# Ruelle, David

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### The Stability of many-particle systems

Abstract : It is shown that a quantal or classical system of N particles of distinct species ?,? = 1, 2, … ? interacting through pair potentials ???(r) are stable, in the sense that the total energy is always bounded below by ?NB, provided ???(r)…

### Extremal invariant states

Abstract : A number of results are derived which are pertinent to the description of physical systems by states on C*-algebras invariants under a symmetry group. In particular an integral decomposition relevant to the study of lower symmetry is…

### The States of classical statistical mechanics

Abstract : a state of an infinite system in classical statistical mechanics is usually described by its correlation functions. We discuss here other description in particular as 1) a state on a B*-algebras, 2) a collection of density distributions,…

### Integral représentations of invariant states on B*-algebras

Abstract : Let ?? be a B* algebra with a group G of automorphisms and K be the set of G?invariant states on ??. We discuss conditions under which a G?invariant state has a unique integral representation in terms of extremal points of K, i.e.,…

### Mean entropy of States in classical statistical mechanics

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…

### A Variational formulation of equilibrium statistical mechanics and the Gibbs phase rule

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.

### Almost periodic states on C*-algebras *)

Abstract : Given a C*-algebra O with a group of automorphisms, we define and study almost periodic states on O. A natural decomposition of such states is introduced and discussed.

*) These notes are a result of discussions between S. Dpolicher,…

*) These notes are a result of discussions between S. Dpolicher,…

### Statistical mechanics of a one-dimensional lattice gas

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…

### Some Remarks on the ground state of infinite systems in statistical mechanics

Abstract : We investigate the ground states of infinite quantum lattice systems. It is shown in particular that a positive energy operator is associated with these states.

### Symmetry breakdown in statistical mechanics

Lecture given at the Ecole d'Eté de Physique Théorique. Cargèse, Corsica, 1969.

Abstract : We discuss the general problem of symmetry beakdown in the algebraic approach to statistical mechanics. We consider in particular the case of classical…

Abstract : We discuss the general problem of symmetry beakdown in the algebraic approach to statistical mechanics. We consider in particular the case of classical…

### Integral representation of states on a C*-algebras

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…

### Superstable interactions in classical statistical mechanics

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…

### On the Nature of turbulence

Abstract : A mechanism for the genetration of turbulence and related phenomena in dissipative systems is proposed.

### Bifurcations in the presence of a symmetry group

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…

### Some Remarks on the location of zeroes of the partition function for lattice systems

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.

### Zeta-functions for expanding maps and Anosov flows

Abstract : Given a real-analytic expanding endomorphism of a compact manifold M, a meromorphic zeta function is defined on the complex-valued real-analytic functions on M. A zeta function for Anosov flows is shown to be meromorphic if the flow and…

### On Manifolds of phase coexistence

Abstract : Using a theorem on convex functions due to Israel, it is shown that a point of coexistence of n+1n+1 phases cannot be isolated in the space of interactions, but lies on some infinite dimensional manifold.

### Probability estimates for continuous spin systems

Abstract : Probability estimates for classical systems of particles with superstable interactions [1] are extended to continuous spin systems.

### A Heuristic theory of phase transitions

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…

### Applications conservant une mesure absolument continue par rapport à dx sur [0, 1]

Abstract : Sufficient conditions are given such that a differentiable, noninvertible, map g : [0,1]~[0,1] leaves invariant a measure absolutely continuous with respect to the Lebesgue measure. In particular, this is shown to be the case for…

### Dynamical systems with turbulent behavior

[Text of a talk presented at the International Mathematical Physics Conference in Rome, 1977]

### Sensitive dependence on initial conditions and turbulent behavior of dynamical systems

Abstract : The asymptotic behavior of differentiable dynamical systems is analyzed. We discuss its descriptoin by asymptotic measures and the turbulent behavior with senditive dependence on initial condition.

### Ergodic theory of differentiable dynamical systems

Abstract : If f is a C1+? diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost verywhere with respect to every f-invariant probability measure on M. These stable manifolds are smooth but do not in general…

### On the Measures which describe turbulence

Abstract : One expects that the average behavior (over large times) for hydordynamics and other natural phenomena is described by certain asymptotic measures on phase space. If initial conditions in a set of zero Lebesgu measure ar discarded, the…

### Measures describing a turbulent flow

Abstract :Recent attempts at understanding hydrodynamic turbulence have used the ideas of strange attractors, characteristic exponents and stable manifolds for differentiable dynamical systems in finite dimensional spaces. This use was somewhat…

### Characteristic exponents and invariant manifolds in Hilbert space

Abstract : The multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds (Pesin theory) are extended to differentiable dynamical systems on Hilbert manifolds under some compactness assumptions. The results…

### Small random perturbations of dynamical systems and the definition of attractors

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…

### Do there exist turbulent crystals ?

Abstract : We discuss the possibility that, besides periodic and quasiperiodic crystals, there exist turbulent crystals as thermodynamic equilibrium states at non-zero temperature. Turbulent crystals would not be invariant under translation, but…