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Information theory and stochastics for multiscale nonlinear systems. (English) Zbl 1082.60002
CRM Monograph Series 25. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3843-1/hbk). viii, 133 p. (2005).
Loosely speaking, the fluctuation-dissipation theorem (FDT) states that for a system which is in statistical equilibrium, the mean value of its response to small external perturbations can be obtained by means of suitable correlation functions of the nonperturbed system. This theorem is used to analyze nonlinear systems with complex multiscale structure, and more especially climate systems. A first chapter displays a brief introduction to equilibrium statistical mechanics. In the second chapter, the main one, one recalls the derivation of FDT (Risken), and then one deepens its application to complex nonlinear systems, and above all, one tests the validity of the FDT in such a case. In the third and last chapter, the authors show the kinds of results one may obtain on using a stochastic mode reduction expanded recently by one of them. The Haken’s books on synergetics are nowhere referred to, but their way of thought is present everywhere.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
86A10 Meteorology and atmospheric physics
82Bxx Equilibrium statistical mechanics