more of the system's properties change in time. Figure 2 shows various thermal systems classified according to their size and typical dissipation energy rate, along with a couple of macroscopic systems for comparison. Most of the small systems are characterized by length scales in the nanometer-to-micrometer range and dissipation rates of 10–1000 kBT/s. The state of a small system Figure 3 External variables such as temperature, pressure, and chemical potential specify the different ensembles in statistical mechanics. All such ensembles yield the same equation of state in the large-volume or thermodynamic limit. For small systems, the equation of state and the spectrum of fluctuations are fully determined by so-called control parameters. Figure 3 illustrates two control parameters that may be used to define the state of a small stretched polymer. By giving direct access to control parameters of single microscopic systems, micromanipulation technology has opened up new opportunities to study nonequilibrium small systems. By varying such parameters, one can perform controlled experiments in which the system is driven away from its initial state of equilibrium and its subsequent response is observed.1 Consider, for example, the system illustrated in figure 3a, and imagine that a tethered polymer initially at equilibrium is driven out of equilibrium by the action of an external perturbation that moves the two walls farther apart. In that situation, the control parameter is the distance between the two walls, and the nonequilibrium protocol is fully specified by giving the wall-to-wall distance as a function of time X(t). Since the system is small and is placed in a thermal bath, its dynamics will be effectively random: Even if the nonequilibrium protocol is exactly repeated, the trajectory followed by the system will be different. Each trajectory may be represented by the time evolution of the positions of all atoms {xi(t)}. As the control parameter X evolves, the total energy of the system, U({xi}, X), varies in a manner that has two distinct contributions, (1) The first term is the variation of the energy resulting from the change in the internal configuration—that is, a change in heat Q—and the second term

is the variation of the energy resulting from the perturbation applied while varying the control parameter—that is, a change in work W. If the control parameter changes from 0 to Xf, the total work done on the system is given by (2) where (δU/δX){xi} is the force applied to move the walls. The heat Q exchanged in the nonequilibrium process may be expressed as ΔU − W, where ΔU is the variation of the energy. Random fluctuations dominate the thermal behavior in small systems. Since the force is a fluctuating quantity, W, Q, and ΔU will also fluctuate for different trajectories, and the amount of heat or work exchanged with the bath will fluctuate in magnitude and even sign. For a given nonequilibrium protocol, the work and heat probability distributions P(W) and P(Q) characterize the work and heat collected over an infinite number of experiments. In general, those distributions will depend on the details of the experimental protocol. Distributions such as P(W) and P(Q) are important for providing detailed information about how a system responds when subjected to a particular experimental process. Fluctuation theorems Nonequilibrium systems are characterized by irreversible heat losses between the system and its environment, typically a thermal bath. Fluctuation theorems embody recent developments toward a unified treatment of arbitrarily large fluctuations in small systems. In equilibrated, time-reversal-invariant systems, no net heat is transferred from the system to the bath. Therefore the probability of absorbing a given amount of heat must be identical to that of releasing it, and the ratio P(Q)/P(−Q) equals 1. The probability ratio becomes different from 1 under nonequilibrium conditions. We assume time-reversal invariance, but note that for equilibrated, noninvariant systems—if, for example, magnetic fields are present—it is the total probability for heat absorption that is equated with the total probability for heat release. The mid-1990s saw the introduction of two important FTs. Denis Evans and Debra Searles derived an FT for systems evolving from equilibrium toward a nonequilibrium steady state, and Giovanni Gallavotti and Eddie Cohen developed an FT for steady-state systems.2 The two works were based on numerical evidence obtained previously.3 In steady-state systems, an external agent continuously produces heat that is transferred to the bath. The average amount of heat <Q> so produced implies an increase in the total average entropy of the system plus environment equal to <S> = <Q>/T.

The rate at which the system exchanges heat with the bath is called the entropy production, σ = Q/Tt, where t is the interval of time over which the system exchanges the heat Q. Associated with the entropy production is a time-dependent probability distribution Pt(σ). Gallavotti and Cohen established an explicit mathematical expression that holds under very general conditions for the ratio Pt(σ)/Pt(−σ) in steady states,

