Exactly Solvable Models for Many-Body Systems Far from Equilibrium

G.M. Schütz    Institut für Festkörperforschung, Forschungszentrum Jülich, 52425 Jülich, Germany

1 Introduction

1.1 Stochastic dynamics of interacting particle systems

Many complex systems of interacting particles that one encounters in nature behave on a phenomenological level in some random fashion. Therefore the theoretical treatment of these systems has to employ statistical approaches. Examples come from areas as diverse as the growth of surfaces (involving atoms or molecules) or the growth of biological systems (involving macroscopic cells), spin-relaxation dynamics, reaction–diffusion processes or, reaching into the sphere of sociological behaviour, the study of traffic flow.

There is a general framework for the statistical description of equilibrium systems, but equally general concepts for systems far from thermal equilibrium are lacking. Hence one usually investigates specific model systems, hoping to gain insight into either the general behaviour of classes of systems or into the specific properties of the system under investigation. Indeed, compared to ‘completed’ classical theories such as electromagnetism or thermodynamics the current state of nonequilibrium statistical mechanics may be seen as a treasure of accumulated knowledge, but with relatively little profound understanding. Yet, some structure has emerged in the recent past, most notably in the concept of universality. It provides a theoretical framework for the observation that often quantities such as critical exponents or certain amplitude ratios do not depend on the specific details of the interactions between the basic constituents of the system. Intimately related is the idea of scaling, expressing the self-similarity of a system if observed on different length scales. These notions apply not only to equilibrium systems, but also to nonequilibrium behaviour of microscopically very different random processes.

Naturally, the discovery of universality and other concepts has largely come from the study of specific systems and of simple models. These possess only those mechanisms that are deemed essential for the understanding of what one observes in real complex systems. Examples of nonequilibrium random processes include driven lattice gases (Spohn, 1991; Schmittmann and Zia, 1995) and reaction–diffusion mechanisms (Privman, 1997; Mattis and Glasser, 1998). They play an important role in the theoretical understanding not only of chemical systems and purely diffusive physical systems. Such models are able, through various mappings and different physical interpretations of the observables, to describe a wide variety of phenomena in physics and beyond. Thus they shed light on the mechanisms leading to universality in particle systems with short-ranged interactions, the emergence of simple collective behaviour which allows for a description of the many-body dynamics in terms of a few relevant variables, and other generic features of systems far from thermal equilibrium.

Usually even simple models are not amenable to exact mathematical analysis. However, it has long been known that there are classes of nontrivial one- and two-dimensional equilibrium statistical mechanical models which can be solved exactly. These have considerably advanced our understanding of critical phenomena in general and of the physics of low-dimensional systems in particular (Baxter, 1982). Recent work has shown that by using a different interpretation of the variables such models may describe nonequilibrium behaviour as well. It is the aim of this work to provide an introduction into exactly solvable nonequilibrium models and to survey some of the insights that have been achieved through the detailed understanding that exact analysis has made possible.

1.2 Integrability

Randomness – which may result from effectively stochastic forces or which may be intrinsic in the underlying microscopic theory – leads to the description of observables in terms of random variables and expectation values (Feller, 1950; van Kampen, 1981; Liggett, 1985; Spohn, 1991). The problem posed by the random behaviour of nonequilibrium systems is to develop tools beyond classical and quantum thermodynamics which allow for a theoretical investigation of these quantities.

The oldest approach to the treatment of reaction–diffusion systems is to formulate rate equations for the reactants in a mean-field approximation. One ignores correlations between particles and often obtains reasonable results by invoking the old law of mass action: the rate of reaction of two species of particles is proportional to the product of their concentrations. However, particularly in low-dimensional systems, mean-field methods tend to be inadequate due to inefficient diffusive mixing. Moreover, in one dimension even short-ranged repulsive interactions represent obstacles seriously blocking the diffusive motion. As a result, large fluctuations persist and rate equation or other mean-field approaches fail (Schmittmann and Zia, 1995; Privman, 1997). This shortcoming was realized quite long ago and to some extent accounted for in Smoluchowski’s theory of diffusion-limited reactions (von Smoluchowski, 1917). This correlation-improved mean-field theory is successful for many problems of interest, but both verification of the assumptions made in this theory and other still untractable problems involving fluctuations require the application of more sophisticated techniques.

Progress may be achieved by adding a suitably chosen noise function to an otherwise deterministic differential equation as in the Langevin approach, or by a Fokker–Planck description, or through the formulation of the stochastic dynamics in terms of a master equation (see, e.g. van Kampen (1981) for these methods). In this type of modelling of a real system usually three approximations are made: the first approximation consists, as in the rate equation approach, in the identification of a few coarse-grained observables such as particle density, magnetization etc. with an effective interaction between these quantities. The second approximation concerns the mathematical prescription of the nature of the random forces which leads to the full dynamical equation describing the system. Solving these equations is a formidable task and therefore usually a third approximation is necessary for the solution of these equations. For instance, in recent years, Monte Carlo simulations on increasingly powerful computers have become a widely applicable numerical technique. Moreover, in the context of critical phenomena, the renormalization group has emerged as an extremely fruitful approach in the study of stochastic processes. Really exact solutions of the dynamical equations for complex systems are comparatively rare. Besides some isolated exact results for various reaction–diffusion systems derived in the past, the only general framework which can produce exact and rigorous results has traditionally been the mathematical treatment using the tools of probability theory (Feller, 1950; van Kampen, 1981; Liggett, 1985, 1999; Spohn, 1991; Kipnis and Landim, 1999).

The past few years have seen an exciting new development which has led to a series of remarkable exact solutions for the stochastic dynamics of interacting particle systems and also to an understanding of the mathematical structure underlying some of the already existing exact, numerical and renormalization group results. At the heart of this development is the close relationship between the Markov generator of the stochastic time evolution in the master equation approach on the one hand and the Hamiltonians for quantum spin systems (or the transfer matrices of statistical mechanics models respectively) on the other. The master equation for the probability distribution of a many-body system is a linear equation of a form similar to the quantum mechanical Schrödinger equation. It can be written as a vector equation with a time-translation operator T (for discrete time evolution) or H (for continuous time evolution) acting on a many-particle Fock space (Kadanoff and Swift, 1968; Doi, 1976; Grassberger and Scheunert, 1980; Sandow and Trimper, 1993). The new insight is the somewhat surprising observation that for some of the most interesting interacting particle systems the time-translation operators, T or H respectively, turn out to be the transfer matrix or quantum Hamiltonian respectively of well-known equilibrium statistical mechanics models. Moreover, in some important cases, these models are integrable, i.e., have an infinite set of conserved charges like six- or eight-vertex models (Baxter, 1982).

It was recognized in the early 1990s that in this way the toolbox of many-body quantum mechanics becomes available for the study of equilibrium and nonequilibrium stochastic processes. Typical results which one obtains using free-fermion techniques, the Bethe ansatz and related algebraic methods, or global symmetries and similarity transformations include firstly stationary properties of the process. Thus one can study a variety of phenomena including phase transitions with divergent length scales in one-dimensional nonequilibrium systems. Going further, one may investigate spectral properties of the time evolution operator which give...