Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems

A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function we establish two nonequilibrium detailed balance relations, which not only lead to a fluctuation theorem for work but also to one related to energy loss by friction. In addition, we carried out the functional integral for heat explicitly, leading to the extended fluctuation theorem for heat. We also present a simple argument for this extended fluctuation theorem in the long time limit. PACS numbers: 05.70.Ln, 05.40.-a, 05.10.Gg.     

On the Fluctuation Relation for Nosé-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nosé-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in (D.J. Searles, et al., J. Stat. Phys. 128:1337, 2007), for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate Λ and of the dissipation function Ω, a similar relaxation regime at shorter averaging times is found. The quantity Ω satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity Λ appears to begin a monotonic convergence after such times. This is consistent with the fact that Ω and Λ differ by a total time derivative, and that the tails of the probability distribution function of Λ are Gaussian.     

Note on Two Theorems in Nonequilibrium Statistical Mechanics

An attempt is made to clarify the difference between a theorem derived by Evans and Searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state.     

Boundary Dissipation in a Driven Hard Disk System

We perform a simulation with the aim of checking the existence of a well defined stationary state for a two dimensional system of driven hard disks when energy dissipation takes place at the system boundaries and no bulk impurities are present. PACS: 02.70.Ns, 05.60.-k, 47.27.ek     

Normal and Abnormal Grain Growth as Transient Phenomena

For the case of uniform grain boundaries the basic equations of the statistical theory of grain growth (GG) of the present authors are shortly reviewed and compared to the classical Hillert model. On the basis of this present theory normal and abnormal 3-D GG is simulated and particularly the effect of the initial grain size distribution function (SDF) on the GG behaviour which may lead to normal or abnormal GG is demonstrated. Finally results of simulations of normal 2-D GG by the statistical method and by a direct model (curvature driven grain boundaries (GBs)) are presented which exhibit good agreement with one another. It is shown by this comparison that the possibility of finding by such direct methods a self similar SDF as long time asymptote can be excluded because, in contrast to the simulations based on the statistical theory, for the direct models the very large computational capacities required for the long simulation times are not available yet. The conclusion repeatedly claimed in literature that the true self similar SDF deviates from the Hillert distribution can thus be shown not to be justified.     

Time Evolution in Macroscopic Systems. I. Equations of Motion

Time evolution of macroscopic systems is re-examined primarily through further analysis and extension of the equation of motion for the density matrix (t). Because contains both classical and quantum-mechanical probabilities it is necessary to account for changes in both in the presence of external influences, yet standard treatments tend to neglect the former. A model of time-dependent classical probabilities is presented to illustrate the required type of extension to the conventional time-evolution equation, and it is shown that such an extension is already contained in the definition of the density matrix.     

Molecular derivation of the hydrodynamic equations for a binary fluid

In this paper, the hydrodynamic equations and the associated transport coefficients are derived for a simple binary fluid from molecular considerations. This is a generalization of the methods of Felderhof and Oppenheim and of Selwyn to multicomponent systems. A linear response formalism is used to describe the relaxation of the binary system from an initial nonequilibrium state. Explicit molecular expressions are given for the transport coefficients in terms of time correlation functions of generalized current densities. These densities have the useful property of not containing a conserved part. The correlation functions are then related to a set of phenomenological coefficients in the theory of nonequilibrium thermodynamics. This explicit identification enables one to relate the correlation functions to experimentally measured transport coefficients.Supported by the National Science Foundation.     

On the weak-noise limit of Fokker-Planck models

The weak-noise limit of Fokker-Planck models leads to a set of nonlinear Hamiltonian canonical equations. We show that the existence of a nonequilibrium potential in the weak-noise limit requires the existence of whiskered tori in the Hamiltonian system and, therefore, the complete integrability of the latter. A specific model is considered, where the Hamiltonian system in the weak-noise limit is not integrable. Two different perturbative solutions are constructed: the first solution describes analytically the breakdown of the whiskered tori due to the appearance of wild séparatrices; the second solution allows the analytic construction of an approximate nonequilibrium potential and an asymptotic expression for the probability density in the steady state.On leave from Institute for Theoretical Physics, Eötvös University, Budapest, Hungary.     

On the transition from ferromagnetism to antiferromagnetism on twofold Cayley tree

Summary A fixed-point conversion theorem which shows the transition from ferromagnetism to antiferromagnetism on twofold Cayley tree is proved. The ferromagnetic and antiferromagnetic maps are shown to be related by an involution and in zero field the stable fixed points of the ferromagnetic map are converted to a stable two-cycle of the antiferromagnetic map. A reduced one-dimensional analysis in zero field yields precisely the same results.     

Fluctuation of the free energy in the Hopfield model

We prove that in the ergodic region [T>J ²(1 + r)] the deviation of the total free energy of the Hopfield neural network converges in distribution asN to a (shifted) Gaussian variable. Moreover, the free energy per site converges in probability to lim(1/N)ln _N .     