(3) Although this expression involves a limit of infinite time, a similar expression without the limit should be valid, to good approximation, as long as t is much greater than the decorrelation time, which is, roughly speaking, the recovery time of the steady state after it is slightly perturbed. Equation 3 indicates that a steady-state system is more likely to deliver heat to the bath (σ is positive) than it is to absorb an equal quantity of heat from the bath (σ is negative). Nonequilibrium steady-state systems always dissipate heat on average. For macroscopic systems, the heat is an extensive quantity and therefore the ratio of probabilities P(σ)/P(−σ) grows exponentially with the system's size. That is to say, the probability of heat absorption by macroscopic systems is insignificant. Our bodies, for example, are maintained in a nonequilibrium state by metabolic processes that dissipate heat all the time. For small systems such as molecular motors that move along a molecular track, however, the probability of absorbing heat can be significant. On average, molecular motors produce heat, but it may be that they move by rectifying thermal fluctuations—a process that would imply the occasional capture of heat from the bath. Fluctuation theorems shed light on Loschmidt's paradox. In 1876, Josef Loschmidt raised an objection to Boltzmann's derivation of the second law of thermodynamics from Newton's laws of motion. According to Loschmidt, since the microscopic laws of mechanics are invariant under time reversal, there must also exist entropy-decreasing evolutions that apparently violate the second law. The FTs show how macroscopic irreversibility arises from time-reversible microscopic equations of motion. Time-reversed trajectories do occur, but they become vanishingly rare with increasing system size. For large systems, the conventional second law emerges. The Jarzynski equality The various FTs that have been reported differ in the details of such considerations as whether the system's dynamics are stochastic or deterministic, whether the kinetic energy or some other variable is kept constant, and whether the system is initially prepared in equilibrium or in a nonequilibrium steady state. A novel treatment of dissipative processes in nonequilibrium systems was introduced in 1997 when Christopher Jarzynski reported a nonequilibrium work relation,4 now called the Jarzynski equality (JE). (See PHYSICS TODAY, September 2002, page 19.) The JE indicates a practical way to determine free-energy differences. Consider a system, kept in contact with a bath at temperature T, whose equilibrium state is determined by a control parameter x. Initially, the control parameter is xA and the system is in an equilibrium state A. The nonequilibrium process is obtained by changing x according to a given protocol x(t), from xA to some final value xB. In general, the final state of the system will not be at equilibrium. It will equilibrate to a state B if it is allowed to further evolve with the control parameter fixed at xB.

The JE states that (4) where ΔG is the free-energy difference between the equilibrium states A and B, and the angle brackets denote an average taken over an infinite number of nonequilibrium experiments repeated under the protocol x(t). Frequently, the JE is recast in the form exp(−Wdis)/kBT = 1, ⟨ ⟩ where Wdis = W − ΔG is the dissipated work along a given trajectory. The exponential average appearing in the JE implies that W ≥ ΔG or, ⟨ ⟩ equivalently, Wdis ≥ 0, which, for macroscopic systems, is the statement ⟨ ⟩ of the second law of thermodynamics in terms of free energy and work. An important consequence of the JE is that, although on average Wdis ≥ 0, the equality can only hold if there exist nonequilibrium trajectories with Wdis ≤ 0. Those trajectories, sometimes referred to as transient violations of the second law, represent work fluctuations that ensure the microscopic equations of motion are time-reversal invariant. The remarkable JE implies that one can determine the free-energy difference between initial and final equilibrium states not just from a reversible or quasi-static process that connects those states, but also via a nonequilibrium, irreversible process that connects them. The ability to bypass reversible paths is of great practical importance. In 1999, Gavin Crooks related various FTs by deriving a generalized theorem for stochastic microscopically reversible dynamics.5 The box below gives details. The past six years have seen further consolidation, and physicists now understand that neither the details of just which quantities are maintained constant during the dynamics nor the somewhat differing interpretations of entropy production, entropy production rate, dissipated work, exchanged heat, and so forth lead to fundamentally distinct FTs. Computer simulations have played an essential role in the development of the FTs.6 Indeed, the first paper on the subject, the 1993 report by Evans, Cohen, and Gary Morriss,3 included molecular-dynamics simulations of a two-dimensional gas of disks. Over suitably short times, their computer runs showed spontaneous ordering of the gas, in agreement with the expression the authors had derived for the probability of fluctuations of a nonequilibrium steady-state fluid's sheer stress. Computer simulations of nonequilibrium systems continue to be important primarily due to the difficulty of setting up and characterizing suitable nonequilibrium small systems. Conversely, FTs, and especially the JE, can potentially be used to improve the performance of molecular-dynamics simulations. Experimental tests of fluctuation theorems Theorists who consider small systems have greatly benefited from advances in micromanipulation that make it possible to measure energy fluctuations in nonequilibrium small systems. With such measurements, experimenters can test the validity of FTs and scrutinize some fundamental assumptions of

