Statistical Mechanics

Aspects of Ergodic, Qualitative and Statistical Theory of Motion

“…The theory of the ergodic decomposition exposed here is substantially taken from Ruelle, [Ru66]; see also the book of Ruelle, [Ru69]. Continuing the analysis of general structural properties of motions of an invertible discrete dynamical system (Ω, S) we shall now discuss the foundations of the notion of complexity of motions on Ω and of its theory.…”

“…The properties discussed in this section are well known, see for instance p. 178 in [Ru69]. There are various other properties of entropy and mainly its extensions to "non-commutative cases" that are less simple and at times quite deep; see [We79], for a review.…”

“…The contents of this section are taken from various sources, for instance from Ch. 7, §3,4 of [Ru69], p. 450 of [DS58], [LR68]. See also the bibliographical note to §5.1.…”

“…The following key restriction will be assumed A natural adaptation, see [GK70], of an original method of proof of smoothness of the functions in (7.1.1), [Ru69], led to the following algebraic approach, which has become widely known as cluster expansion after having been rediscovered several times in the span of a decade or two.…”

“…and in the East (thanks mainly to the works of Dobrushin, Minlos, Sinai, etc.). The more or less definitive formulation of the notion and of the basic properties of a Gibbs distribution can be found in the classical papers [Do68a], [Do68b], [Do68c], [Do69], [Ru69] and [LR69].…”