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It is well known that ergodic theory can be used to formally prove a weak form of relaxation to equilibrium for finite, mixing, Hamiltonian systems. In this Letter we extend this proof to any dynamics that preserves a mixing equilibrium distribution. The proof uses an approach similar to that used in umbrella sampling, and demonstrates the need for a form of ergodic consistency of the initial and final distribution. This weak relaxation only applies to averages of physical properties. It says nothing about whether the distribution of states relaxes towards the equilibrium distribution or how long the relaxation of physical averages takes., Comment: 18 pages

The issue of relaxation has been addressed in terms of ergodic theory in the past. However, the application of that theory to models of physical interest is problematic, especially when dealing with relaxation to nonequilibrium steady states. Here, we consider the relaxation of classical, thermostatted particle systems to equilibrium as well as to nonequilibrium steady states, using dynamical notions including decay of correlations. We show that the condition known as {\Omega}T-mixing is necessary and sufficient to prove relaxation of ensemble averages to steady state values. We then observe that the condition known as weak T-mixing applied to smooth observables is sufficient for relaxation to be independent of the initial ensemble. Lastly, weak T-mixing for integrable functions makes relaxation independent of the ensemble member, apart from a negligible set of members enabling the result to be applied to observations from a single physical experiment. The results also allow us to give a microscopic derivation of Prigogine's principle of minimum entropy production in the linear response regime. The key to deriving these results lies in shifting the discussion from characteristics of dynamical systems, such as those related to metric transitivity, to physical measurements and to the behaviour of observables. This naturally leads to the notion of physical ergodicity., Comment: 44 pages, 1 figure

We examine the question of whether fluids and crystals are differentiated on the basis of their zero frequency shear moduli or their limiting zero frequency shear viscosity. We show that while fluids, in contrast with crystals, do have a zero value for their shear modulus, in contradiction to a widespread presumption, a crystal does not have an infinite or exceedingly large value for its limiting zero frequency shear viscosity. In fact, while the limiting shear viscosity of a crystal is much larger than that of the liquid from which it is formed, its viscosity is much less than that of the corresponding glass that may form assuming the liquid is a good enough glass former., Comment: 13 pages, 3 figures

Using the Dissipation Theorem and a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we give a first-principles derivation of Boltzmann's postulate of equal a priori probability in phase space for the microcanonical ensemble. We show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble., Comment: 12 pages

Using the recently derived Dissipation Theorem and a corollary of the Transient Fluctuation Theorem (TFT), namely the Second Law Inequality, we derive the unique time independent, equilibrium phase space distribution function for an ergodic Hamiltonian system in contact with a remote heat bath. We prove under very general conditions that any deviation from this equilibrium distribution breaks the time independence of the distribution. Provided temporal correlations decay, and the system is ergodic, we show that any nonequilibrium distribution that is an even function of the momenta, eventually relaxes (not necessarily monotonically) to the equilibrium distribution. Finally we prove that the negative logarithm of the microscopic partition function is equal to the thermodynamic Helmholtz free energy divided by the thermodynamic temperature and Boltzmann's constant. Our results complement and extend the findings of modern ergodic theory and show the importance of dissipation in the process of relaxation towards equilibrium., Comment: 18 pages, no figures

A simple model featuring a double well potential is used to represent a liquid that is quenched from an ergodic state into a history dependent glassy state. Issues surrounding the application of the Jarzynski Equality to glass formation are investigated. We demonstrate that the Jarzynski Equality gives the free energy difference between the initial state and the state we would obtain if the glass relaxed to true thermodynamic equilibrium. We derive new variations of the Jarzynski Equality which are relevant to the history dependent glassy state rather than the underlying equilibrium state. It is shown how to compute the free energy differences for the nonequilibrium history dependent glassy state such that it remains consistent with the standard expression for the entropy and with the second law inequality., Comment: 16 pages, 5 figures

