Section: New Results

Statistical physics and hydrodynamic limits

Participant : Guy Fayolle.

These last years, having in mind a global project concerning the analysis of complex systems, we did focus on the interplay between discrete and continuous description: in some cases, this recurrent question can be addressed quite rigorously via probabilistic methods (see previous activity reports).

To describe the systems of interest, which are in touch with many application domains, we started from paradigmatic elements, namely discrete curves subject to stochastic deformations. Up to some convenient mappings, it appears that most models can be set in terms of interacting exclusion processes, the ultimate goal being to derive hydrodynamic limits after proper scalings.

The key ideas can be found in [56] , where the basic ASEP system on the torus is the toy model. In this case, the usual sequence of empirical measures, converges in probability to a deterministic measure, which is the unique weak solution of a Cauchy problem.

The Gordian knot is indeed the analysis of a family of specific partial differential operators in infinite dimension. Indeed, the values of functions at given points play here the role of usual variables, their number becoming infinite. The method presents some new theoretical features, involving path integrals, promeasures (as introduced by Bourbaki), variational calculus, and the construction of generalized measures. In [56] , we present a detailed analysis of the asep system on the torus $\mathbb{Z}/N\mathbb{Z}$. Then we claim that most of the arguments a priori for multi-type exclusion processes, and should lead to systems of coupled partial differential equations of Burgers' type. At the moment, this claim is being proved for the famous ABC model, reformulated in terms of the dynamics of a random walk on the triangular lattice.