statistical mechanics. Sergio Ciliberto and Claude Laroche, in their 1998 study of Rayleigh–Benard convection, performed the first experimental test of the Gallavotti–Cohen FT.7 In 2002, Evans's group verified an integrated form of equation 3 in an experiment that used an optical trap to repeatedly drag microscopic beads through water.8 They computed the entropy production for each bead trajectory and found that the likelihood of entropy-consuming trajectories relative to entropy-producing trajectories was precisely as theory predicted. For short times in the millisecond range, the researchers readily observed entropy-consuming trajectories. And as expected, the classical bulk behavior was recovered for longer times on the order of seconds. Figure 4 Gerhard Hummer and Attila Szabo noted the biophysical relevance of the JE and showed how free energies could be extracted via single-molecule experiments carried out under nonequilibrium conditions.9 Soon thereafter, a group led by one of us (Bustamante) tested the JE by mechanically stretching a single molecule of RNA, both reversibly and irreversibly, between its folded and unfolded conformations.10 Figure 4 illustrates the group's experimental design. When the RNA was unfolded slowly, the average forward and reverse trajectories could be superimposed; that is, the reaction was reversible. When the RNA was unfolded more rapidly, the mean unfolding force increased and the mean refolding force decreased. The folding–unfolding cycle was thus hysteretic, an indication that work was dissipated. When Bustamante and coworkers applied the JE to the irreversible work trajectories, they recovered the free energy of the unfolding process to within kBT/2 of its best independent estimate—the work needed to reversibly stretch the RNA. Their experimental test is an example of how the JE bridges the statistical mechanics of equilibrium and nonequilibrium systems. Experimenters have continued to progress in their ability to test FTs. Technical improvements have recently enabled Evans's group to test equation 3, rather than its integrated form.11 Nicolas Garnier and Ciliberto have used electrical circuits as the driven, dissipative system.12 They injected current to maintain an electrical dipole, composed of a resistor and a capacitor, in a nonequilibrium steady state; collected the probability distributions of work and heat; and showed those distributions to be in very good agreement with the appropriate FT. Figure 5 Compared to tests involving trapped beads or stretched polymers, experiments with electrical circuits are less prone to drift and other systematic biases, and they permit much greater numbers of trajectories. Those advantages allow for the investigation of systems with larger dissipation rates. By recording several hours of fluctuation data from their driven electrical dipole, Garnier and Ciliberto investigated the

exchanged heat and work with unusually high resolution, and detected the non-Gaussian tails in the heat distribution. Such tails, predicted by Ramses van Zon and Cohen for the heat distribution in linear systems,13 are also characteristic of work distributions in systems with a nonlinear response to change in a control parameter. For further details see figure 5. Physicists have also deepened their understanding of the JE. Jarzynski's result—and Crooks's generalization as well—applies to systems that start at equilibrium and are then driven out of equilibrium by some external influence. For many systems of interest, however, including biological molecular machines and nanophotonics devices, the JE does not apply. Such systems often execute irreversible transitions between nonequilibrium steady states. In 1998, Yoshitsugu Oono and Marco Paniconi proposed a general phenomenological framework that encompassed nonequilibrium steady states and transitions between such states. Three years later, Takahiro Hatano and Shin-ichi Sasa built on that work and generalized the JE to arbitrary transitions between nonequilibrium steady states.14 Their result was tested and confirmed last year by a measurement of the dissipation and fluctuations of microspheres optically driven through water.15 Those theoretical and experimental advances represent steps toward a complete theory of steady-state thermodynamics. Such a theory would have a profound effect on how scientists describe nonequilibrium steady-state systems such as molecular machines and cells. Although we have not discussed them, several quantum versions of the classical FTs have appeared in the physics literature. To date, no quantum FT has yet received experimental scrutiny, but such experiments might show interesting surprises. For example, quantum coherence may allow large fluctuations to be observed and FTs to be tested in much larger systems than would be possible in a classical world. That and many other exciting challenges remain for scientists continuing to work with small systems, a fertile ground where physics, chemistry, and biology converge.

Carlos Bustamante is a Howard Hughes Medical Institute investigator and a professor of molecular and cell biology, chemistry, and physics at the University of California, Berkeley. Jan Liphardt is an assistant professor of biophysics, also at the University of California, Berkeley. Felix Ritort is a professor of statistical physics at the University of Barcelona in Spain.

Box: The Crooks Fluctuation Theorem Gavin Crooks provided a significant generalization of an important fluctuation theorem (FT) obtained earlier by Christopher Jarzynski. As

described in the text, the Jarzynski equality (JE) relates the change ΔG in free energy of two equilibrium states to an appropriate work average calculated with an irreversible path. In the Jarzynski scenario, and also in Crooks's generalized FT, the system is initially in thermal equilibrium but then driven out of equilibrium by the action of an external agent. Let xF(s) denote a time-dependent nonequilibrium "forward" process for which the variable s runs from 0 to some final time t. The forward process initially acts on an equilibrium state A and it and ends at a state B that is not at equilibrium. In the reverse process, the initial state B is allowed to reach equilibrium and the system evolves to a nonequilibrium state A. The nonequilibrium protocol for the reverse process xR(s) is time-reversed with respect to the forward one, xR(s) = xF(t − s), so that both processes last for the same time t. Let PF(W) and PR(W) stand for the work probability distributions along the forward and reversed processes respectively. Then the Crooks FT asserts The Crooks FT can be manipulated to yield the JE. It also resembles the Gallavotti–Cohen FT (equation 3) derived for steady-state systems if one identifies σt with Wdis/T = (W − ΔG)/T. The main difference is that the Gallavotti–Cohen relation is asymptotically valid, whereas the Crooks theorem holds for any finite time t. Return to text

