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Zbl 0401.28016
Ruelle, David
Thermodynamic formalism. The mathematical structures of classical equilibrium. Statistical mechanics. With a foreword by Giovanni Gallavotti. (English)
[B] 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.

Display scanned Zentralblatt-MATH page with this review.
[K.-E.Hellwig (Berlin)]

MSC 2000:: *82-01 Textbooks (statistical mechanics)
37D35 Thermodynamic formalism, variational principles, equilibrium states
37-01 Instructional exposition (Dynamical systems and ergodic theory)
28D20 Entropy and other measure-theoretic invariants
37D20 Uniformly hyperbolic systems
37A60 Dynamical systems in statistical mechanics
37C10 Vector fields, flows, ordinary differential equations
54H20 Topological dynamics
82B05 Classical equilibrium statistical mechanics (general)
82B20 Lattice systems
81T08 Constructive quantum field theory

Keywords: entropy; constructive quantum fluid theory; Smale space; spin systems; thermodynamic formalism; Gibbs states; thermodynamic limits; equilibrium states; phase transitions

Cited in: Zbl 1165.37007 Zbl 1062.82001 Zbl 1034.60085 Zbl 0964.37005 Zbl 0930.54029 Zbl 0896.60079 Zbl 0889.28007 Zbl 0816.58030 Zbl 0733.58035 Zbl 0657.58026 Zbl 0668.62059 Zbl 0641.54035 Zbl 0628.58043 Zbl 0618.58028 Zbl 0609.58033 Zbl 0591.60100 Zbl 0575.47009 Zbl 0542.58028 Zbl 0526.28014 Zbl 0438.60087

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	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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