Encyclopedia of Mathematics and its Applications. Vol. 5. Reading, Massachusetts: Addison-Wesley Publishing Company. XIX, 183 p. $ 21.50 (1978).

As aim by the editors of the encyclopedia, the mathematical framework of the title problem, here formulated for classical lattice spin systems, is presented in a pure and rigorous manner. Physical motivations are only shortly sketched and must be taken from elsewhere. – In the introduction some basic ideas of the thermodynamic formalism are reviewed and related to the main context. The first two chapters develop the structure of Gibbs states, their relation to thermodynamic limits, and their behaviour under lattice morphisms as well as conditioning. Chapter three introduces equilibrium states under assumption of translational invariance and contains some general results on phase transitions. The link between Gibbs states and equilibrium states is established in chapter four. Application to one-dimensional lattice systems is given in chapter five. The last two chapters extend the formalism to the case when the configuration space is a general compact metrizable space with homeomorphic action of

${Z}^{v}$. Some further background and open problems are collected in appendices. – Concise presentation is balanced by clear notation. The level is generally advanced and especially some knowledge in functional analysis is presupposed. Since wide-spread literature of the last twenties has been elaborated and is represented in an unified way, this book is a valuable tool for anybody working on this and related fields.