Recommended: 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", cond-mat/0508089 = Journal of Statistical Physics 123 (2006): 571--584 ["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."] J. R. Dorfman, Introduction to Chaos in Nonequilibrium Statistical Mechanics S. F. Edwards, "New Kinds of Entropy", Journal of Statistical Physics 116 (2004): 29--42 [I need to think about how his last kind of entropy is related to Lloyd-Pagels thermodynamic depth.] Gregory L. Eyink, "Action principle in nonequilbrium statistical dynamics," Physical Review E 54 (1996): 3419--3435 K. H. Fischer and J. A. Hertz, Spin Glasses 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!] Pierre Gaspard, Chaos, Scattering and Statistical Mechanics Josef Honerkamp, Stochastic Dynamical Systems Mark Kac, Probability in Physical Sciences and Related Topics Joel Keizer, Statistical Thermodynamics of Nonequilibrium Processes [Review: Molecular Fluctuations for Fun and Profit] Rolf Landauer, "Motion Out of Noisy States," Journal of Statistical Physics 53 (1988): 233--248 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.] L. Onsager and S. Machlup, "Fluctuations and Irreversible Processes," Physical Review 91 (1953): 1505--1512 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, "Thermodynamic dual structure of linear-dissipative driven systems", Physical Review E 72 (2005): 036130 Hyung-June Woo, "Statistics of nonequilibrium trajectories and pattern selection", Europhysics Letters 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: D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A.

Petrosyan, "Entropy Production and Time Asymmetry in Nonequilibrium Fluctuations", Physical Review Letters 98 (2007): 150601 G. Baez, H. Larralde, F. Leyvraz and Rafael A. Mendez-Sanchez, "Fluctuation-Dissipation Theorem for Metastable Systems," cond-mat/0303281 [forthcoming in PRL] Francis J. Alexander and Gregory L. Eyink, "Rayleigh-Ritz Calculation of Effective Potential Far from Equilibrium," Physical Review Letters 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, J. R. Cressman Jr., W. I. Goldburg, "Test of the Fluctuation Relation in compressible turbulence on a free surface", nlin.CD/0607037 M. M. Bandi, W. I. Goldburg, J. R. Cressman Jr, "Measurement of entropy production rate in compressible turbulence", nlin.CD/0607036 A. Bandrivskyy, S. Beri and D. G. Luchinsky, "Non-equilibrium distributions at finite noise intensities," cond-mat/0301173 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 Ludovic Bellon and Sergio Ciliberto, "Experimental study of the Fluctuation-Dissipation-Relation during an aging process," cond-mat/0201224 Ludovic Berthier ,Juan P. Garrahan, "Non-topographic description of inherent structure dynamics in glass formers," cond-mat/0303451 Eric Bertin, Kirsten Martens, Olivier Dauchot, and Michel Droz, "Intensive thermodynamic parameters in nonequilibrium systems", Physical Review E 75 (2007): 031120 Eric Bertin, Olivier Dauchot, Michel Droz, "Definition and relevance of nonequilibrium intensive thermodynamic parameters", cond-mat/0512116 = Physical Review Letters 96 (2006): 120601 L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim "Current Fluctations in Stochastic Lattice Gases", Physical Review Letters 94 (2005): 030601 = cond-mat/0407161 "Fluctuations in Stationary non Equilibrium States," cond-mat/0104153 "Macroscopic fluctuation theory for stationary non equilibrium states," cond-mat/0108040 Richard A. Blythe, "An introduction to phase transitions in stochastic

