
Author(s) 
Petrina, D.Ya. Gerasimenko, V.I. Malyshev, P.V. 

Title  Mathematical foundations of classical statistical mechanics, Continuous systems 
Publisher  Gordon and Breach 
Year of publication  1989 
Reviewed by  Günter Vojta 
The foundations of statistical mechanics are a topic which has been in the focus of scientific interest since the times of Boltzmann. The authors of this monograph treat one distinct problem with one method. The problem is the foundation of classical statistical mechnics of continuous systems, i.e. of systems of many particles moving in a continuous phase space as opposed to lattice systems; the field of statistical continuum mechanics, viz. the statistical theory of continuous media, is not touched here. The method consists in the solution of the Bogolyubov equations, an infinite system of integrodifferential equations for the manyparticle distribution functions; in literature this chain of equations is generally known as BogolyubovBornGreenKirkwoodYvon (BBGKY) hierarchy.
The text is divided into five chapters. The first chapter contains basic facts on the dynamics of finite particle systems: classical Hamiltonian mechanics, the Liouville equation, and the evolution operator. In the second chapter the Bogolyubov equations are introduced and the Cauchy problem for them is formally solved. Next there follow two chapters on equilibrium states of systems of infinitely many particles in the frame of the canonical and the grand canonical ensembles. Main subjects are here the existence and the uniqueness of limit distribution functions as well as spectral and topological properties of evolution operators. Finally the fifth chapter treats (for the first time in book form) the thermodynamic limit for nonequilibrium systems. This is possible, however, only for spatially homogeneous systems, the whole field of proper transport processes as a paradigm for nonequilibrium systems being thus outside the scope of the book.
The valuable list of references contains 170 citations with titles. The subject index is too short, a list of symbols is inadequate and by far too short.
The text is written in a lively style, well translated from the Russian original and well printed. It contains much physical motivation and discussion. Unfortunately, recent developments of chaos theory and ergodic theory so important for the foundations of statistical mechanics are not taken into account. Likewise, other methodologies e.g. of the Prigogine school, modern projection operator techniques or the information theoretical approach are not mentioned at all. Within these restrictions, the book gives a good introduction and thorough discussion of an important topic of theoretical physics. It is certainly of interest for research workers, teachers and postgraduate students of theoretical physics and mathematics and should be bought at least by libraries.