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.

In the previous chapter we studied the kinematics of many-body systems, and the form of operators representing the physical properties of a system, all of which were embodied by the quantum field. In this chapter we shall study the quantum dynamics of many-body systems, which can also be embodied by the quantum fields. We shall employ the fact that the quantum dynamics of a system, instead of being described in terms of the dynamics of the states or the evolution operator, i.e. as previously done through the Schrödinger equation, can instead be carried by the quantum fields. The quantum dynamics is therefore expressed in terms of the correlation functions or Green's functions of the quantum fields evaluated with respect to some state of the system. In particular we shall consider the general case of quantum dynamics for arbitrary non-equilibrium states. After introducing various types of Green's functions and relating them to measurable quantities, we will discuss the simplifications reigning for the special case of equilibrium states.

Quantum dynamics

Quantum dynamics can be described in different ways since quantum mechanics is a linear theory and the dynamics described by a unitary transformation of states. This will come in handy in the next chapter when we study a quantum theory in terms of its perturbative expansion using the so-called interaction picture. Here we first discuss the Schrödinger and Heisenberg pictures.

The Schrödinger picture

Having the Hamiltonian on the multi-particle space at hand we can consider the dynamics described in the multi-particle space.

The contour-ordered Green's function considered in the previous chapter was ideal for discussing general closed time path properties such as the perturbative diagrammatic structure for non-equilibrium states. However, the contour-ordered Green's function lacks physical transparency and does not appeal to intuition. We need a different approach, which brings quantities back to real time. To accomplish this we introduce a representation where forward and return parts of the closed time path are ordered by numbers, specifying the position of a contour time by two indices, i = 1, 2. Next is the diagrammatic perturbation theory in the real-time technique then formulated in a fashion where the aspects of non-equilibrium states emerge in the physically most appealing way. In particular, we shall construct the representation where spectral properties and quantum statistics show up on a different footing in the diagrams. Lastly, we consider the connection to the imaginary-time treatment of non-equilibrium states, and establish its equivalence to the real-time approach propounded in this chapter.

Real-time matrix representation

To let our physical intuition come into play; we need to get from contour times back to real times. This is achieved by labeling the forward and return contours of the closed time path, depicted in Figure 4.5, by numbers, specifying the position of a contour time by an index. The forward contour we therefore label c1 and the return contour c2, i.e. a contour time variable gets tagged by the label 1 or 2 specifying its belonging to forward or return contour, respectively.

Sand in stasis or in motion – the image these words conjure up is one of lifelong familiarity and intuitive simplicity. Despite appearances, however, matter in the granular state combines some of the most complex aspects of known physical systems; to date, a detailed understanding of its behaviour remains elusive.

Granular media are neither completely solid-like nor completely liquid-like in their behaviour – they pack like solids, but flow like liquids. They can, like liquids, take the shape of their containing vessel, but unlike liquids, they can also adopt a variety of shapes when they are freestanding. This leads to the everyday phenomenon of the angle of repose, which is the angle that a sandpile makes with the horizontal. The angle of repose can take values between θr (the angle below which the sandpile is stationary) and θm (the angle above which avalanches spontaneously flow down the slope); in the intervening range of angles, the sandpile manifests bistability, in that it can either be at rest or have flowing down it. This avalanche flow is such that all the motion occurs in a relatively narrow boundary layer, so that granular flow is strongly non-Newtonian.

Sandpiles are not just disordered in their geometry – the shape and texture of the grains, on which physical parameters like friction and restitution depend, are also sources of disorder. These features, along with their amorphous packings, have important consequences for granular statics and dynamics.

In this chapter, we are interested in static pilings of cohesionless grains. For example, we would like to be able to describe how forces or stresses are distributed in these systems. As a matter of fact, this is not a simple issue as, for instance, two apparently identical sandpiles but prepared in different ways can show rather contrasted bottom pressure profiles.

The aim is to be as complementary as possible to the existing books on granular media. There are indeed numerous ones which deal with Janssen's model for silos, Mohr–Coulomb yield criterion or elasto-plasticity of granular media or soils, see e.g. We shall then sum up only the basics of that part of the literature and spend more time with a review of the more recent experiments, simulations and modellings performed and developed in the last decade. This chapter is divided into two main sections. The first one is devoted to microscopic results, concerning in particular the statistical distribution of contact forces and orientations, while, in the second part, more macroscopic aspects are treated with stress profile measures and distribution. Finally, let us remark that, although the number of papers related to this field is very large, we have tried to cite a restricted number of articles, excluding in particular references written in another language than English, as well as conference proceedings or reviews difficult to access.

Statics at the grain scale

Static solutions

Equilibrium conditions

Let us consider a single grain in a granular piling at rest.

Granular compaction is characterised by a competition between fast and slowp degrees of freedom; far from the jamming limit, individual grains can quickly move into suitable voids in their neighbourhood. As the jamming limit is approached, however, voids which can accommodate whole grains become more and more rare; a cooperative rearrangement of grain clusters is required to fill the partial voids which remain. Such collective processes are necessarily slow, and eventually lead to dynamical arrest.

The modelling of granular compaction has been the subject of considerable effort. Early simulations of shaken hard sphere packings, carried out in close symbiosis with experiment, were followed by lattice-based theoretical models; the latter could not, of course, incorporate the reality of a disordered substrate. Mean-field models which could incorporate such disorder could not, on the other hand, impose the finite connectivity of grains included in Refs. It was to answer the need of an analytically tractable model which incorporated finitely connected grains on fully disordered substrates that random graph models of granular compaction were first introduced by Berg and Mehta.

A random graph consists of a set of nodes and bonds, with the bonds connecting each node at random to a finite number of others, thus, from the point of view of connectivity, appearing like a finite-dimensional structure. Each bond may link two sites (a graph) or more (a so-called hypergraph).

Introduction

From a general introduction to phenomena in vibrated sand, we focus now on (granular) matter in the jammed state, which has become a focus of interest for physicists in recent years. Glasses manifest jamming, in addition to densely packed granular media; while the mechanisms of jamming in each case show strong similarities, the ineffectiveness of temperature as a dynamical motor in granular media leads to vastly more surprising effects.

A direct consequence of such athermal behaviour in sandpiles is the stable formation of cooperative structures such as bridges, or indeed the very existence of an angle of repose; neither would be possible in the presence of Brownian motion. In this chapter and the next, we show that the dynamics of bridge formation is a typical result of collective dynamics, as is the relaxation of the angle of repose; competition between density fluctuations and external driving forces can, on the other hand, result in sandpile collapse. The chapters which follow these, with their focus on jamming, unify aspects of glasses and granular media; one of them uses random graphs to illustrate competitive and cooperative effects in granular compaction, while the other makes clear how asymmetric grains can orient themselves suitably so as to waste less space, as the jamming limit is approached.