Gibbs Measures and Phase Transitions, Second Edition by Hans-Otto Georgii

By Hans-Otto Georgii

From a evaluation of the 1st version: ""This booklet […] covers extensive a large diversity of themes within the mathematical concept of part transition in statistical mechanics. […] it's in truth one of many author's acknowledged goals that this entire monograph should still serve either as an introductory textual content and as a reference for the expert."" (F. Papangelou, Zentralblatt MATH) the second one variation has been prolonged by means of a brand new part on huge deviations and a few reviews at the more moderen advancements within the quarter.

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The ^-specification y* = (y^Ae^ = P*7'-. i s called the Gibbsian specification for and X. Each random field /i e ^() = ^(y*) is called a Gifefes measure or a Gj'fefes random field for ) of all Gibbs measures for O can, and should, be regarded as a proper mathematical description of the set of all possible equilibrium states for a physical system that consists of a huge number of components which are coupled together by O.

By hypothesis there is a local function u with || JïA — u|| ^ e. We put v = e~u. Then XAv :g e£ZA < oo. We consider the local function g = v/XAv. (|l-exp(fl*-u)|) ^ 2(e£ - 1). Now if / e if then XA(fg) is local and htf - AA(/0)|| ^ 11/11 ||AA(|p* - Çe£A and we have l{ffA=ç} e i£ and yAl{ffA=C} e S. Thus pA e if'. 31) below. 24). 25) Example. Let X e Ji{E,é>) and $ be a uniformly convergent Xadmissible potential. 24) (b) shows that y® is quasilocal.

29) 9(y) = {n e 0>(Q, &)\\L = p A (Mv) for all A e ^ } . This follows immediately from the equation pA(fiXA)(A) = j p(dco) J XA(dC\œ)pA(0 = MM) A which holds for each fi e ^(Q, IF), A e IF, and A e if. In particular, if p is positive then for all \i e <&(y), A e if, and B e SA we have n(aA e B) = 0 if and only if ^A(B) = 0. Thus any two measures in ^(y) are mutually absolutely continuous on J*A for all A e if. (3) If I = rX for some measurable function r > 0 on E then p defined by pA(co) = pA(co) / F] rfa) (A e if, œ e Q) / leA is a 1-modification with pl_ = p^,..

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