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Nonlinear nonequilibrium thermodynamics II. Advanced theory. (English) Zbl 0808.73002

Springer Series in Synergetics. 59. Berlin: Springer-Verlag. xii, 223 p. (1994).
This second volume (“Advanced theory”) of R. Stratonovich’s book “Nonlinear nonequilibrium thermodynamics”, together with the first volume [R. Stratonovich, Nonlinear nonequilibrium thermodynamics I. Linear and nonlinear fluctuation – dissipation theorems (1992; Zbl 0786.73002)], represents a new revised and enlarged edition of the original work [R. Stratonovich, Nonlinear nonequilibrium thermodynamics (1985; Zbl 0656.73003)].
The author – one of the founders of this new branch of statistical physics – offers the first coherent, systematic, rigorous and detailed overview of the complex and rapid-developing interdisciplinary area of (non)linear nonequilibrium thermodynamics. While the first volume deals with the derivation and application of different fluctuation-dissipation theorems, this second volume is devoted to various theoretical aspects of the physics of classical and quantum nonlinear (open) systems.
The first chapter (“Generating equations of non-Markov theory”) presents (non)Markov generating classical and quantum equations from which various systems of (non)linear fluctuation-dissipation relations (FDRs) may be obtained. It is also discussed the difficult problem of a quantum generalization of the concept of Markov process, and the author suggests two different solutions to it.
Chapter 2 (“Nonequilibrium thermodynamics of open systems”) is mainly based upon Markov techniques. It covers a wide variety of open systems that are analyzed from various points of view (fluctuations, stability, generating equations, \(H\)-theorems, nonequilibrium stationary states, nonequilibrium kinetic phase transitions etc.). It can be noticed that the theory may no longer draw on the familiar results of equilibrium thermodynamics, and that new and specific concepts and physical quantities have to be defined.
The third chapter (“The Kirchhoff’s form of fluctuation-dissipation relations”) is devoted to a very spectacular application of the general nonequilibrium thermodynamic laws: the nonlinear interaction of waves with reflectors and scatterers. There are obtained generalizations of the Kirchhoff law to coherent waves during nonlinear interaction processes (scattering, reflection, absorption). These generalized laws are in fact linear, quadratic and cubic FDRs for the particular system of coherent waves interacting with a physical body.
The last chapter (“Method of projection in space of phase-space distributions and irreversible processes”) analyses the motion in phase space for a physical system. A very modern and interesting point of view is adopted: to express the projection operator of the physical Markov process in terms of microdynamics (in keeping with the chosen space-time coarse-graining techniques). The method of projection operators in phase space allows the derivation of some universal thermodynamic relations and is a very performant way to connect microdynamics with nonequilibrium macroscopic theory.
In order not to interrupt the continuity of the exposition, some mathematical results are given in the appendices.
We have to underline the level of rigour, the modernity and the great variety of the mathematical tools used in the book (that are borrowed from probability theory, stochastic processes, (partial) differential equations, Fourier and Laplace transform theory, distribution theory, operator theory, etc.); they correspond to the highest level encountered in theoretical physics courses. We have to notice that the very general nature of statistical thermodynamics (the possibility to apply it to the most various electrical, thermal, mechanical, chemical systems) demands such a variety and “elasticity” of the mathematical methods that are employed.
The bibliography to each chapter is not exhaustive, but it is essential for the considered problems. Many of the papers indicated are original contributions of the author.
The clarity of the exposition, as well as the numerous illustrative examples recommend this book not only to theoretical physicists, but also to applied scientists, who will have the possibility to directly apply the (major) results here obtained to various problems of practical interest. On the other hand, theoretical physicists will be “spoiled” by the rigour, precision and fineness in raising and solving the problems of this very difficult and complex field of nonequilibrium nonlinear thermodynamics.
Reviewer: I.Bena (Iaşi)

MSC:

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74A15 Thermodynamics in solid mechanics
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
82Cxx Time-dependent statistical mechanics (dynamic and nonequilibrium)
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