We give a proof of transient fluctuation relations for the entropy production (dissipation function) in nonequilibrium systems, which is valid for most time reversible dynamics. We then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax, producing a unique nonequilibrium steady state. Although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown, if such a steady state exists, the generation of the steady state fluctuation relation from the transient relation is shown to be very general. It is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation, which is needed also for, e.g., the existence of transport coefficients. Because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities. The steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor, whose convergence is problematic, for systems close to equilibrium. We examine some model dynamics that have been considered previously, and show how they are described in the context of this work., Comment: 30 pages, 1 figure

We examine the question of whether the formal expressions of equilibrium statistical mechanics can be applied to time independent non-dissipative systems that are not in true thermodynamic equilibrium and are nonergodic. By assuming the phase space may be divided into time independent, locally ergodic domains, we argue that within such domains the relative probabilities of microstates are given by the standard Boltzmann weights. In contrast to previous energy landscape treatments, that have been developed specifically for the glass transition, we do not impose an a priori knowledge of the inter-domain population distribution. Assuming that these domains are robust with respect to small changes in thermodynamic state variables we derive a variety of fluctuation formulae for these systems. We verify our theoretical results using molecular dynamics simulations on a model glass forming system. Non-equilibrium Transient Fluctuation Relations are derived for the fluctuations resulting from a sudden finite change to the system's temperature or pressure and these are shown to be consistent with the simulation results. The necessary and sufficient conditions for these relations to be valid are that the domains are internally populated by Boltzmann statistics and that the domains are robust. The Transient Fluctuation Relations thus provide an independent quantitative justification for the assumptions used in our statistical mechanical treatment of these systems., Comment: 17 pages, 4 figures, minor amendments

Recently there has been considerable interest in the Fluctuation Theorem (FT). The FT shows how time reversible microscopic dynamics leads to irreversible macroscopic behavior as the system size or observation time increases. We show that the argument of the Evans-Searles FT, the dissipation function, plays a central role in nonlinear response theory and derive the Dissipation Theorem, giving exact relations for nonlinear response of classical N-body systems. These expressions should be verifiable experimentally. When linearized they reduce to the Green-Kubo expressions for linear response., Comment: 14 pages

Linear irreversible thermodynamics asserts that the instantaneous local spontaneous entropy production is always nonnegative. However for a viscoelastic fluid this is not always the case. Given the fundamental status of the Second Law, this presents a problem. We provide a new derivation of the Second Law, from first principles, which is valid for the appropriately time averaged entropy production allowing the instantaneous entropy production to be negative for short intervals of time. We show that time averages (rather than instantaneous values) of the entropy production are nonnegative. We illustrate this using molecular dynamics simulations of oscillatory shear., Comment: 5 pages 2 figures

The nonequilibrium free energy theorems show how distributions of work along nonequilibrium paths are related to free energy differences between the equilibrium states at the end points of these paths. In this paper we develop a natural way of barostatting a system and give the first deterministic derivation of the Crooks and Jarzynski relations for these isothermal isobaric systems. We illustrate these relations by applying them to molecular dynamics simulations of a model polymer undergoing stretching., Comment: 9 pages, 2 figures

A thermostatted dynamical model with five degrees of freedom is used to test both the Evans-Searles and the Gallavotti-Cohen fluctuation relations (ESFR and GCFR respectively). In the absence of an external driving field, the model generates a time independent ergodic equilibrium state with two conjugate pairs of Lyapunov exponents. Each conjugate pair sums to zero. The fluctuation relations are tested numerically both near and far from equilibrium. As expected from previous work, near equilibrium the ESFR is verified by the simulation data while the GCFR is not confirmed by the data. Far from equilibrium where a positive exponent in one of these conjugate pairs becomes negative, we test a conjecture regarding the GCFR made by Gallavotti and co-workers. They conjectured that where the number of nontrivial Lyapunov exponents that are positive becomes less than the number of such negative exponents, then the form of the GCFR needs to be corrected. We show that there is no evidence for this conjecture in the empirical data. In fact as the field increases, the uncorrected form of the GCFR appears to become more accurate. The real reason for this observation is likely to be that as the field increases, the argument of the GCFR more and more accurately approximates the argument of the ESFR. Since the ESFR works for arbitrary field strengths, the uncorrected GCFR appears to become ever more accurate as the field increases. The final point of evidence against the conjecture is that when the smallest positive exponent changes sign, the conjecture predicts a discontinuous change in the "correction factor" for GCFR. We see no evidence for a discontinuity at this field strength; only a gradual improvement of degree of agreement as the field increases., Comment: 19 pages, 5 figures

