91e rencontre entre mathématiciens et physiciens théoriciens : Systèmes dynamiques et physique statistique

Classical equilibrium statistical mechanics deals with the average behavior of a, typically large, Hamiltonian dynamical system. Differential calculus provides a mathematical tool to model this behavior in the so-called micro-canonical ensemble, and may be used to construct thermodynamic quantities like temperature and pressure. We show how such quantities may be observed numerically within the dynamical system itself under the ergodic hypothesis and without resorting to an axiomatic theory of thermodynamics. We also discuss more recent developments on non-equilibrium statistical mechanics where results at present are sparse and incomplete.

(joint work with S. Rosay, LPT-ENS, Paris)
Understanding the mechanisms by which space gets represented in the cortex and the hippocampus is a fundamental problem in neuroscience. The experimental discovery of so-called "place cells" and "grid-cells", encoding specific positions in space, provide essential elements in this context. How a spatial chart ie a relation between different points in space, may be built and memorised? In this talk, we will present a model involving binary neurons (that ay be either active or silent), making possible to store several spatial charts. This model may be solved with the help of the techniques of Statistical Physics of Disordered Systems. We will discuss the different possible phases of the system and the essential features of its dynamics (activated diffusion in one chart and transitions between charts).

I will start with a broad introduction intended for a general audience. Then I will specialize to dynamical systems and present some applications of concentration inequalities. Finally, I will sketch how these inequalities can be proven.

This is a joint work with M. Benedicks and D. Schnellmann.
For a smooth one-parameter family of smooth hyperbolic discrete-time dynamics (i.e., Anosov systems, which are structurally stable), the "physical" (SRB) measure depends differentiably on the parameter, say t, and the derivative is given by an explicit "linear response" formula (Ruelle, 1997). When structural stability does not hold, the linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where a modulus of continuity t log(t) is possible for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case) and Baladi-Smania (slow recurrence case) recently obtained linear response for fully tangential families (confined within a topological class). In this talk we focus on the transversal case (e.g. the quadratic family), where we obtain Holder upper and lower bounds (in the sense of Whitney, along suitable classes of parameters).

Computing the large N expansion of the eigenvalue statistics of a random matrix, has been an important question for many years, with applications ranging from quantum gravity to electronics or even finance. Recently was found a recursion relation which allows to compute recursively all terms in the asymptotic expansion of any correlation function. The initial data for the recursion is a Riemann surface embedded in CxC, called spectral curve. Beyond random matrices, this recursion can be applied to any embedded Riemann surface, and generates a sequence of differential forms, called "invariants" of the surface. Those invariants posses fascinating mathematical properties (quasi-modular forms, satisfy Hirota equations, special geometry relations,...), and have many applications in physics and mathematics. In particular, many known invariants of enumerative geometry, can be recovered as specializations, for instance Gromov-Witten invariants of Calabi-Yau manifolds, or knot polynomials (Jones polynomial). The talk will introduce this recursion, and present some examples of applications.

The Frenkel-Kontorova model describes how a chain of atoms minimizes its total energy on interaction with a substrat. We consider in a joint work with E. Garibaldi and P. Thieullen the case where the environment is almost periodic. A minimizing configuration in the Aubry sense may not detect the configurations at the lowest energy. We introduce a stronger notion of calibrated configurations and prove in the case of a one dimensional quasi-periodic environment,  (e.g. the Fibonacci case), their existence. The main tool we use is the Mather set, the set of minimizing measures, and the space of tilings.