Since 1990, when the first edition appeared, there has been a significant advance in the development of nonequilibrium systems. The centerpiece of the first edition was the nonequilibrium molecular-dynamics methods and their theoretical analysis, the connections between linear and nonlinear response theory, and the design of the simulation methods. This is now a mature field with only one significant addition, which is the new method for elongational flows.

Chapter 10 in the first edition was called “Towards a thermodynamics of steady states.” This contained an introduction to deterministic chaotic systems. The second edition has the same title for Chapter 10, but the contents are now completely different. The application of the ideas of modern dynamical-systems theory to nonequilibrium systems has grown enormously with all of Chapter 8 devoted to this. However, this still constitutes the barest of introductions with whole books (Gaspard, 1998; Dorfman, 1999; Ott, 2002; and Sprott, 2003) devoted to this theme. The theoretical advances in this area are some of the biggest. The development of methods to study the time evolution using periodic orbits, and the use of periodic orbits to develop SRB measures for nonequilibrium systems are exciting steps forward.

Based on the dynamical properties, Lyapunov exponents in particular, there have been great strides made in the development of the study of fluctuations in nonequilibrium systems.

Introduction

Linear response theory can be used to design computer simulation algorithms for the calculation of transport coefficients. There are two types of transport coefficients: mechanical and thermal, and we will show how thermal transport coefficients can be calculated using mechanical methods.

In Nature nonequilibrium systems may respond essentially adiabatically, or depending upon circumstances, they may respond approximately isothermally — the quasi-isothermal response. No natural systems can be precisely adiabatic or isothermal. There will always be some transfer of the dissipative heat produced in nonequilibrium systems towards thermal boundaries. This heat may be radiated, convected, or conducted to the boundary reservoir. Provided this heat transfer is slow on a microscopic timescale and provided that the temperature gradients implicit in the transfer process lead to negligible temperature differences on a microscopic length scale, we call the system quasi-isothermal. We assume that quasi-isothermal systems can be modelled microscopically in computer simulations, as isothermal systems.

In view of the robustness of the susceptibilities and equilibrium time-correlation functions to various thermostatting procedures (see Sections 5.2 and 5.4), we expect that quasi-isothermal systems may be modeled using Gaussian or Nosé—Hoover thermostats or enostats. Furthermore, since heating effects are quadratic functions of the thermodynamic forces, the linear response of nonequilibrium systems can always be calculated by analyzing the adiabatic, isothermal, or isoenergetic response.

Classical mechanics

In nonequilibrium statistical mechanics we seek to model transport processes beginning with an understanding of the motion and interactions of individual atoms or molecules. The laws of classical mechanics govern the motion of atoms and molecules, so in this chapter we begin with a brief description of the mechanics of Newton, Lagrange, and Hamilton. It is often useful to be able to treat constrained mechanical systems. We will use a principle due to Gauss to treat many different types of constraint — from simple bond-length constraints, to constraints on kinetic energy. As we shall see, kinetic energy constraints are useful for constructing various constant temperature ensembles. We will then discuss the Liouville equation and its formal solution. This equation is the central vehicle of nonequilibrium statistical mechanics. We will then need to establish the link between the microscopic dynamics of individual atoms and molecules and the macroscopic hydrodynamical description discussed in the last chapter. We will discuss two procedures for making this connection. The Irving and Kirkwood procedure relates hydrodynamic variables to nonequilibrium ensemble averages of microscopic quantities. A more direct procedure, which we will describe, succeeds in deriving instantaneous expressions for the hydrodynamic field variables.

Newtonian mechanics

Classical mechanics (Goldstein, 1980) is based on Newton's three laws of motion.

Introduction

Nonequilibrium steady states are fascinating systems to study. Although there are many parallels between these states and equilibrium states, a convincing theoretical description of steady states, particularly far from equilibrium, has yet to be found. Close to equilibrium, linear response theory and linear irreversible thermodynamics provide a relatively complete treatment, (Sections 2.1 to 2.3). However, in systems where local thermodynamic equilibrium has broken down, and thermodynamic properties are not the same local functions of thermodynamic state variables that they are at equilibrium, our understanding is very primitive indeed.

In Section 7.3 we gave a statistical-mechanical description of thermostatted, nonequilibrium steady states far from equilibrium — the transient time-correlation function (TTCF) and Kawasaki formalisms. The transient time-correlation function is the nonlinear analog of the Green—Kubo correlation functions. For linear transport processes the Green—Kubo relations play a role which is analogous to that of the partition function at equilibrium. Like the partition function, Green—Kubo relations are highly nontrivial to evaluate. They do, however, provide an exact starting point from which one can derive exact interrelations between thermodynamic quantities. The Green—Kubo relations also provide a basis for approximate theoretical treatments as well as being used directly in equilibrium molecular-dynamics simulations.