Deterministic thermostats are frequently employed in non-equilibrium molecular dynamics simulations in order to remove the heat produced irreversibly over the course of such simulations. The simplest thermostat is the Gaussian thermostat, which satisfies Gauss's principle of least constraint and fixes the peculiar kinetic energy. There are of course infinitely many ways to thermostat systems, e.g. by fixing $\sum\limits_i{|{p_i}|^{\mu + 1}}$. In the present paper we provide, for the first time, convincing arguments as to why the conventional Gaussian isokinetic thermostat ($\mu=1$) is unique in this class. We show that this thermostat minimizes the phase space compression and is the only thermostat for which the conjugate pairing rule (CPR) holds. Moreover it is shown that for finite sized systems in the absence of an applied dissipative field, all other thermostats ($\mu=1$) perform work on the system in the same manner as a dissipative field while simultaneously removing the dissipative heat so generated. All other thermostats ($\mu=1$) are thus auto-dissipative. Among all $\mu$-thermostats, only the $\mu=1$ Gaussian thermostat permits an equilibrium state., Comment: 27 pages including 10 figures; submitted for publication Journal of Chemical Physics

Recently van Zon and Cohen [1-3] proposed an extension of the Fluctuation Theorems (FTs) of Evans and Searles [4]. For dissipative nonequilibrium systems, Cohen and van Zon studied the fluctuations of the heat absorbed, over a period of time t, by a surrounding thermostat. They showed theoretically that for thermostatted systems their extension does not exhibit the standard form expected for FTs and In the present paper we show that for thermostatted nonequilibrium steady states modeled by Langevin dynamics, the heat function is in fact identical to the time integral of the phase space compression factor which appears in the Gallavotti-Cohen FT (GCFT). Thus the work of van Zon and Cohen confirms at least for Langevin systems, that the GCFT does not apply to thermostatted steady states., Comment: 10 pages. submitted to Phys Rev Letts

The fluctuation relation of the Gallavotti-Cohen Fluctuation Theorem (GCFT) concerns fluctuations in the phase space compression rate of dissipative, reversible dynamical systems. It has been proven for Anosov systems, but it is expected to apply more generally. This raises the question of which non-Anosov systems satisfy the fluctuation relation. We analyze time dependent fluctuations in the phase space compression rate of a class of N-particle systems that are at equilibrium or in near equilibrium steady states. This class does not include Anosov systems or isoenergetic systems, however, it includes most steady state systems considered in molecular dynamics simulations of realistic systems. We argue that the fluctuations of the phase space compression rate of these systems at or near equilibrium do not satisfy the fluctuation relation of the GCFT, although the discrepancies become somewhat smaller as the systems move further from equilibrium. In contrast, similar fluctuation relations for an appropriately defined dissipation function appear to hold both near and far from equilibrium., Comment: 46 pages, for publication in PRE

The Fluctuation Theorem (FT) is a generalisation of the Second Law of Thermodynamics that applies to small systems observed for short times. For thermostatted systems it gives the probability ratio that entropy will be consumed rather than produced. In this paper we derive the Transient and Steady State Fluctuation Theorems using Lyapunov weights rather than the usual Gibbs weights. At long times the Fluctuation Theorems so derived are identical to those derived using the more standard Gibbs weights., Comment: 26 pages; to appear in Physica D

For thermostatted dissipative systems the Fluctuation Theorem gives an analytical expression for the ratio of probabilities that the time averaged entropy production in a finite system observed for a finite time, takes on a specified value compared to the negative of that value. Hitherto it had been thought that the presence of some thermostatting mechanism was an essential component of any system which satisfies a Fluctuation Theorem. In the present paper we show that a Fluctuation Theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields., Comment: 10 pages including 4 figures, submitted for publication

