Ruelle D. Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics

Cambridge: Cambridge University Press. – 2004. – 194 p. (Cambridge Mathematical Library) Thermodynamics is still, as it always was, at the center of physics, the standardbearer of successful science. As happens with many a theory, rich in applications, its foundations have been murky from the start and have provided a traditional challenge on which physicists and mathematicians alike have tested their latest skills. Ruelle’s book is perhaps the first entirely rigorous account of the foundations of thermodynamics. It makes heavier demands on the reader’s mathematical background than any volume published so far. It is hoped that ancillary volumes in time will be published which will ease the ascent onto this beautiful and deep theory; at present, much of the background material can be gleaned from standard texts in mathematical analysis. In any case, the timeliness of the content shall be ample reward for the austerity of the text. Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, and the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the computational basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on physics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this edition by the author. Contents
Foreword to the first edition page
Preface to the first edition
Preface to the second editionGibbs states: complements
Translation invariance. Theory of equilibrium states
Connection between Gibbs states and Equilibrium states
One-dimensional systems
Extension of the thermodynamic formalism
Statistical mechanics on Smale spaces
Miscellaneous definitions and results
Topological dynamics
Convexity
Measures and abstract dynamical systems
Integral representations on convex compact sets
Open problems
Flows
Update of open problems