Quantitative studies and Hydrodynamical limits for interacting particle systems

Abstract

The main results of my work contribute to the mathematical study of microscopic non-equilibrium systems that were first introduced in order to derive macroscopic physical laws such as Fourier's law. In particular the main objective is to determine the scaling of the spectral gap, i.e. the relaxation rate, in terms of the number of the particles for a paradigmatic model describing heat transport, the chain of oscillators. The mathematical study of this model started at the end of $90$s and it is challenging due to the degeneracy of the dynamics as the noise is not assumed to act to all the degrees of freedom, leading to lack of ellipticity and coercivity. We give bounds on the spectral gap for weak nonlinearities of the chain, i.e. perturbations around linear homogeneous chains and also a complete answer for the linear, homogeneous and disordered, chain of oscillators as well as $d$-dimensional grids of oscillators. The methods range from hypocoercivity inspired techniques, in the sense of Villani, to spectral analysis of discrete Schrödinger operators. Moreover we study heat conduction in gases addressing, with both analytic and probabilistic techniques, the question of the existence, and properties, of a non-equilibrium steady state for the nonlinear BGK model, introduced by Bhatnagar, Gross and Krook, with diffusive boundary conditions. The case that we address concerns large boundary temperatures away from the equilibrium case. Furthermore, besides non-equilibrium phenomena in many particle systems, this thesis deals with the question of deriving nonlinear diffusion equations from microscopic stochastic processes. We present a new, quantitative, unified method to show that the particle densities of one-dimensional processes on a periodic lattice, including the zero-range and simple exclusion jump processes as well as diffusion processes of Ginzburg-Landau type, converge to the solution of a nonlinear diffusion equation with an explicit, uniform in time, convergence rate. We discuss how we can extend the result to all the dimensions. Finally a study of the scaling of the spectral gap for all the mean field $\mathcal{O}(n)$ models of Ginzburg-Landau type using semiclassical tools, is included in this thesis. This concerns the spectral gap as a function of the number of particles, spins, for the dynamics below and at the critical temperature, with and without an external magnetic field.