A number of relations between the Kaplan-Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the transport coefficients in terms of the Kaplan-Yorke dimension is derived. Alternatively, the Kaplan-Yorke dimension for a dissipative macroscopic system can be expressed in terms of the transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good., Comment: 12 pages, 5 figures, submitted to J. Stat. Phys

We develop a General Fluctuation Formula for phase variables that are odd under time reversal. Simulations are used to verify the new formula., Comment: 10 pages, 5 figures, submitted to Procedings of the 3rd Tohwa University International Conference of Statistical Physics, Nov 8-12, 1999 (AIP Conferences Series)

We study the use of the Evans Nonequilibrium Molecular Dynamics (NEMD) heat flow algorithm for the computation of the heat conductivity in one-dimensional lattices. For the well-known Fermi-Pasta-Ulam (FPU) model, it is shown that when the heat field strength is greater than a certain critical value (which depends on the system size) solitons can be generated in molecular dynamics simulations starting from random initial conditions. Such solitons are stable and travel with supersonic speeds. For smaller heat fields, no solitons are generated in the molecular dynamics simulations; the heat conductivity obtained via the NEMD algorithm increases monotonically with the size of the system., Comment: 13 pages + 9 figures

We show that arbitrary phase space vector fields can be used to generate phase functions whose ensemble averages give the thermodynamic temperature. We describe conditions for the validity of these functions in periodic boundary systems and the Molecular Dynamics (MD) ensemble, and test them with a short-ranged potential MD simulation., Comment: 21 pages, 2 figures, Revtex. Submitted to Phys. Rev. E

The Fluctuation Theorem gives an analytical expression for the probability of observing second law violating dynamical fluctuations, in nonequilibrium systems. At equilibrium statistical mechanical fluctuations are known to be ensemble dependent. In this paper we generalise the Transient and Steady State Fluctuation Theorems to various nonequilibrium dynamical ensembles. The Transient and Steady State Fluctuation Theorem for an isokinetic ensemble of isokinetic trajectories is tested using nonequilibrium molecular dynamics simulations of shear flow., Comment: 16 pages, 1 table, 4 figures; presentation of generalised formulae and discussion clarified

Non-equilibrium molecular dynamics simulations are used to demonstrate the asymptotic convergence of the Transient and Steady State forms of the Fluctuation Theorem. In the case of planar Poiseuille flow, we find that the Transient form, valid for all times, converges to the Steady State form on microscopic time scales. Further, we find that the time of convergence for the two Theorems coincides with the time required for satisfaction of the asymptotic Steady State Fluctuation Theorem. PACS numbers: 05.20.-y, 05.70.Ln, 47.10.+g, 47.40.-n, Comment: 8 pages, submitted to Phys Rev Letts

Green-Kubo and Einstein expressions for the transport coefficients of a fluid in a nonequilibrium steady state can be derived using the Fluctuation Theorem and by assuming the probability distribution of the time-averaged dissipative flux is Gaussian. These expressions are consistent with those obtained using linear response theory and are valid in the linear regime. It is shown that these expressions are however, not valid in the nonlinear regime where the fluid is driven far from equilibrium. We advance an argument for why these expression are only valid in the linear response, zero field limit., Comment: 32 pages, inc. 6 figures Discussion and notation improved

The Fluctuation Theorem describes the probability ratio of observing trajectories that satisfy or violate the second law of thermodynamics. It has been proved in a number of different ways for thermostatted deterministic nonequilibrium systems. In the present paper we show that the Fluctuation Theorem is also valid for a class of stochastic nonequilibrium systems. The Theorem is therefore not reliant on the reversibility or the determinism of the underlying dynamics. Numerical tests verify the theoretical result., Comment: Minor changes; typos corrected; accepted by Physical Review E

The Fluctuation Theorem (FT) gives an analytic expression for the probability, in a nonequilibrium system of finite size observed for a finite time, that the dissipative flux will flow in the reverse direction to that required by the Second Law of Thermodynamics. In the present letter a Local version of the Fluctuation Theorem (LFT), is derived. We find that in the case of planar Poiseuille flow of a Newtonian fluid between thermostatted walls, non-equilibrium molecular dynamics simulation results support LFT., Comment: Submitted to Physical Review Letters