This conference will gather a mixed audience composed of mathematicians and theoretical physicists. We believe that carefully chosen mini-courses can be of great use not only for graduate students, but also for researchers who became more and more specialized due to the complexity of this field.

We shall mainly focus on classical deterministic systems and stochastic systems such as interacting particle systems. Besides we want to organize one specific day focused on three specialized themes namely: KPZ universality, self-organized criticality and the Boltzmann equation.

The two basic paradigms for deterministic (dynamical system) non-equilibrium statistical mechanics are the so-called thermostated systems and open systems.
Thermostated systems are Hamiltonian systems (with finitely many degrees of freedom) driven away from equilibrium by an external (non-Hamiltonian) force and constrained by a deterministic thermostating force to stay on a surface of constant energy.
Open systems are Hamiltonian systems consisting of a « small » Hamiltonian system (with finitely many degrees of freedom) interacting with, say two, ‘large’ reservoirs which are infinitely extended Hamiltonian systems. The reservoirs are initially in thermal equilibrium at distinct temperatures and the temperature differential leads to a steady heat flux from the hotter to the colder reservoir across the small system.

A more abstract approach consists in describing the microscopic time evolution by a general smooth dynamical system, identify nonequilibrium steady states (NESS), and study how these vary under perturbations of the dynamics.
Another important role is played by the so-called Fluctuation Theorems, such as the Evans-Searles and Gallavotti-Cohen Fluctuation Theorems, related to entropy production.

Another modeling of nonequilibrium consists in introducing stochasticity in the micro-dynamics. This leads to markovian models of interacting particles driven by reservoirs and/or bulk forces.  In this context, there are exactly solvable models, such as the symmetric and asymmetric exclusion processes coupled to boundary reservoirs, where one can explicitly compute correlations in the NESS. There is also a general macroscopic approach in which one studies the fluctuations of the density profiles in non-equilibrium, via the theory of hydrodynamic limits and associated large deviations.