dynamical systems", cond-mat/0511627 G. Boffetta, G. Lacorata, S. Musacchio and A. Vulpiani, "Relaxation of finite perturbations: Beyond the fluctuation-response relation", Chaos 13 (2003): 806--811 Richard Brak, Jan de Gier and Vladimir Rittenberg, "Nonequilibrium stationary states and equilibrium models with long range interactions", cond-mat/0311149 Stephen G. Brush, The Kind of Motion We Call Heat: Statistical Physics and Irreversible Processes A. A. Budini and M.O. Caceres, "Functional characterization of generalized Langevin equations", cond-mat/0402311 [Abstract: "We present an exact functional formalism to deal with linear Langevin equations with arbitrary memory kernels and driven by any noise structure characterized through its characteristic functional. No others hypothesis are assumed over the noise, neither the fluctuation dissipation theorem. We found that the characteristic functional of the linear process can be expressed in terms of noise's functional and the Green function of the deterministic (memory-like) dissipative dynamics. This object allow us to get a procedure to calculate all the Kolmogorov hierarchy of the non-Markov process. As examples we have characterized through the 1-time probability a noise-induced interplay between the dissipative dynamics and the structure of different noises. Conditions that lead to non-Gaussian statistics and distributions with long tails are analyzed. The introduction of arbitrary fluctuations in fractional Langevin equations have also been pointed out."] Giovanni Bussi, Alessandro Laio and Michele Parrinello, "Equilibrium Free Energies from Nonequilibrium Metadynamics", Physical Review Letters 96 (2006): 090601 C. Bustamante, J. Liphardt, and F. Ritort, "The Nonequilibrium Thermodynamics of Small Systems", Physics Today 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", Physical Review E 75 (2007): 031109 M. E. Cates and M. R. Evans (eds.), Soft and Fragile Matter: Nonequilibrium Dynamics, Metastability and Flow [Scottish Universities Summer School in Physics, vol. 53] Vladimir Y. Chernyak, Mcihael Chertkov and Christopher Jarzynski, "Path-integral analysis of fluctuation theorems for general Langevin processes", cond-mat/0605471 Philippe Chomaz, Francesca Gulminelli and Olivier Juillet, "Generalized Gibbs ensembles for time dependent processes", cond-mat/0412475 Leonardo Crochik and Tania Tome, "Entropy production in the majority-vote model", Physical Review E 72 (2005): 057103 Gavin E. Crooks, "Measuring Thermodynamic Length", Physical Review Letters 99 (2007): 100602 ["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."] Karen E. Daniels, Christian Beck and Eberhard Bodenschatz, "Defect turbulence and generalized statistical mechanics," cond-mat/0302623 P. De Gregorio, F. Sciortino, P. Tartaglia, E. Zaccarelli, K. A. Dawson, "Slowed Relaxational Dynamics Beyond the Fluctuation-Dissipation Theorem," cond-mat/0111018 S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics 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 Roderick C. Dewar, "Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states," cond-mat/0005382 Deepak Dhar, "Pico-canonical ensembles: A theoretical description of metastable states," cond-mat/0205011 Ronald Dickman and Ronaldo Vidigal, "Path Integrals and Perturbation Theory for Stochastic Processes", cond-mat/0205321 Gregor Diezemann, "Fluctuation-dissipation relations for Markov processes", Physical Review E 72 (2005): 0111104 J. R. Dorfman, Pierre Gaspard and T. Gilbert, "Entropy production of diffusion in spatially periodic deterministic systems," nlin.CD/0203046 Jean-Pierre Eckmann, "Non-equilibrium steady states", math-ph/0304043 Andreas Eibeck and Wolfgang Wagner, "Stochastic Interacting Particle Systems and Nonlinear Kinetic Equations", Annals of Applied Probability 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 Debra J. Searles, "On Irreversibility, Dissipation and Response Theory", cond-mat/0612105 Denis J. Evans, Debra J. Searles, Lamberto Rondoni, "On the Application of the Gallavotti-Cohen Fluctuation Theorem to Thermostatted Steady States Near Equilibrium", cond-mat/0312353 M. R. Evans and R. A. Blythe, "Nonequilibrium Dynamics in Low Dimensional Systems," cond-mat/0110630 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 Gregory Eyink, "Fluctuation-response relations for multitime correlations," Physical Review E 62 (2000): 210--220 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 ["We introduce a high dimensional symplectic map, modeling a large system consisting of weakly interacting chaotic subsystems, as a toy model to analyze the interplay between single-particle chaotic dynamics and particles interactions in thermodynamic systems. We study the growth with time of the Boltzmann entropy, S_B, in this system as a function of the coarse graining resolution. We show that a characteristic scale emerges, and that the behavior of S_B vs t, at variance with the Gibbs entropy, does not depend on the coarse graining resolution, as far as it is finer than this scale. The interaction among particles is crucial to achieve this result, while the rate of entropy growth depends essentially on the single-particle chaotic dynamics (for t not too small). It is possible to interpret the basic features of the dynamics in terms of a suitable Markov approximation."] 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"] Suzanne Fielding and Peter Sollich, "Observable-dependence of fluctuation-dissipation relations and effective temperatures," cond-mat/0107627 Roger Filliger and Max-Olivier Hongler, "Relative entropy and efficiency measure for diffusion-mediated transport processes", Journal of Physics A: Mathematical and General 38 (2005): 1247--1255 ["We propose an efficiency measure for diffusion-mediated transport processes including molecular-scale engines such as Brownian motors.... Ultimately, the efficiency measure can be directly interpreted as the relative entropy between two probability distributions, namely: the distribution of the particles in the presence of the external rectifying force field and a reference distribution describing the behavior in the absence of the rectifier". Interesting for the link between relative entropy and energetics.] J. F. Fontanari and P. F. Stadler, "Fractal Geometry of Spin-Glass Models," SFI Working Paper 01-06-034 Silvio Franz, "How glasses explore configuration space," cond-mat/0212091 Henryk Fuks and Nino Boccara, "Convergence to equilibrium in a class of interacting particle systems evolving in discrete time," nlin.CG/0101037 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

