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.

The purpose of this book is to provide an introduction to the applications of quantum field theoretic methods to systems out of equilibrium. The reason for adding a book on the subject of quantum field theory is two-fold: the presentation is, to my knowledge, the first to extensively present and apply to non-equilibrium phenomena the real-time approach originally developed by Schwinger, and subsequently applied by Keldysh and others to derive transport equations. Secondly, the aim is to show the universality of the method by applying it to a broad range of phenomena. The book should thus not just be of interest to condensed matter physicists, but to physicists in general as the method is of general interest with applications ranging the whole scale from high-energy to soft condensed matter physics. The universality of the method, as testified by the range of topics covered, reveals that the language of quantum fields is the universal description of fluctuations, be they of quantum nature, thermal or classical stochastic. The book is thus intended as a contribution to unifying the languages used in separate fields of physics, providing a universal tool for describing non-equilibrium states.

Chapter 1 introduces the basic notions of quantum field theory, the bose and fermi quantum fields operating on the multi-particle state spaces. In Chapter 2, operators on the multi-particle space representing physical quantities of a many-body system are constructed. The detailed exposition in these two chapters is intended to ensure the book is self-contained. In Chapter 3, the quantum dynamics of a many-body system is described in terms of its quantum fields and their correlation functions, the Green's functions.

There exists a regime of overlap between the equilibrium and non-equilibrium behavior of a system, the non-equilibrium behavior of weakly perturbed states. When a system is perturbed ever so slightly, its response will be linear in the perturbation, say the current of the conduction electrons in a metal will be proportional to the strength of the applied electric field. This regime is called the linear response regime, and though the system is in a non-equilibrium state all its characteristics can be inferred from the properties of its equilibrium state. In the next chapter we shall go beyond the linear regime by showing how to obtain quantum kinetic equations. The kinetic-equation approach to transport is a general method, and allows in principle nonlinear effects to be considered. However, in many practical situations one is interested only in the linear response of a system to an external force. The linear response limit is a tremendous simplification in comparison with general non-equilibrium conditions, and is the subject matter of this chapter. In particular the linear response of the density and current of an electron gas are discussed. The symmetry properties of response functions, and the fluctuation–dissipation theorem are established. Lastly we demonstrate how correlation functions can be measured in scattering experiments, as illustrated by considering neutron scattering from matter. Needless to say, in measurements of (say) the current in a macroscopic body, far less information in the current correlation function is probed.

Linear response

In this section we consider the response of an arbitrary property of a system to a general perturbation.

The methods of quantum field theory, originally designed to study quantum fluctuations, are also the tool for studying the thermal fluctuations of statistical physics, for example in connection with understanding critical phenomena. In fact, the methods and formalism of quantum fields are the universal language of fluctuations. In this chapter we shall capitalize on the universality of the methods of field theory as introduced in Chapters 9 and 10, and use them to study non-equilibrium phenomena in classical statistical physics where the fluctuations are those of a classical stochastic variable. We shall show that the developed non-equilibrium real-time formalism in the classical limit provides the theory of classical stochastic dynamics.

Newton's law, which governs the motion of the heavenly bodies, is not the law that seems to govern earthly ones. They sadly seem to lack inertia, get stuck and feebly ramble around according to Brownian dynamics as described by the Langevin equation. Their dynamics show transient effects, but if they are on short time scale too fast to observe, dissipative dynamics is typically specified by the equation v ∝ F where the proportionality constant could be called the friction coefficient. This is Aristotelian dynamics, average velocity proportional to force, believed to be correct before Galileo came along and did thorough experimentation. If a sponge is dropped from the tower of Pisa, it will almost instantly reach its saturation final velocity. If a heavier sponge is dropped simultaneously, it will fall faster reaching the ground first.