New features appear in the kinetics of phase ordering and phase separation in systems where long-range repulsive interactions (LRRI) compete with the shortrange attractive interactions considered earlier. Competing interactions can lead to the emergence of modulated phases, where a particular symmetry, wavelength, and amplitude are selected (Seul and Andelman, 1995). Both in equilibrium and nonequilibrium systems such modulated phases have domain structures with various shapes, patterns, and morphologies. Figure 13.1 shows some domain structures seen in systems displaying modulated phases. Modulated phases in materials are important in technological applications (Park et al., 1997; Black et al., 2000). An understanding of such phases is crucial in order to be able to design materials with specific properties and control their morphology.

Many systems in nature can be modeled through the inclusion of long-range interactions. Examples of such systems are uniaxial ferromagnetic films, ferromagnetic surface layers, ferrofluid films, ferroelectrics, Langmuir (lipid) monolayers, block copolymers, and cholesteric liquid crystals. A uniaxial ferromagnetic film in the presence of an external magnetic field can be modeled by augmenting the standard scalar order parameter model A with an additional long-range interaction arising from the parallel orientation of magnetic dipoles (Roland and Desai, 1990). This repulsive interaction competes with the attractive domain wall energy. An external magnetic field makes the film's magnetization a nonconserved quantity so that a description based on model A is appropriate. Block copolymers and Langmuir monolayers are examples of conserved order parameter (model B) systems where the connectivity between the covalently bonded blocks of the polymer chains results in an effective LRRI (Sagui and Desai, 1994).

The late stages of model B dynamics for asymmetric quenches, where the initial condition places the post-quench system just inside and quite near to the coexistence curve, exhibit characteristic features. For an asymmetric system, ψ0 is a measure of the extent of off-criticality of the system. For ψ0 > 0 the majority phase equilibrates at ψ+ = +1 and the minority phase at ψ–= –1. At late times, the minority-phase clusters have a characteristic radius R(τ) which is much larger than the interface width ξ. An important coupling exists between the interface and the majority phase through the surface tension σ. The conservation law dictates that the minority phase will occupy a much smaller “volume” fraction than the majority phase in the final equilibrium state. The dynamics is governed by interactions between the different domains of the minority phase. At late times, these domains have spherical and circular shapes for three- and twodimensional systems, respectively. Late-stage coarsening is referred to as Ostwald ripening.

The late-stage dynamics may be mapped onto a diffusion equation with sources and sinks (domains) whose boundaries are time dependent. The classic papers by Lifshitz and Slyozov (1961) and Wagner (1961) form the theoretical cornerstone for the description of domain coarsening dynamics for model B. The Lifshitz– Slyozov–Wagner (LSW) theory of coarsening is based on the assumption that each interface between a minority phase domain and the majority phase background is infinitely sharp. It describes the diffusive interactions between the domains through a mean-field treatment with precisely defined boundary conditions at each of the interfaces.

The kinetics of first-order phase transitions involves the separation of an initially one-phase system into two coexisting phases. The formation and coarsening of domains of the coexisting phases as the system evolves are of central interest. The phase segregation process is usually studied by first preparing the system in a region of the phase diagram where the homogeneous state is stable. The system is then suddenly quenched into the two-phase region, and segregation into domains of the two stable phases takes place. Such phase segregation arises in a variety of physical contexts, including binary alloys and fluid mixtures, ferromagnetic systems, superfluids, polymer mixtures, and chemically reacting fluids. The temperature– composition (T, c) phase diagram for a binary mixture composed of constituents A and B is shown in Fig. 2.1. For low enough temperatures, in the region bounded by the coexistence curve the binary mixture will segregate into A-rich and B-rich phases.

A quench that takes the system from a homogeneous to a two-phase region is often performed by changing temperature suddenly at fixed concentration. Such a quench from the one-phase state at high temperatures may be carried out either along the critical isoconcentration line that passes through the critical point (path a), or along off-critical paths (path b). Phase segregation may be monitored by the changes in the local concentration of the binary mixture. In general, the variable that signals the passage from the one-phase to two-phase regions is called the order parameter ø.

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.