The TTCF and Kawasaki expressions may be used as nonlinear, nonequilibrium partition functions.

Mechanics provides a complete microscopic description of the state of a system. When the equations of motion are combined with initial conditions and boundary conditions, the subsequent time evolution of a classical system can be predicted. In systems with more than just a few degrees of freedom such an exercise is impossible. There is simply no practical way of measuring the initial microscopic state of, for example, a glass of water, at some instant in time. In any case, even if this was possible we could not then solve the equations of motion for a coupled system of 1023 molecules.

In spite of our inability to fully describe the microstate of a glass of water, we are all aware of useful macroscopic descriptions for such systems. Thermodynamics provides a theoretical framework for correlating the equilibrium properties of such systems. If the system is not at equilibrium, fluid mechanics is capable of predicting the macroscopic nonequilibrium behaviour of the system. In order for these macroscopic approaches to be useful, their laws must be supplemented, not only with a specification of the appropriate boundary conditions, but with the values of thermophysical constants such as equation-of-state data and transport coefficients. These values cannot be predicted by macroscopic theory.

The thermodynamic temperature

Equilibrium thermodynamics provides a very useful connection between mechanical and thermal properties of fluids and solids. The predicted relationships between different quantities measured under different thermodynamic conditions are a fundamental consequence of thermodynamics. It is natural to attempt to develop a similar thermodynamic treatment of non-equilibrium systems, at least for steady states. At present, there are a number of different treatments: the extended irreversible thermodynamics (Jou et al., 2001); the approach to microscopic relaxation processes (Öttinger, 2005); and the approach that we follow here. It is fair to say that, at present, there is no consensus on the correctness of any of these approaches, and indeed some debate about whether it is even possible to define the usual thermodynamic quantities for a nonequilibrium system. Clearly then, it is necessary to limit the types of nonequilibrium processes to which we apply thermodynamics. As an example of a system where a thermodynamic treatment may be successful, consider a steady-state Poiseuille flow system where we can define a local temperature and local shear rate at each point in the fluid. There will be gradients in both the shear rate and the temperature that determine the local streaming velocity profile and the conduction of heat to the boundary.

During the 1980s there have been many new developments regarding the nonequilibrium statistical mechanics of dense classical systems. These developments have had a major impact on the computer simulation methods used to model nonequilibrium fluids. Some of these new algorithms are discussed in the recent book by Allen and Tildesley (1987), Computer Simulation of Liquids. However, that book was never intended to provide a detailed statistical mechanical backdrop to the new computer algorithms. As the authors commented in their preface, their main purpose was to provide a working knowledge of computer simulation techniques. The present volume is, in part, an attempt to provide a pedagogical discussion of the statistical mechanical environment of these algorithms.

There is a symbiotic relationship between nonequilibrium statistical mechanics on the one hand and the theory and practice of computer simulation on the other. Sometimes, the initiative for progress has been with the pragmatic requirements of computer simulation and at other times, the initiative has been with the fundamental theory of nonequilibrium processes. Although progress has been rapid, the number of participants who have been involved in the exposition and development, rather than with application, has been relatively small.

The formal theory is often illustrated with examples involving shear flow in liquids.

Adiabatic linear response theory

In this chapter we will discuss how an external field Fe, perturbs an N-particle system. We assume that the field is sufficiently weak that only the linear response of the system need be considered. These considerations will lead us to equilibrium fluctuation expressions for mechanical transport coefficients such as electrical conductivity. These expressions are formally identical to the Green—Kubo formulae that were derived in the last chapter. The difference is that the Green—Kubo formulae pertain to thermal transport processes where boundary conditions perturb the system away from equilibrium — all Navier—Stokes processes fall into this category. Mechanical transport coefficients, on the other hand, refer to systems where mechanical fields which appear explicitly in the equations of motion for the system, drive the system away from equilibrium. As we will see, it is no coincidence that there is such a close similarity between the fluctuation expressions for thermal and mechanical transport coefficients. In fact one can often mathematically transform the nonequilibrium boundary conditions for a thermal transport process into a mechanical field. The two representations of the system are then said to be congruent.

A major difference between the derivations of the equilibrium fluctuation expressions for the two representations is that in the mechanical case one does not need to invoke Onsager's regression hypothesis.