3.3.3Relaxation Times of Kinetically Constrained Spin Models with Glassy Dynamics

Cristina Toninelli

Université Paris Sud—CNRS

Cristina.Toninelli@lpt.ens.fr

We discuss kinetically constrained spin models (KCSM), that is interacting particle systems with Glauber-like dynamics in which the creation/destruction of a particle can occur only if the configuration satisfies some local constraints. KCSM were introduced in physical literature to model liquid/glass transition. Numerical simulations show that, as density ρ is increased, they display an anomalously slow dynamics and glassy features including stretched exponential relaxation. We present a new probabilistic technique through which we determine the scaling with the system size of the relaxation time, τ , and we obtain upper and lower bounds for its dependence on ρ. On the one hand, we prove that τ diverges for some models faster than any power law of 1 − ρ as ρ 1. On the other hand, we establish exponential decay of spin-spin time auto-correlation functions for all the models in the ergodic regime. This excludes the stretched exponential relaxation conjectured from simulations, which is due to the rapid divergence of τ .

3.4 Non-equilibrium Statistical Mechanics

Organizers G. Jona-Lasinio (Rome), B. Nachtergaele (Davis)

3.4.1 Current Fluctuations in Boundary Driven Interacting Particle Systems

Claudio Landim

IMPA landim@impa.br

We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.

well known approaches (Weak coupling or master equation approach, LandauerBuettiker scattering approach to independent electron systems, Keldysh formalism and Meir-Wingreen approach to locally interacting Fermions).

3.4.5Derivation of the Gross-Pitaevski Equation for the Dynamics of Bose-Einstein Condensates

Benjamin Schlein

Harvard schlein@math.harvard.edu

In this talk I am going to report on a recent result obtained in collaboration with L. Erdoes and H.-T. Yau. We consider a system of N interacting bosons in the GrossPitaevskii limit, where N tends to infinity and the scattering length a of the pair potential tends to zero so that Na remains constant. In this limit we prove that the macroscopic dynamics of the system is correctly described by the time-dependent Gross-Pitaevskii equation.

3.4.6Energy Transport in One-Dimensional Chains: Predictions from the Phonon Kinetic Equation

Herbert Spohn

TU Muenchen spohn@ma.tum.de

For low density gases in one space-dimension the Boltzmann collision term vanishes. In contrast, for the phonon Boltzmann equation the wave number space is a one-dimensional torus and the kinetic energy is a periodic function. This allows for non degenerate phonon collisions. We investigate the spectrum of the linearized collision operator. For an on-site potential this operator has a spectral gap implying diffusive energy transport, while for the FPU β chain we prove the non integrable decay as t −3/5 for the energy current correlation function. This is joint work with Jani Lukkarinen.