Mathematical Sciences: Multiple Scales for Interacting Particle Systems: Mesoscopic and Macroscopic Equations

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PARAGRAPH 1 In the context of Nonequilibrium Statistical Mechanics, the evolution of physical systems is modelled at a microscopic level by Interacting Particle Systems. Such processes involve a large number of components (particles) with either deterministic or stochastic dynamics, governed by local laws of interaction. It appears that large groups of particles, which in principle evolve in an unorderly manner, tend to organize themselves in a coherent pattern at some larger space or time scale. In particular, these rescaled, possibly stochastic, microscopic models are expected to give rise to deterministic Nonlinear Partial Differential Equations describing the evolution of macroscopic quantities. One of the problems addressed here, is the rigorous derivation of nonlinear partial differential equations from interacting particle systems . Proceeding a step further, one would hope that for finer scales of the particle systems, the local random fluctuations are preserved in the macroscopic equation, while microscopic details still remain hidden. In this framework, we propose to study (i) Interacting Particle Systems with long range interactions and weakly propagating interfaces, and (ii) Stochastic discrete velocity models and fluid dynamics systems of equations. We are particularly interested in understanding the convergence of particle systems to weak solutions of the macroscopic equations, past the formation of shocks for the fluids equations and past all possible geometric singularities of the propagating interfaces. The proposed techniques are drawn from Nonlinear Partial Differential Equations, as well as, Probability Theory and Stochastic Processes. A crucial part in the analysis of the interacting particle systems, lies in the interplay of the different space-time rescaling regimes present, each one with its own limiting deterministic partial differential equation. PARAGRAPH 2 This research project is concerned with the mathematical analysis of microscopic and macroscopic models of non-equilibrium phenomena arising in material science, biology and fluid mechanics. The first part of the proposal is concerned with the derivation of transport properties for materials undergoing phase transitions (for example alloys changing crystallographic structure). Similar techniques will hopefully prove useful in the prediction of nonequilibrium behavior in biological processes. The mathematical tools employed here, are expected to provide new detailed quantitative information about our physical systems, going one step beyond phenomenology. In the second part of the project, we study the convergence of discrete random models, to gas and fluid dynamics equations. In addition to the issues proposed in the first part, here we hope that such discrete models may yield robust algorithms for the numerical computation of gas dynamics flow, particularly in the presence of shock waves.