abstract:
Non-equilibrium statistical mechanics aims at describing the macroscopic
properties of systems which are in contacts with two thermal baths starting
from simple microscopic models made of interacting particle systems. We will
give an introduction to this largely open problem by considering models in
one spatial dimensions, i.e. chains of particles with nearest neighbor
interaction.
First we will describe the problem of deriving Fourier law in Hamiltonian
models and we will show how a diverging heat conductivity arises in
non-linear oscillators chains. A major issue concerns the role played by
conservation laws. We will exhibit a family of Hamiltonian models for heat
conduction (with and without momentum conservation) for which Fourier law is
verified in one dimension.
In the high temperature limit, from the Hamiltonian model a stochastic
model is derived which is analytically solvable through the construction of
a dual representation. The dual particle process describes the evolution of
all correlation functions. An exact expression for the covariance of energy
exhibits long-range correlations in the presence of a current. If time
permits we will also discuss the connection of this model with the simple
symmetric exclusion process. These are evident in the language description
of quantum spin chains.
(Joint work with J. Kurchan and F. Redig).