Chaotic Hypothesis, Fluctuation Theorem and Singularities

The chaotic hypothesis has several implications which have generated interest in the literature because of their generality and because a few exact predictions are among them. However its application to Physics problems requires attention and can lead to apparent inconsistencies. In particular there are several cases that have been considered in the literature in which singularities are built in the models: for instance when among the forces there are Lennard-Jones potentials (which are infinite in the origin) and the constraints imposed on the system do not forbid arbitrarily close approach to the singularity even though the average kinetic energy is bounded. The situation is well understood in certain special cases in which the system is subject to Gaussian noise; here the treatment of rather general singular systems is considered and the predictions of the chaotic hypothesis for such situations are derived. The main conclusion is that the chaotic hypothesis is perfectly adequate to describe the singular physical systems we consider, ıe deterministic systems with thermostat forces acting according to Gauss' principle for the constraint of constant total kinetic energy (“isokinetic Gaussian thermostats”), close and far from equilibrium. Near equilibrium it even predicts a fluctuation relation which, in deterministic cases with more general thermostat forces (ıe not necessarily of Gaussian isokinetic nature), extends recent relations obtained in situations in which the thermostatting forces satisfy Gauss' principle. This relation agrees, where expected, with the fluctuation theorem for perfectly chaotic systems. The results are compared with some recent works in the literature. PACS: 47.52.+j, 05.45.-a, 05.70.Ln, 05.20.-y     

Fast-ionic-conductor behavior of driven lattice-gas models

We discuss driven diffusive lattice-gas systems as a model for fast ionic conductors, derive associated hydrodynamic equations and expressions for transport coefficients, and compare mean-field theory, Monte Carlo results and experimental observations. The comparison between model and real behaviours helps to understand some properties of those materials which seem to be characterized by the occurrence of nonequilibrium steady states and phase transitions. In particular, our study suggests the existence in Nature of a novel (nonequilibrium) universality class.     

Non-Markovian Quantum Kinetics and Conservation Laws

A link between memory effects in quantum kinetic equations and nonequilibrium correlations associated with the energy conservation is investigated. In order that the energy be conserved by an approximate collision integral, the one-particle distribution function and the mean interaction energy are treated as independent nonequilibrium state parameters. The density operator method is used to derive a kinetic equation in second-order non-Markovian Born approximation and an evolution equation for the nonequilibrium quasi-temperature which is thermodynamically conjugated to the mean interaction energy. The kinetic equation contains a correlation contribution which exactly cancels the collision term in thermal equilibrium and ensures the energy conservation in nonequilibrium states. Explicit expressions for the entropy production in the non-Markovian regime and the time-dependent correlation energy are obtained.     

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results

We consider two single-species reaction-diffusion models on one-dimensional lattices of lengthL: the coagulation-decoagulation model and the annihilation model. For the coagulation model the system of differential equations describing the time evolution of the empty interval probabilities is derived for periodic as well as for open boundary conditions. This system of differential equations grows quadratically withL in the latter case. The equations are solved analytically and exact expressions for the concentration are derived. We investigate the finite-size behavior of the concentration and calculate the corresponding scaling functions and the leading corrections for both types of boundary conditions. We show that the scaling functions are independent of the initial conditions but do depend on the boundary conditions. A similarity transformation between the two models is derived and used to connect the corresponding scaling functions.     

A Remark on the Equivalence of Isokinetic and Isoenergetic Thermostats in the Thermodynamic Limit

The Gaussian isokinetic and isoenergetic thermostats of Hoover and Evans are formally equivalent, as remarked by Gallavotti, Rondoni, and Cohen. But outside of equilibrium the fluctuations are uncontrolled and might break the equivalence. We show that equivalence is ensured if we consider an infinite system assumed to be ergodic under space translations.     

The statistical mechanical resolution of a thermodynamic “paradox”

We discuss a thermodynamic paradox suggested by Fallows and Greenleaf. Using quantum statistical mechanics, we analyze the problem in detail, showing why no paradox arises.This work was carried out some time ago, circulated as a MIT internal report, and has since enjoyed a modest popularity. We are pleased to publish it more broadly in this volume dedicated to the memory of such an outstanding researcher as Prof. P. Résibois. (MOS)     

Convergence of dynamical zeta functions

I study poles and zeros of zeta functions in one-dimensional maps. Numerical and analytical arguments are given to show that the first pole of one such zeta function is given by the first zero ofanother zeta function: this describes convergence of the calculations of the first zero, which is generally the physically interesting quantity. Some remarks on how these results should generalize to zeta functions of dynamical systems with pruned symbolic dynamics and in higher dimensions follow.     

Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti–Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations. At equilibrium one recovers in particular the Onsager reciprocity relations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.     

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part II. Numerical methods

The scaling exponent and the scaling function for the 1D single-species coagulation model (A+AA) are shown to be universal, i.e., they are not influenced by the value of the coagulation rate. They are independent of the initial conditions as well. Two different numerical methods are used to compute the scaling properties of the concentration: Monte Carlo simulations and extrapolations of exact finite-lattice data. These methods are tested in a case where analytical results are available. To obtain reliable results from finite-size extrapolations, numerical data for lattices up to ten sites are sufficient.     

Transient and cyclic behavior of cellular automata with null boundary conditions

One-dimensional cellular automata (CA) over finite fields are studied in which each interior cell is updated to contain the sum of the previous values of its two nearest neighbors. Boundary cells are updated according to null boundary conditions. For a given initial configuration, the CA evolves through transient configurations to an attracting cycle. The dependence of the maximal transient length and maximal cycle length on the number of cells is investigated. Both can be determined from the minimal polynomial of the update matrix, which in this case satisfies a useful recurrence relation. With cell values from a field of characteristic two, the explicit dependence of the maximal transient length on the number of cells is determined. Extensions and directions for future work are presented.Deceased.