By analogy with the formalism of differential and integral calculus, one could think that thermodynamic formalism is a set of relationships between thermodynamic quantities, such as, for example, the equation of state or the variational principle. However, the content of the book by David Ruel, who was probably the first to use this term, shows that we are talking about mathematical methods of statistical physics based on the one introduced in the late 60s by R.L. Dobrushin and, independently, O. Lanford and D. Ruel, the concept of the Gibbs state (synonyms: DLR state, Gibbs measure, Gibbs random field). But that’s not all: today thermodynamic formalism is more likely perceived not even as a part of statistical physics, but as an ideologically close branch of the theory of dynamical systems.

The proposed book by one of the creators of thermodynamic formalism, D. Ruelle, is based on the course lectures given by the author at universities in the USA and France (“Thermodynamic formalism”, 1978). It discusses from a mathematical point of view both traditional issues of classical equilibrium statistical mechanics - the Gibbs distribution, phase transitions and others, as well as related issues of the theory of dynamical systems (symbolic and topological dynamics, entropy, variational principle). In the form of the last two chapters, the publication also included D. Ruel’s later book on dynamic zeta functions (“Dynamical zeta functions for piecewise monotone maps of the interval”, 1994).

It will be useful for mathematicians and physicists , specializing in the field of statistical mechanics and the theory of dynamical systems.

ContentsTheory of Gibbsian statesGibbsian states: additionsTranslational invariance. Theory of equilibrium statesRelation between Gibbsian and equilibrium statesOne-dimensional systemsGeneralization of thermodynamic formalismStatistical mechanics on Smale spacesIntroduction to dynamic zeta functionsPiecewise monotonic mappings

Trans. from English B.M. Gurevich and S.V. Savchenko. - M. - Izhevsk: Institute of Computer Research, 2002. - 288 p. - (Modern Mathematics) - ISBN 5-93972-115-X.