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 Letters 96 (2006): 114101 Pierre Gaspard, "Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes", Journal of Statistical Physics 117 (2004): 599--615 T. Gilbert, J. R. Dorfman and P. Gaspard, "Entropy Production, Fractals, and Relaxation to Equilibrium," Physical Review Letters 85 (2000): 1606--1609 A. Giuliani, F. Zamponi and G. Gallavotti, "Fluctuation Relation beyond Linear Response Theory", cond-mat/0412455 C. Godreche and J. M. Luck, "Nonequilibrium dynamics of urn models," cond-mat/0109213 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 Alexander N. Gorban A. N. Gorban and I. V. Karlin, "Quasi-Equilibrium Closure Hierarchies for The Boltzmann Equation", cond-mat/0305599 William T. Grandy "Time Evolution In Macroscopic Systems. I: Equations of Motion," cond-mat/0303290 "Time Evolution In Macroscopic Systems. II: The Entropy," cond-mat/0303291 A. Greven, G. Keller and G. Warnecke (eds.), Entropy T. Hanney and R. B. Stinchcombe, "Real-space renormalisation group approach to driven diffusive systems", cond-mat/0606515 Takahiro Harada and Shin-ichi Sasa "Energy dissipation and violation of the fluctuation-response relation in non-equilibrium Langevin systems", cond-mat/0510723 "Fluctuations, Responses and Energetics of Molecular Motors", cond-mat/0610757 R. J. Harris, A. Rákos, G. M. Schuetz, "Breakdown of Gallavotti-Cohen symmetry for stochastic dynamics", cond-mat/0512159 Kumiko Hayashi and Shin-ichi Sasa, "Linear response theory in stochastic many-body systems revisited", cond-mat/0507719 ["The Green-Kubo relation, the Einstein relation, and the fluctuation-response relation are representative universal relations among measurable quantities that are valid in the linear response regime. We provide pedagogical proofs of these universal relations for stochastic many-body systems. Through these simple proofs, we characterize the three relations as follows. The Green-Kubo relation is a direct result of the local detailed balance condition, the fluctuation-response relation represents the dynamic extension of both the Green-Kubo relation and the fluctuation relation in equilibrium statistical mechanics, and the Einstein

relation can be understood by considering thermodynamics. We also clarify the interrelationships among the universal relations."] 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 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, "Discrete breathers in transient processes and thermal equilibrium", Physica D 198 (2004): 120--135 Christopher Jarzynski, "Comparison of far-from-equilibrium work relations", cond-mat/0612305 Owen Jepps, Denis J. Evans and Debra J. Searles, "The fluctuation theorem and Lyapunov weights," cond-mat/0311090 Henry E. Kandrup, Ioannis V. Sideris, and C. L. Bohn, " Chaos, ergodicity, and the thermodynamics of lower-dimensional Hamiltonian systems," astro-ph/0108038

Dragi Karevski, "Foundations of Statistical Mechanics: in and out of Equilibrium", cond-mat/0509595 ["The first part of the paper is devoted to the foundations, that is the mathematical and physical justification, of equilibrium statistical mechanics. It is a pedagogical attempt, mostly based on Khinchin's presentation, which purpose is to clarify some aspects of the development of statistical mechanics. In the second part, we discuss some recent developments that appeared out of equilibrium, such as fluctuation theorem and Jarzynski equality."] R. Kawai, J. M. R. Parrondo, C. Van den Broeck, "Dissipation: The phase-space perspective", cond-mat/0701397 Michal Kurzynski, The Thermodynamic Machinery of Life [Blurb] Hernan Larralde, Francois Leyvraz, and David P. Sanders, "Metastability in Markov processes", cond-mat/0608439 = Journal of Statistical Mechanics (2006): P08013 Raphael Lefevere, "On the local space-time structure of non-equilibrium steady states", math-ph/0609049 Dino Leporini and Roberto Mauri, "Fluctuations of non-conservative systems", Journal of Statistical Mechanics: Theory and Experiment 2007: P03002 Francois Leyvraz, Hernan Larralde, and David P. Sanders, "A Definition of Metastability for Markov Processes with Detailed Balance", cond-mat/0509754 Katja Lindenberg and Bruce West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems S. Lubeck, "Universal scaling behavior of non-equilibrium phase transitions", cond-mat/0501259 [160 pp. review]

