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.

In the previous chapter we introduced the one-particle irreducible effective action by collecting the one-particle irreducible vertex functions into a generator whose argument is the field, the one-state amplitude in the presence of the source. The effective action thus generates the one-particle irreducible amputated Green's functions. We shall now enhance the usability of the non-equilibrium effective action by establishing its relationship to the sum of all one-particle irreducible vacuum diagrams. To facilitate this it is convenient to add the final mathematical tool to the arsenal of functional methods, viz. functional integration or path integrals over field configurations. We are then following Feynman and instead of describing the field theory in terms of differential equations, we get its corresponding representation in terms of functional or path integrals. This analytical condensed technique shall prove powerful when unraveling the content of a field theory. The loop expansion of the non-equilibrium effective action is developed, and taken one step further as we introduce the two-particle irreducible effective action valid for non-equilibrium states. As an application of the effective action approach, we consider a dilute Bose gas and a trapped Bose–Einstein condensate.

Functional integration

Functional differentiation has its integral counterpart in functional integration. We shall construct an integration over functions and not just numbers as in elementary integration of a function. We approach this infinite-dimensional kind of integration with care (or, from a mathematical point of view, carelessly), i.e. we base it on our usual integration with respect to a single variable and take it to a limit.

Quantum corrections to the classical Boltzmann results for transport coefficients in disordered conductors can be systematically studied in the expansion parameter ħ /pFl, the ratio of the Fermi wavelength and the impurity mean free path, which typically is small in metals and semiconductors. The quantum corrections due to disorder are of two kinds, one being the change in interactions effects due to disorder, and the other having its origin in the tendency to localization. When it comes to an indiscriminate probing of a system, such as the temperature dependence of its resistivity, both mechanisms are effective, whereas when it comes to the low-field magneto-resistance only the weak localization effect is operative, and it has therefore become an important diagnostic tool in material science. We start by discussing the phenomena of localization and (especially weak localization) before turning to study the influence of disorder on interaction effects.

Localization

In this section the quantum mechanical motion of a particle at zero temperature in a random potential is addressed. In a seminal paper of 1958, P. W. Anderson showed that a particle's motion in a sufficiently disordered three-dimensional system behaves quite differently from that predicted by classical physics according to the Boltzmann theory [71]. In fact, at zero temperature diffusion will be absent, as particle states are localized in space because of the random potential. A sufficiently disordered system therefore behaves as an insulator and not as a conductor. By changing the impurity concentration, a transition from metallic to insulating behavior occurs, the Anderson metal–insulator transition.

Superconductivity was discovered in 1911 by H. Kamerlingh Onnes. Having succeeded in liquefying helium, transition temperature 4.2K, this achievement in cryogenic technology was used to cool mercury to the man-made temperature that at that time was closest to absolute zero. He reported the observation that mercury at 4.2K abruptly entered a new state of matter where the electrical resistance becomes vanishingly small. This extraordinary phenomenon, coined superconductivity, eluted a microscopic understanding until the theory of Bardeen, Cooper and Schrieffer in 1957 (BCS-theory). The mechanism responsible for the phase transition from the normal state to the superconducting state at a certain critical temperature is that an effective attractive interaction between electrons makes the normal ground state unstable. As far as conventional or low-temperature superconductors are concerned, the attraction between electrons follows from the form of the phonon propagator, Eq. (5.45), viz. that the electron–phonon interaction is attractive for frequencies less than the Debye frequency, and in fact can overpower the screened Coulomb repulsion between electrons, leading to an effective attractive interaction between electrons. The original BCS-theory was based on a bold ingenious guess of an approximate ground state wave function and its low-energy excitations describing the essentials of the superconducting state. Later the diagrammatic Green's function technique was shown to be useful to describe more generally the properties of superconductors, such as under conditions of spatially varying magnetic fields and especially for general non-equilibrium conditions.

At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. According to Feynman, as described in Chapter 4, instead of formulating quantum theory in terms of operators, the canonical formulation, for calculational purposes quantum dynamics can conveniently be formulated in terms of a few simple stenographic rules, the Feynman rules for propagators and interaction vertices.

In Chapters 4 and 5, we showed how to arrive at the Feynman rules of diagrammatic perturbation theory for non-equilibrium states starting from the Hamiltonian defining the theory. The feature of non-equilibrium states, originally carried by the dynamical indices, could be expressed in terms of two simple universal vertex rules for the RAK-components of the matrix Green's functions. We are thus well acquainted with diagrammatics even for the description of non-equilibrium situations. However, for the situations studied using the quantum kinetic equations in Chapters 7 and 8, only the Dyson equation was needed, i.e. the self-energy, the 2-state one-particle irreducible amputated Green's function. No need for higher-order vertex functions was required, and the full flourishing diagrammatics was not put into action. In this chapter we shall proceed the other way around. We shall show that the diagrammatics of a physical theory, including the description of non-equilibrium states, can be obtained by simply stating quantum dynamics, the superposition principle, as the two exclusive options for a particle: to interact or not to interact! From this simple Shakespearean approach we shall construct the Feynman diagrammatics of non-equilibrium dynamics.