STATISTICAL MECHANICS AND LARGE DEVIATION THEORY

1) Large deviation theory applied to statistical mechanics and complex dynamical systems,

2) Kinetic theory,

3) Computation of large deviations and rare events,

4) Phase transition theory.

### Large deviation
theory applied to statistical mechanics and complex
dynamical systems

Large deviation theory is a key framework for modern
statistical mechanics. It allows to rephrase elegantly
equilibrium statistical mechanics, where thermodynamic
potential (entropy, free energy, and so on) then
appear as large deviation rate functions. It also
generalizes powerfully those concepts for
non-equilibrium problems. Large deviation theory for
complex dynamical systems is a key tool to understand
their statistics: attractors, most probable states,
typical fluctuations, rare events and extremes. We
work at developing large deviation theory and its
applications to statistical mechanics and complex
dynamical systems through various projects:

b) Beyond large deviations: generalisation of the Eyring-Kramers transition rate formula to irreversible diffusion processes. [.pdf]

c) Large deviations in fast–flow systems (systems with two time scales). arXiv:1510.02227, [.pdf]

d) Perturbative approach and Taylor series expansion of the quasi-potential for Freidlin--Wentzell type large deviations (dynamical systems with small noises). [.pdf]

e) Beyond Freidlin-Wentzell theory: Non-classical large deviations for a noisy system with non-isolated attractors, [.pdf]

f) Large deviation theory for the equilibrium statistical mechanics of geophysical flows. (review: F. Bouchet and A. Venaille, Physics Reports, 2009.)

g) Large deviation theory for the equilibrium statistical mechanics of systems with long range interactions. (publication: Barre-Bouchet-Dauxois-Ruffo-JStatPhys-2005.)

Some statistical physics applications of large deviations:

a) Large deviations beyond the Freidlin-Wentzell regime for front dynamics and the Allen--Cahn equation. arXiv:1507.05577, [.pdf]

b) Large deviations for non equilibrium systems with long range interactions, large deviations for the Shinomoto-Kuramoto model of coupled rotators. [.pdf]

c) Large deviations and instantons for bistable turbulent systems. (publications: Bouchet--Laurie--2015 and Bouchet--Laurie-Zaboronski--J StatPhys--2014)

### Kinetic theory

Kinetic theory is a natural way to approach relaxation to equilibrium or non-equilibrium statistical mechanics of physical systems when a small parameter (often the inverse number of particle) gives a natural time scale separation. We have studied fundamental problems in kinetic theory, as well as applications in systems with long range interactions, two dimensional turbulence, and geostrophic turbulence problems. Example of past and current projects follow:

### Phase transition theory.

To be completed