James F. Lutsko, "Chapman-Enskog expansion about nonequilibrium states: the sheared granular fluid", cond-mat/0510749 Michael C. Mackey and Marta Tyran-Kaminska, "Temporal Behavior of the Conditional and Gibbs' Entropies", cond-mat/0509649 [Weirdly, what Mackey calls "conditional entropy" is what everyone else calls "relative entropy" or "Kullback-Leibler divergence", and not at all what everyone else calls "conditional entropy".] 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 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 Christian Maes and Maarten H. van Wieren, "Time-Symmetric Fluctuations in Nonequilibrium Systems", Physical Review Letters 96 (2006): 240601 = cond-mat/0601299 Ferenc Markús and Katalin Gambár, "Generalized Hamilton-Jacobi equation for simple dissipative processes", Physical Review E 70 (2004): 016123 [link] Joaquin Marro and Ronald Dickman, Nonequilibrium Phase Transitions in Lattice Models [Blurb] Alan McKane and Martin Tarlie, "Unstable decay and state selection II," cond-mat/0107327 Paul Meakin, Fractals, Scaling and Growth Far from Equilibrium S. S. Melnyk, O. V. Usatenko, and V. A. Yampol'skii, "Memory Functions of the Additive Markov chains: Applications to Complex Dynamic Systems", physics/0412169 Mauro Merolle, Juan P. Garrahan and David Chandler, "Space-time thermodynamics of the glass transition", PNAS 102 (2005): 10837--10840 ["We consider the probability distribution for fluctuations in dynamical action and similar quantities related to dynamic heterogeneity. We argue that the so-called 'glass transition' is a manifestation of low action tails in these distributions where the entropy of trajectory space is subextensive in time. These low action tails are a consequence of dynamic heterogeneity and an indication of phase coexistence in trajectory space. The glass transition,

where the system falls out of equilibrium, is then an order-disorder phenomenon ... occurring at a temperature ... which is a weak function of measurement time." This sounds interesting, and Chandler is certainly very good --- I grew up on his stat. mech. book --- but if this is right why is it self-contributed by Chandler to PNAS?] Emil Mittag and Denis J. Evans, "Time-dependent fluctuation theorem," Physical Review E 67 (2003): 026113 Sutapa Mukherji and Somendra M. Bhattacharjee, "Nonequilibrium Growth Problems," cond-mat/9908330 Geza Odor, "Phase transition universality classes of classical, nonequilibrium systems," cond-mat/0205644 = Reviews of Modern Physics 76 (2004): 663--724 [145pp. review] Hans Christian Ottinger, "Weakly and Strongly Consistent Formulations of Irreversible Processes", Physical Review Letters 99 (2007): 130602 Giorgio Parisi, "Local overlaps, heterogeneities and the local fluctuation dissipation relations," cond-mat/0211608 Agusti Perez-Madrid, "Molecular Theory of Irreversibility", cond-mat/0509491 Hong Qian "A Gallavotti-Cohen-Type Symmetry in the Steady-state Kinetics of Single Enzyme Turnover Reactions", cond-mat/0507659 "Relative Entropy: Free Energy Associated with Equilibrium Fluctuations and Nonequilibrium Deviations", math-ph/0007010 = Physical Review E 63 (2001): 042103 Hong Qian and Timothy C. Reluga, "Nonequilibrium Thermodynamics and Nonlinear Kinetics in a Cellular Signaling Switch", Physical Review Letters 94 (2005): 028101 Saar Rahav and Christopher Jarzynski, "Fluctuation relations and coarse-graining", arxiv:0708.2437 = Journal of Statistical Mechanics (2007): P09012 Jorgen Rammer, Quantum Field Theory of Non-equilibrium States [blurb] J. C. Reid, D. M. Carberry, G. M. Wang, E. M. Sevick, Denis J. Evans and Debra J. Searles, "Reversibility in nonequilibrium trajectories of an optically trapped particle", Physical Review E 70 (2004): 016111 [link] Pedro M. Reis, Rohit A. Ingale, Mark D. Shattuck, "Universal velocity distributions in an experimental granular fluid", cond-mat/0611024 [Measurable departures from the Maxwell-Boltzmann distribution, in accordance with theory...] F. Ritort, "Single molecule experiments in biophysics: exploring the thermal behavior of nonequilibrium small systems", cond-mat/0509606 [Review] Felix Ritort and Peter Sollich, "Glassy dynamics of kinetically constrained models," cond-mat/0210382 L. Rondoni and E. G. D. Cohen, "Gibbs Entropy and Irreversible Thermodynamics," cond-mat/9908367 Lamberto Rondoni, Carlos Mejia-Monasterio, "Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms", arxiv:0709.1976 [review]

