Macroscopic Non-Equilibrium Behavior of Chains of Oscillators
The purpose of the course is to give an introduction to the
mathematical problem of the deduction of the macroscopic behavior
of
chain of anharmonic oscillators. Some rigorous
mathematical results
can be obtained if the dynamics of these systems are perturbed by
noise (eventually conservative of energy and momentum).
Introduction: chain of anharmonic oscillators,
thermodynamic
properties, equilibrium ensembles (microcanonical and
grancanonical
measure, pressure etc.). Historical of the subject: Fourier, Debye,
Peierls, Fermi-Pasta-Ulam,
Toda, ... Examples: harmonic, FPU,
rotors.... Stochastic perturbations of the dynamics.
Hydrodynamic behavior: Euler equations from
hyperbolic space-time
scaling by relative entropy method.
Stationary non-equilibrium states. Thermal
conductivity. Linear
response and Green-Kubo formulas. Fourier's Law and Equilibrium
fluctuations.
Anomalous thermal conductivity, superdiffusion.
Numerical and
rigorous results.
Phonon Boltzmann equation. Some rigorous results.
Stefano Olla
CEREMADE, UMR-CNRS 7534
Université
Place du Maréchal De Lattre De Tassigny
75775 PARIS CEDEX 16 -
phone: (33)(0)1 44054616
fax: (33)(0)1 44054599
olla@ceremade.dauphine.fr
http://www.ceremade.dauphine.fr/~olla