zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Thermodynamic formalism. The mathematical structures of classical equilibrium. Statistical mechanics. With a foreword by Giovanni Gallavotti. (English) Zbl 0401.28016
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.

MSC:
82-01Textbooks (statistical mechanics)
37D35Thermodynamic formalism, variational principles, equilibrium states
37-01Instructional exposition (Dynamical systems and ergodic theory)
28D20Entropy and other measure-theoretic invariants
37D20Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37A60Dynamical systems in statistical mechanics
37C10Vector fields, flows, ordinary differential equations
54H20Topological dynamics
82B05Classical equilibrium statistical mechanics (general)
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
81T08Constructive quantum field theory