Probability estimates for continuous spin systems

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

Superstable interactions in classical statistical mechanics

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 $$\varrho (x_1 ,...,x_n ) \leqq \xi ^n$$ . 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.     

Dimension of the commutant for theSU(N) affine algebras

Explicit formulae are obtained, giving the number of independent matrices which commute with the matricesS andT describing the modular transformations of theSU(N) affine characters.Chercheur IISN     

Zeta-functions for expanding maps and Anosov flows

Given a real-analytic expanding endomorphism of a compact manifoldM, a meromorphic zeta function is defined on the complex-valued real-analytic functions onM. A zeta function for Anosov flows is shown to be meromorphic if the flow and its stable-unstable foliations are real-analytic.     

Statistical mechanics of a one-dimensional lattice gas

We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan 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 aK-system.     

Large volume limit of the distribution of characteristic exponents in turbulence

For spatially extended conservative or dissipative physical systems, it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity. In the case of a turbulent viscous fluid, however, this simple idea is complicated by the phenomenon of intermittency. In the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation. These bounds have a reasonable large volume behavior. For two-dimensional fluids a particularly striking result is obtained: the total information creation is bounded above by a fixed multiple of the total energy dissipation (at fixed viscosity). The distribution of characteristic exponents is estimated in an intermittent model of turbulence (see [7]), and it is found that a change of behavior occurs at the valueD=2.6 of the self-similarity dimension.