From Dynamics to Thermodynamics
Stefano Olla (University of Paris-Dauphine)

Thermodynamics is one of the most established and successful physical theory, applied to most macroscopic system that satisfy the 0-the principle, i.e. the existence of equilibrium states. But the "connection" to microscopic dynamics following the laws of mechanics (classical or quantum) is still controversial.   The classical approach is to understand thermodynamics as a limiting process, where time is rescaled with space, that follows the evolution of slow observables (energy, volume,...). These observables typically characterize the equilibrium states. Such limits are usually called hydrodynamics limits and quasi-static limits. The main point is to use the ergodicity and the mixing properties of the "large" microscopic dynamics in order to establish this separation of scales and the corresponding local equilibrium, described by statistical mechanics. I will illustrate the mathematical program (still open) to obtain such limits for the most simple model, the one dimensional Fermi-Pasta-Ulam chain of oscillators.  Depending on the external agents acting on the system (heat bath or forces) we obtain in the large space-time limit, the thermodynamic isothermal or adiabatic transformations from one equilibrium to an other. The completion of this program will require main mathematical advances in ergodic theory, and in the analysis of non-linear partial differential equations for conservation laws. Some results are obtained by stochastic perturbations of the microscopic dynamics that provide the ergodic properties required. These stochastic perturbations can be interpreted as the result of chaotic behavior of other degrees of movements in a faster time scale.