Quantum Theory of Irreversibility: Open Systems and Continuum Mechanics

William C. Schieve, Lawrence Paul Horwitz

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 A recent critical overview of the “modern view” is in the unpublished thesis of B. C. Bishop of the University of Texas Philosophy Department (Bishop, 1999).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).

A. Pérez-Madrid

Chiara Marletto, Vlatko Vedral, Laura T. Knoll +7 more

Is it possible to fit irreversibility exactly in a universe whose laws are time-reversal symmetric? This vexed open problem, dating back to the origin of statistical mechanics, is crucial for the realisability of logically irreversible computations such as erasure. A well-known type of irreversibility is epitomised by Joule's experiment, involving a volume of fluid and a stirrer powered by mechanical means, e.g. a suspended weight: while it is possible to construct a cycle heating up the water by mechanical means only, it is impossible to cool the water down via a cycle that likewise utilises mechanical means only. This irreversibility is of particular interest because, unlike others, it is based on a system that performs some transformation, or task, operating in a cycle. First introduced via a thermodynamic cycle (e.g. Carnot's), this idea was generalised by von Neumann to a constructor - a system that can perform a given task on another system and crucially retains the ability to perform the task again. On this ground, a task is possible if there is no limitation to how well it can be performed by some constructor; it is impossible otherwise. Here we shall define a irreversibility, generalising Joule's, requiring that a task $T$ is possible, while the transpose task (where $T$'s input and output are switched) is not. We shall demonstrate the compatibility of this constructor-based irreversibility with quantum theory's time-reversal symmetric laws, confining attention to an example using the universal quantum homogeniser. This result is an essential further step for understanding irreversibility within quantum theory, one of the most fundamental theories of physics we posses at present. We will also simulate the irreversibility experimentally with high-quality photon qubits.

G. C. Ghirardi

Boris B. Kadomtsev

David W. Snoke

This chapter begins the final section of the book, which presents both review and new results of original research on decoherence and measurement theory. In this chapter, it is shown that normal quantum mechanics can lead to irreversible behavior in an open system, in contrast to the expectation of the Poincaré theorem that predicts repeating, cyclical behavior for all closed systems. The quantum Boltzmann equation, which implies the famous H-theorem that underlies all statistical mechanics, is derived.