18 - Quantum statistical mechanics, extended

Summary

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 A recent critical overview of the “modern view” is in the unpublished thesis of B. C. Bishop of the University of Texas Philosophy Department (Bishop, 1999).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