David Ruelle "Extending the definition of entropy to nonequilibrium steady states," cond-mat/0303156 "Natural Nonequilibrium States in Quantum Statistical Mechanics," math-ph/9906005 Himadri S. Samanta and J. K. Bhattacharjee, "Non equilibrium statistical physics with fictitious time", cond-mat/0509563 Henrik Sandberg, Jean-Charles Delvenne, John C. Doyle, "The Statistical Mechanics of Fluctuation-Dissipation and Measurement Back Action", math.DS/0611628 ["We show that a linear macroscopic system is dissipative if and only if it can be approximated by a linear lossless microscopic system, over arbitrarily long time intervals. As a by-product, we obtain mechanisms explaining Johnson-Nyquist noise as initial uncertainty in the lossless state as well as measurement back action and a trade off between process and measurement noise."] Shin-ichi Sasa and Teruhisa S. Komats "Steady state thermodynamics", cond-mat/0411052 [82pp. tome] "Thermodynamic Entropy and Excess Information Loss in Dynamical Systems with Time-Dependent Hamiltonian," chao-dyn/9807010 "Thermodynamic Irreversibility from High-Dimensionl Hamiltonian Chaos," cond-mat/9911181 B. Schmittmann and R. K. P. Zia "Driven Diffusive Systems: An Introduction and Recent Developments," cond-mat/9803392 Statistical Mechanics of Driven Diffusive Systems Frank Schweitzer, Werner Ebeling and Benno Tilch, "Statistical Mechanics of Canonical-Dissipative Systems and Applications to Swarm Dynamics," cond-mat/0103360 Debra J. Searles and Denis J. Evans "Fluctuation Theorem for Stochastic Systems," Physical Review E 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

Udo Seifert, "Entropy production along a stochastic trajectory and an integral fluctuation theorem", cond-mat/0503686 = Physical Review Letters 95 (2005): 040602 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] "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) P. Shiktorov, E. Starikov, V. Gruzinskis, L. Reggiani, L. Varani and J.C. Vaissiere, "Common Origin of Quantum Regression and Quantum Fluctuation Dissipation Theorems," cond-mat/0011420 Yair Shokef, Guy Bunin, and Dov Levine, "Fluctuation-dissipation relations in driven dissipative systems", cond-mat/0511409 = Physical Review E 73 (2006): 046132 T. Speck and U. Seifert, "The Jarzynski relation, fluctuation theorems, and stochastic thermodynamics for non-Markovian processes", Journal of Statistical Mechanics (2007) L09002 P. Sollich, S. Fielding and P. Mayer, "Fluctuation-dissipation relations and effective temperatures in simple non-mean field systems," cond-mat/0111241 Dieter Straub, Alternative Mathematical Theory of Non-Equilibrium Phenomena Sauro Succi, Iliya V. Karlin and Hudong Chen, "Role of the H theorem in lattice Boltzmann hydrodynamic simulations," cond-mat/0205639 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 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 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 M. H. Vainstein, I. V. L. Costa and F. A. Oliveira, "Mixing, Ergodicity and the Fluctuation-Dissipation Theorem in complex systems", cond-mat/0501448 Raul O. Vallejos and Celia Anteneodo, "Theoretical estimates for the largest Lyapunov exponent of many-particle systems," cond-mat/0204412 N. G. van Kampen, Stochastic Processes in Physics and Chemistry Ramses van Zon, H. van Beijeren and J. R. Dorfman, "Kinetic Theory of Dynamical Systems," chao-dyn/9906040

Ramses van Zon and Jeremy Schofield, "Mode coupling theory for multi-point and multi-time correlation functions," cond-mat/0108029 Aurea R. Vasconcellos, J. Galvao Ramos and Roberto Luzzi, "Nonlinear Higher-Order Thermo-Hydrodynamics: Generalized Approach in a Nonequilibrium Ensemble Formalism", cond-mat/0412227 B. von Haeften, G. Izus, and H. S. Wio, "System Size Stochastic Resonance: General Nonequilibrium Potential Framework", cond-mat/0504131 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 Qiuping A. Wang, "Action principle and Jaynes' guess method", cond-mat/0407515

Bruce J. West and Bill Deering, The Lure of Modern Science Stephen R. Williams, Debra J. Searles, Denis J. Evans, "Numerical study of the Steady State Fluctuation Relations Far from Equilibrium", cond-mat/0601328 Horacio S. Wio, An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics Daniel K. Wojcik and J. R. Dorfman, "Quantum Multibaker Maps: Extreme Quantum Regime," cond-mat/0203494 Hyung-June Woo [Thanks to Dr. Woo for reprints], "Variational formulation of nonequilibrium thermodynamics for hydrodynamic pattern formations," Physical Review E 66 (2002) 066104 Wen-an Yong and Li-Shi Luo, "Nonexistence of H Theorem for Some Lattice Boltzmann Models", Journal of Statistical Physics 121 (2005): 91--103 V. I. Yukalov "Expansion Exponents for Nonequilibrium Systems," cond-mat/0303384 "Irreversibility of Time for Quasi-Isolated Systems," Physics Letters A 308 (2003): 313--318 = cond-mat/0303497 [Very important if correct.] "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"