This book had its origin in a graduate course in statistical mechanics given by Professor W. C. Schieve in the Ilya Prigogine Center for Statistical Mechanics at the University of Texas in Austin.

The emphasis is quantum non-equilibrium statistical mechanics, which makes the content rather unique and advanced in comparison to other texts. This was motivated by work taking place at the Austin Center, particularly the interaction with Radu Balescu of the Free University of Brussels (where Professor Schieve spent a good deal of time on various occasions). Two Ph.D. candidate theses at Austin, those of Kenneth Hawker and John Middleton, are basic to Chapters 3 and 4, where the master equations and quantum kinetic equations are discussed. The theme there is the dominant and fundamental one of quantum irreversibility. The particular emphasis throughout this book is that of open systems, i.e. quantum systems in interaction with reservoirs and not isolated. A particularly influential work is the book of Professor A. McLennan of Lehigh University, under whose influence Professor Schieve first learned non-equilibrium statistical mechanics.

An account of relatively recent developments, based on the addition in the Schrödinger equation of stochastic fluctuations of the wave function, is given in Chapter 13. These methods have been developed to account for the collapse of the wave function in the process of measurement, but they are deeply connected as well with models for irreversible evolution.

The first six chapters of the present work set forth the theme of our book, particularly extending the entropy principle that was first introduced by Boltzmann, classically. These, with equilibrium quantum applications (Chapters 7, 8, 9 and possibly also Chapters 14 and 15), represent a one-semester advanced course on the subject.

As frequently pointed out in the text, quantum mechanics introduces special problems to statistical mechanics. Even in Chapter 1, written by the coauthor of this work, Professor Lawrence P. Horwitz of Tel Aviv, the idea of a density operator is required which is not a probability distribution, as in the classical case.

Introduction to black holes

In 1783 John Mitchell wrote, “If the semi-diameter of a sphere of the same density as the sun were to exceed that of the sun in the proportion of five hundred to one, and supposing light to be attracted by the same force in proportion to its vis-inertia with other bodies, all light emitted from such a body would be made to return towards it, by its own gravity” (Mitchell, 1783). Much later, in a prophetic paper, Oppenheimer and Snyder (1939) described the nature of “continued gravitational contraction” of a neutron star. With the nuclear heat gone, the core of the dead star becomes incapable of supporting itself under its own gravitational pull. The final phase is that the high density of the remaining core prevents the escape of the last light. The star disappears from view. Wheeler, later, coined the term black hole for such an object in the cosmos (Misner et al., 1973).

What is most remarkable is that today astronomers/astrophysicists have identified, with modern technical skills, numbers (uncountable) of these black holes. There seems no empirical doubt as to their existence. See the incredible visual treat in the volume The Universe, edited by Martin Rees (2005). Frolov and Novikov (1998) have given a condensed list of objects, eleven in number, which are binary systems that contain black holes. This comes from X-ray studies of binaries. As pointed out by them, the central arguments for the existence of black holes are: (a) the emission has a compact nature, and (b) the emission makes possible the analysis of the orbital motion, and one obtains the mass of the compact partner. If it is of the order of three solar masses, it is a black hole. See the resultant discussion of Cherepaschuk (1996). The strongest black hole candidates are three in number: GS2023+338, GS2000+25 and XN oph 1997. The first has a period of 6.5 days, a mass of the compact companion is of the order of 10 solar masses, and its luminosity is 6 × 1038 erg/sec.

Introduction

Mathematically, given a linear differential operator Lx,

one encounters the solution to the inhomogeneous differential equation

Here ρ(x) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let

G(x, y) is the Green's function. This is a function of x with y a parameter. Take G(x, y) to satisfy the same boundary conditions as φ(x). Then

since

An example of Lx is, of course, the Schrödinger operator

We take ρ(x) = V(x)ψ(x, t),ψ(x, t) being the wave function and V(x) the potential operator.

We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al., 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000).

One- and two-time quantum Green's functions and their properties

Let us introduce the creation operator ψ†(r, t) and annihilation operator ψ(r, t) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles:

The Hamiltonian operator for the particles is

and the number density of particles at rt is the operator

Note that r = (r1 … rN) for (1, 2, 3, …, N) and V(|r − r′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.

Intrinsic theory of irreversibility

In connection with the previous chapter on Wigner–Weisskopf, quantum irreversibility, Gamov states and the Lax–Phillips theory of decay, we shall examine the recent program in statistical mechanics carried forward by Ilya Prigogine and his colleagues since his early book in 1962 (Prigogine, 1962). This discussion is somewhat out of the focus of the present book, since the work to be discussed has principally been devoted to isolated quantum systems, not those in interaction with “reservoirs.” In addition, Prigogine's work is very classical in its content. We shall not attempt the task of reviewing the many changes that have taken place between 1962 and the present. Fine texts describing some of this work are those of Balescu (1963, 1975). A recent critical overview of the “modern view” is in the unpublished thesis of B. C. Bishop of the University of Texas Philosophy Department (Bishop, 1999).

