Turbulent motion and the structure of chaos. A new approach to the statistical theory of open systems. Transl. from the Russian by Alexander Dubroslavsky
The aim of the author is to develop explanations of two classes of phenomena: chaos and organization, which occur in many systems. In systems consisting of a large number of individual components, these phenomena are somehow associated with the loss of correlation or coherence between the motions of the individuals on one hand, and on the other, with the development of correlation or coherence between the motions of the individuals of the system. Of course, the whole problem is to find the right order parameters, i.e., the right functions of the microscopic or individual coordinates, in terms of which chaos and organization can be characterized. The present book could be regarded as a review of the work of the Russian school of statistical physicists, written with the goal mentioned above in mind. It is written in physicist style: lots of hand waving, at times quite imprecise let alone nonrigorous. (Mathematicians remember that the issue is not the lack of rigour. Statements should be precise and the aim is to explain phenomena, not to prove theorems.) A long first chapter is devoted to two issues: to establish kinetic equations (i.e. equations for the time evolution of the probability density of finding a particle with given position and velocities) starting from the exact evolution equations describing the motion of the whole collection, plus perhaps some phenomenological by hypothesis as to how to account for collisions. The mysterious notion of (Gibbs) ensemble is mentioned but never explained. I'd bet (a dinner) that very few books in statistical physics ever explain it. It corresponds to the way probability theory was done in the pre-Kolmogorov days. The second topic dealt with is the various notions of entropy and their possible import for understanding organization or the lack of it. In the second chapter the analysis of the kinetic equations is carried out and the third chapter is devoted to various aspects of fluctuation- dissipation relations. When the rate of change of some relevant coordinate is subject to a linear damping plus a random forcing, a relation exists between the deviations of the mean (fluctuations), the decay rate (due to the damping (dissipation)) and the mean equilibrium value of an ``energy'' like quantity. (If you press me, I'd interpret these relationships as a way of obtaining the mean equilibrium value of the ``energy''.) Chapter 4 contains a description of the relationship between Langevin- like equations and Fokker-Planck equations. My only comment is that physicists should learn the basics of stochastic calculus. The gist of Chapters 5 and 6 is to provide specific examples in which the increase or decrease of entropy, as some controlling parameter is varied keeping the mean value of the energy constant, corresponds with the increase of chaos or organization in the system. You read it and decide yourself whether the goals are archieved.
relationship between Langevin-like equations and Fokker-Planck equations
scientific article; zbMATH DE number 53322