Small random perturbations of dynamical systems and the definition of attractors
RESEAUX
SYSTEMES DYNAMIQUES
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
THEORIES NON LINEAIRES
Abstract : The strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction. In view of this we study a new definition of attractors based on ideas of Conley. We argue that the attractors observed in the presence of small random perturbations correspond to this new definition.
RUELLE
P/81/23
IHES
03/1981
A4
15 f.
EN
TEXTE
PREPUBLICATION
P_81_23.pdf
1981
A Heuristic theory of phase transitions
RESEAUX
SYSTEMES COMPLEXES
ESPACES DE BANACH
TRANSITIONS DE PHASE
Abstract : Let Z be a suitable Banach space of interactions for a lattice spin system. If n+1 thermodynamic phases coexist for ?0 ?Z, it is shown that a manifold of codimension n of coexistence of (at least) n+1 phases passes through ?0. There are also n+1 manifolds of codimension n?1 of coexistence of (at least) n phases; these have a common boundary along the manifold of coexistence of n+1 phases. And so on for coexistence of fewer phases. This theorem is proved under a technical condition (R) which says that the pressure is a differentiable function of the interaction at ?0 when restricted to some codimensionn affine subspace of Z. The condition (R) has not been checked in any specific instance, and it is possible that our theorem is useless or vacuous. We believe however that the method of proof is physically correct and constitutes at least a heuristic proof of the Gibbs phase rule.
RUELLE
P/76/149
IHES
10/1976
A4
25 f.
EN
TEXTE
PREPUBLICATION
P_76_149.pdf
1976
Some Remarks on the location of zeroes of the partition function for lattice systems
RESEAUX
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
SYSTEMES NON LINEAIRES
THEORIE DES TREILLLIS
Abstract : We use techniques which generalize the Lee-Yang circle theorem to investigate the distribution of zeroes of the partition function for various classes of classical lattice systems.
RUELLE
P/72/29
IHES
1972
A4
23 f.
EN
TEXTE
PREPUBLICATION
P_72_29.pdf
1972
Bifurcations in the presence of a symmetry group
RESEAUX
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
SYSTEMES NON LINEAIRES
GROUPES DE SYMETRIE
ELECTROMAGNETISME
Abstract : Let the origin O of a Banach space E be a fixed point of a diffeomorphism or a critical point of a vector, assuming equivariance under a linear group of isometries of E. Explicit techniques are presented to handle the generalization of the Hopf bifurcation to this equivariant situation.
RUELLE
P/72/20
IHES
07/1972
A4
33 f.
EN
TEXTE
PREPUBLICATION
P_72_20.pdf
1972
On the Nature of turbulence
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
INFORMATIQUE QUANTIQUE
SYSTEMES NON LINEAIRES
Abstract : A mechanism for the genetration of turbulence and related phenomena in dissipative systems is proposed.
RUELLE
TAKENS
P/70/X036
IHES
05/1970
A4
16 f.
EN
TEXTE
PREPUBLICATION
P_70_X036.pdf
1970
Superstable interactions in classical statistical mechanics
RESEAUX
PHYSIQUE STATISTIQUE
FONCTIONS CONTINUES
FONCTIONS DE CORRELATION
EQUILIBRE
Abstract : We consider classical systems of particles inv dimensions. For a very large class of pair potentials (superstable lower regular potentials) it is shown that the correlation functions have bounds of the form ?(x1,...,xn)??n. Using these and further inequalities one can extend various results obtained by Dobrushin and Minlos [3] for the case of potentials which are non-integrably divergent at the origin. In particular it is shown that the pressure is a continuous function of the density. Infinite system equilibrium states are also defined and studied by analogy with the work of Dobrushin [2a] and of Lanford and Ruelle [11] for lattice gases.
RUELLE
P/70/X033
IHES
1970
A4
23 f.
EN
TEXTE
PREPUBLICATION
P_70_X033.pdf
1970
Integral representation of states on a C*-algebras
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
Abstract : Let E be the compact set of states on a C?-algebra U with identity. We discuss the representations of a state ? as barycenter of a probability measure ? on E. Examples of such representations are the central decomposition and the ergodic decomposition. They are associated with an Abelian von Neumann algebra B in the commutant ?(U)? of the image of U in the representation canonically associated with ?. This situation is studied in general and a number of applications are discussed.
RUELLE
P/70/X029
IHES
1970
A4
30 f.
EN
TEXTE
PREPUBLICATION
P_70_X029.pdf
1970
Statistical mechanics of a one-dimensional lattice gas
MECANIQUE STATISTIQUE
RESEAUX
GAZ
Abstract : We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result of Van Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be a K-system.
RUELLE
P/67/X016
IHES
1967
A4
10 f.
EN
TEXTE
PREPUBLICATION
P_67_X016.pdf
1967
A Variational formulation of equilibrium statistical mechanics and the Gibbs phase rule
ENTROPIE
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
Abstract : It is shown that for an infinite lattice system, thermodynamic equilibrium is the solution of a variational problem involving a mean entropy of states introduced earlier [2]. As an application, a version of the Gibbs phase rule is proved.
RUELLE
P/67/X012
IHES
1967
A4
6 f.
EN
TEXTE
PREPUBLICATION
P_67_X012.pdf
1967
Condensation of lattice gases
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
CONDENSATION
GAZ
RUELLE
GINIBRE
GROSSMANN
P/66/X005
IHES
1966
A4
8 f.
EN
TEXTE
PREPUBLICATION
P_66_X005.pdf
1966
Mean entropy of States in classical statistical mechanics
ENTROPIE
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
Abstract : The equilibrium states for an infinite system of classical mechanics may be represented by states over Abelian C* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated : linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with the KOLMOGOROV-SINAI invariant of ergodic theory.
RUELLE
ROBINSON
P/66/04
IHES
1966
A4
25 f.
EN
TEXTE
PREPUBLICATION
P_66_04.pdf
1966
Polymers and g ? ? ? 4 theory in four dimensions
POLYMERES
DIMENSIONS
RESEAUX
GROUPE DE RENOMALISATION
PERTURBATIONS
ANALYSE NUMERIQUE
METHODE DE MONTE-CARLO
FROHLICH
ARAGAO DE CARVALHO
CARACCIOLO
P/82/39
IHES
07/1982
A4
36 f.
EN
TEXTE
PREPUBLICATION
P_82_39.pdf
1982