In many ways this is related to the use of Gel'fand triplets (Gel'fand and Vilenkin, 1968), mentioned in Chapter 17.6, to describe irreversible quantum states, i.e. Gamov states. See Chapter 17, the book of Bohm and Gadella (1989) and also the more recent review article by Bohm and colleagues (Bohm et al., 1997). The object in statistical mechanics is to extend the range of the Liouville operator (classical and quantum) such that its eigenvalues are complex and intrinsically irreversible. Statistical mechanics has always been focused on many particle systems and the appearance of a continuum spectrum in the thermodynamic limit, already used in many places in this book. In the classical case Antoniou and Tasaki carried out the adaptation of the Gel'fand triplet approach to the Liouville operator, but this is, apparently, not possible quantum mechanically (Antoniou and Tasaki, 1993). Thus, another route must be taken in what Prigogine has termed, for emphasis, “large Poincaré systems” which have a continuum spectrum (Prigogine, 1997).

Introduction

We will now turn to quantum entanglements and their contemporary, possibly practical, interest. Entanglements, first discussed in the E.P.R. paradox (Einstein, Podolsky, Rosen, 1935) are a perplexing nonlocal feature of quantum mechanics. This was immediately and succinctly discussed by Schrödinger (1935). The long history of this apparent paradox is outlined in the wonderful book of Jammer (1974). We will not focus on the central issue of “hidden variables” and their resolution by the Bell inequalities (Bell, 1964) and the test, nor the failure, of these in experiment (see Fry, 1998).

The distinctive quantum nature of entanglements has led to two quantum effects which we will discuss: quantum information teleportation (Zeilenger, 1998) and quantum computation by means of entangled states. A nice, recent, elementary introduction to the latter is in the Los Alamos reports of James and Kwiat (2002). The quantum correlations or entanglements are sensitive to environmental destruction. This was pointed out early by Zurek, who termed this “decoherence.” The loss of coherence may occur on a short time scale. Recent discussions have been given by Zurek (2002, 2003). We have already given a theoretical example early in Chapter 2. It is pertinent to discuss this here, since it is a property of opensystem quantum master equations that is a central part of our study in this book. In a sense, the decoherence of correlations has turned out to be a “practical” application of these theoretical notions. Are there remedies for unwanted decoherence? Quantum error correction is a possibility. This will be mentioned also.

Entanglements: foundations

Following the reading of the E.P.R. paper, Schrödinger (1935b) quickly repeated the argument from quantum theory, but more generally. He introduced the term “entanglement” to describe what he seemed to agree was a curious, if not unacceptable, property.

Let us follow his point of view. Given a state ψ (x, y) of a composite system of particles x and y formed in their mutual interaction. ψ (x, y) is not a product state.

Basic notions and theWigner–Weisskopf theory

Although the notions of bound states, scattering and quantum transitions are well defined in the quantum theory, the description of an unstable system, involved in the process of decay, has remained an outstanding issue for many years. The problem is fundamental, since it concerns the nature of irreversible processes, one of the most important issues in statistical mechanics and the theme that is central to this book.

The theory of decay is intimately connected with scattering theory and necessarily contains mathematical ideas and methods. We shall try to explain these points carefully as we get to them.

We treat elsewhere in the book the ideas of Boltzmann, Van Hove and Prigogine on irreversible phenomena. The tools that are developed there are basically approximate, although very useful. One can argue that the basic rigorous characteristic of an irreversible process is that, as represented in terms of the evolution of a state in the Hilbert space of the quantum theory, it must be a semigroup. This type of evolution, resulting in an operation Z(t) on a state ψ, should satisfy the property

Z(t2) Z(t1) = Z(t1 + t2). (17.1)

The argument is as follows. If the system evolves in time t1 and is stopped, then evolves further at time t2, since the process has no memory, the total evolution should be as if the system evolved from the initial state to a state at t1 + t2 independently of the fact that it was done in two stages (Piron, 1976). Since the process is irreversible, the operator Z(t) may have no inverse. Such an evolution is called a semigroup. As we shall see, it is not possible to obtain such an evolution law in the framework of the standard quantum theory (Horwitz et al., 1971), but recently much work has been done, and methods have been developed, based on ideas of Sz.-Nagy and Foias (1976), such as the theory of Lax and Phillips (1967) and its extension to the quantum theory (Strauss et al., 2000) in which semigroup evolution can be achieved.