Showing papers by "David Ruelle published in 2012"

Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics

Abstract: The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems describing units of fluid of smaller and smaller spatial extent. These units are macroscopic but have a few degrees of freedom, and they can be studied by the methods of (microscopic) nonequilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specifically, we obtain the formula for the exponents of the structure functions (). The meaning of the adjustable parameter κ is that when an eddy of size r has decayed to eddies of size , their energies have a thermal distribution. The above formula, with is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture that can thus also be used in related problems.

A Mechanical Model for Fourier’s Law of Heat Conduction

Abstract: Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier’s law for heat conduction from the laws of mechanics remains thus a major unsolved problem. In this note we present a deterministic mechanical model of a heat-conducting chain with nontrivial interactions, where kinetic energy fluctuations at the nodes of the chain are removed. In this model the derivation of Fourier’s law can proceed rigorously.

Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems

Abstract: We consider a class of Ising spin systems on a set Λ of sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the |Λ| Lee-Yang zeros in the complex z = e2βh plane, where β is the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of β. In some cases one obtains, in an appropriately taken β↗ ∞ limit, a gas of hard objects on a set Λ′; the fugacity for the limiting system is a rescaling of z and the Lee-Yang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the negative real axis for all β. One zero-temperature limit o...

Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems

Abstract: We consider a class of Ising spin systems on a set \Lambda of sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the |\Lambda| Lee-Yang zeros in the complex z = exp{2\beta h} plane, where \beta is the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of \beta. In some cases one obtains, in an appropriately taken \beta to infinity limit, a gas of hard objects on a set \Lambda'; the fugacity for the limiting system is a rescaling of z and the Lee-Yang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the negative real axis for all \beta. One zero-temperature limit of this type, for example, is a monomer-dimer system; our results thus generalize, to finite \beta, a well-known result of Heilmann and Lieb that the Lee-Yang zeros of monomer-dimer systems are real and negative.