On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

For Hamiltonian systems subject to an external potential which in the presence of a thermostat will reach a nonequilibrium stationary state Dettmann and Morriss proved a strong conjugate pairing rule (SCPR) for pairs of Lyapunov exponents in the case of isokinetic (IK) stationary states which have a given kinetic energy. This SCPR holds for all initial phases of the system, all times t, and all numbers of particles N. This proof was generalized by Wojtkowski and Liverani to include hard interparticle potentials. A geometrical reformulation of those results is presented. The present paper proves numerically, using periodic orbits for the Lorentz gas, that SCPR cannot hold for isoenergetic (IE) stationary states which have a given total internal energy. In that case strong evidence is obtained for CPR to hold for large N and t, where it can be conjectured that the larger N, the smaller t will be. This suffices for statistical mechanics.     

Adiabatic Theorems and Reversible Isothermal Processes

Isothermal processes of a finitely extended, driven quantum system in contact with an infinite heat bath are studied from the point of view of quantum statistical mechanics. Notions like heat flux, work and entropy are defined for trajectories of states close to, but distinct from states of joint thermal equilibrium. A theorem characterizing reversible isothermal processes as quasi-static processes (“isothermal theorem”) is described. Corollaries concerning the changes of entropy and free energy in reversible isothermal processes and on the 0th law of thermodynamics are outlined.*Supported by the Swiss National Foundation.     

Entropy Production in Nonequilibrium Statistical Mechanics

We consider systems of nonequilibrium statistical mechanics, driven by nonconservative forces and in contact with an ideal thermostat. These are smooth dynamical systems for which one can define natural stationary states μ (SRB in the simplest case) and entropy production e(μ) (minus the sum of the Lyapunov exponents in the simplest case). We give exact and explicit definitions of the entropy production e(μ) for the various situations of physical interest. We prove that e(μ)≥0 and indicate cases where e(μ)>0. The novelty of the approach is that we do not try to compute entropy production directly, but make it depend on the identification of a natural stationary state for the system. Received: 15 July 1996 / Accepted: 30 October 1996     

Magnetic system under a fluctuating field

We study nonequilibrium steady states, phase transitions and critical phenomena in a d-dimensional lattice model which represents a magnetic system under the action of a field fluctuating very rapidly with time. This induces competing kinetics which produces a sort of (dynamical) frustration which might occur also in some natural disordered systems. The exact solution for d = 1, partial exact results for d ≥ 2, and a comparison with some related models are reported.     

Anisotropic finite-size scaling analysis of a two-dimensional driven diffusive system

The standard two-dimensional uniformly driven diffusive model is simulated extensively for much larger systems with a multi-spin coding technique. The nonequilibrium phase transition is analyzed with anisotropic finite-size scaling both at the critical point and off the critical point. The field-theoretic values of critical exponents fit the data well at and aboveT _c . BelowT _c the scaling is rather difficult and the results are not conclusive.     

Violation of the Fluctuation-Dissipation Theorem and Heating Effects in the Time-Dependent Kondo Model

The fluctuation-dissipation theorem (FDT) plays a fundamental role in understanding quantum many-body problems. However, its applicability is limited to equilibrium systems and it does in general not hold in nonequilibrium situations. This violation of the FDT is an important tool for studying nonequilibrium physics. In this paper we present results for the violation of the FDT in the Kondo model where the impurity spin is frozen for all negative times, and set free to relax at positive times. We derive exact analytical results at the Toulouse point, and results within a controlled approximation in the Kondo limit, which allow us to study the FDT violation on all time scales. A measure of the FDT violation is provided by the effective temperature, which shows initial heating effects after switching on the perturbation, and then exponential cooling to zero temperature as the Kondo system reaches equilibrium.     

Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to \(\frac{1}{2} \sqrt{T_{L}T_{R}}\) where T _L and T _R are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier’s Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach.     

Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

We consider systems of moving particles in 1-dimensional space interacting through energy storage sites. The ends of the systems are coupled to heat baths, and resulting steady states are studied. When the two heat baths are equal, an explicit formula for the (unique) equilibrium distribution is given. The bulk of the paper concerns nonequilibrium steady states, i.e., when the chain is coupled to two unequal heat baths. Rigorous results including ergodicity are proved. Numerical studies are carried out for two types of bath distributions. For chains driven by exponential baths, our main finding is that the system does not approach local thermodynamic equilibrium as system size tends to infinity. For bath distributions that are sharply peaked Gaussians, in spite of the near-integrable dynamics, transport properties are found to be more normal than expected.     

Lyapunov Instability for a Hard-Disk Fluid in Equilibrium and Nonequilibrium Thermostated by Deterministic Scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

A Phase Transition in a Quenched Amorphous Ferromagnet

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over \(\mathbb {R}^d\) , \(d\ge 2\) . Each particle bears a real-valued spin with symmetric a priori distribution; the spin-spin interaction is pair-wise and attractive. Two spins are supposed to interact if they are neighbors in the graph defined by a homogeneous Poisson point process. For this model, we prove that with probability one: (a) quenched thermodynamic states exist; (b) they are multiple if the intensity of the underlying point process and the inverse temperature are big enough; (c) there exist multiple quenched thermodynamic states which depend on the realizations of the underlying point process in a measurable way.     

