Luc Rey-Bellet (University of Massachusetts)

Title:  Thermodynamics of non-equilibrium steady states, entropy production and fluctuations.


Physically a non-equilibrium steady state is obtained by driving and maintaining a system out of equilibrium by a combination of thermostats and external non-Hamiltonian driving forces.  Mathematically the non-equilibrium steady state is characterized by time-reversal symmetry breaking, i.e., lack of detailed balance. Entropy production  is a direct observable which measures this symmetry breaking and whose fluctuations turn out to exhibit remarkable universal properties: the Gallavotti-Cohen fluctuation theorem which is a far from equilibrium generalization of Kubo formula and  Onsager relations.

This will discussed in great generality both for deterministic and stochastic dynamics.  Contrary to equilibrium statistical mechanics where the Gibbs ansatz provide a a detailed description of the steady state,  for nonequilibrium steady states a detailed knowledge of the dynamics and its ergodic properties is  necessary to understand  the steady state.  These properties are naturally formulated in terms of ergodic  and probabilistic properties for the dynamics of the  systems: ergodicity and mixing, central limit theorem and large deviations.

The first part of  our lecture will consist of the exposition of a general framework to understand the structure of steady states.  The second part of the lectures will deals with a number of examples which have been analyzed rigorously : a) Nonequilibrium molecular  dynamics. b) Thermostated Hamiltonian systems. c) Jump processes.  Finally in the third part of the lecture we introduces and analyze a large new class of nonequilibrium models based on game theoretic considerations.