Nonequilibrium work fluctuations in a driven Ising model

Monte Carlo simulations of a sheared Ising model are used to study nonequilibrium fluctuations of mechanical work. The validity of the transient (starting from equilibrium) and the steady state fluctuation relations is verified. A fluctuation relation has been also shown to hold for the mechanical work done on the system, during the transition between two nonequilibrium steady states corresponding to different drivings.     

Nonlinear chemical dynamics in low dimensions: An exactly soluble model

Restricting space to low dimensions can cause deviations from the mean-field behavior in certain statistical systems. We investigate, both numerically and analytically, the behavior of the chemical reaction A+2X3X in one and two dimensions. In one dimension, we produce exact results showing that the trimolecular reaction system stabilizes in a nonequilibrium, locally frozen, asymptotic state in which the ratior of A to X particles is a constant number,r=0.38, quite different from the mean-field ratio,r _MF=1. The same trimolecular model, however, reaches the mean-field limit in two dimensions. In contrast, the bimolecular chemical reaction A+X2X is shown to agree with the mean-field predictions in all dimensions. For both models, we show that the adoption of certain types of transition rules in the laws of evolution can lead to oscillatory steady states.     

Decline of minorities in stubborn societies

Opinion compromise models can give insight into how groups of individuals may either come to form consensus or clusters of opinion groups, corresponding to parties. We consider models where randomly selected individuals interact pairwise. If the opinions of the interacting agents are not within a certain confidence threshold, the agents retain their own point of view. Otherwise, they constructively dialogue and smooth their opinions. Persuasible agents are inclined to compromise with interacting individuals. Stubborn individuals slightly modify their opinion during the interaction. Collective states for persuasible societies include extremist minorities, which instead decline in stubborn societies. We derive a mean field approximation for the compromise model in stubborn populations. Bifurcation and clustering analysis of this model compares favorably with Monte Carlo analysis found in the literature.     

Nonequilibrium relaxation method — an alternative simulation strategy

One well-established simulation strategy to study the thermal phases and transitions of a given microscopic model system is the so-called equilibrium method, in which one first realizes the equilibrium ensemble of a finite system and then extrapolates the results to infinite system. This equilibrium method traces over the standard theory of the thermal statistical mechanics, and over the idea of the thermodynamic limit. Recently, an alternative simulation strategy has been developed, which analyzes the nonequilibrium relaxation (NER) process. It is called theNER method. NER method has some advantages over the equilibrium method. The NER method provides a simpler analyzing procedure. This implies less systematic error which is inevitable in the simulation and provides efficient resource usage. The NER method easily treats not only the thermodynamic limit but also other limits, for example, non-Gibbsian nonequilibrium steady states. So the NER method is also relevant for new fields of the statistical physics. Application of the NER method have been expanding to various problems: from basic first- and second-order transitions to advanced and exotic phases like chiral, KT spin-glass and quantum phases. These studies have provided, not only better estimations of transition point and exponents, but also qualitative developments. For example, the universality class of a random system, the nature of the two-dimensional melting and the scaling behavior of spin-glass aging phenomena have been clarified.     

Two-Point Correlations and Critical Line of the Driven Ising Lattice Gas in a High-Temperature Expansion

Based on a high-temperature expansion, we compute the two-point correlation function and the critical line of an Ising lattice gas driven into a nonequilibrium steady state by a uniform bias E. The lowest nontrivial order already reproduces the key features, i.e., the discontinuity singularity of the structure factor and the (qualitative) E dependence of the critical line. Our approach is easily generalized to other nonequilibrium lattice models and provides a simple analytic tool for the study of the high-temperature phase and its boundaries.     

Stationary nonequilibrium states in boundary-driven Hamiltonian systems: Shear flow

We investigate stationary nonequilibrium states of systems of particles moving according to Hamiltonian dynamics with specified potentials. The systems are driven away from equilibrium by Maxwell-demon reflection rules at the walls. These deterministic rules conserve energy but not phase space volume, and the resulting global dynamics may or may not be time reversible (or even invertible). Using rules designed to simulate moving walls, we can obtain a stationary shear flow. Assuming that for macroscopic systems this flow satisfies the Navier-Stokes equations, we compare the hydrodynamic entropy production with the average rate of phase-space volume compression. We find that they are equalwhen the velocity distribution of particles incident on the walls is a local Maxwellian. An argument for a general equality of this kind, based on the assumption of local thermodynamic equilibrium, is given. Molecular dynamic simulations of hard disks in a channel produce a steady shear flow with the predicted behavior.     

Excess Entropy Production in Quantum System: Quantum Master Equation Approach

For open systems described by the quantum master equation (QME), we investigate the excess entropy production under quasistatic operations between nonequilibrium steady states. The average entropy production is composed of the time integral of the instantaneous steady entropy production rate and the excess entropy production. We propose to define average entropy production rate using the average energy and particle currents, which are calculated by using the full counting statistics with QME. The excess entropy production is given by a line integral in the control parameter space and its integrand is called the Berry–Sinitsyn–Nemenman (BSN) vector. In the weakly nonequilibrium regime, we show that BSN vector is described by \(\ln \breve{\rho }_0\) and \(\rho _0\) where \(\rho _0\) is the instantaneous steady state of the QME and \(\breve{\rho }_0\) is that of the QME which is given by reversing the sign of the Lamb shift term. If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, the excess entropy production approximately reduces to the difference between the von Neumann entropies of the system. Additionally, we point out that the expression of the entropy production obtained in the classical Markov jump process is different from our result and show that these are approximately equivalent only in the weakly nonequilibrium regime.