Limiting Properties of a Continuous Local Mean-Field Interacting Spin System

Abstract

A key interest in the study of interacting spin systems is the rigorous analysis of the macroscopic dynamical behaviour of systems that are described by their microscopic evolution. In this dissertation, we investigate unbounded spin systems where the microscopic evolution is modelled by stochastic differential equations (SDE). To each site of the discrete d-dimensional torus a spin is associated. The spins are distributed on the whole real line and evolve randomly according to the SDEs. The interaction between the spins is of local mean-field type, a long-range spatially variable interaction. The strength of the interaction between two spins depends on the difference of their positions on the torus. We aim to understand rigorously the time evolution of random variables as the size of the system increases.<br /> We prove in Chapter I the convergence of (space and spin dependent) empirical processes under proper rescaling to the classical solution of a nonlinear partial differential equation (PDE). This PDE is called hydrodynamic equation. We use the relative entropy method, to show this hydrodynamic limit result. To apply this method, we need to prove the existence of a classical solution of the hydrodynamic equation, which is non-linear and non-elliptic.<br /> In Chapter II we prove the propagation of chaos property of the system. We show that finitely many tagged spins are in the limit mutually independent. They evolve in the limit according to stochastic differential equations, without an interaction term. Instead (compared to the original SDEs), there is a term involving the solution of the hydrodynamic equation.<br /> In Chapter III we derive large deviation principles for the corresponding equilibrium system. We look at random variables that are distributed according to the invariant measure of the stochastic differential equation. For the empirical measure, defined by these random variables, we derive large deviation principles. We use a generalisation of Varadhan’s lemma that is stated and proven in Appendix C.<br /> In Chapter IV we analyse the landscape of the rate function of one of the equilibrium large deviation principles. We interpret this rate function as energy of the system in the limit. This is motivated by the fact that the hydrodynamic equation is the Wasserstein gradient flow of this rate function. We determine minima, critical values, bifurcation properties and lowest paths between minima.<br /> Finally in Chapter V we prove a dynamical large deviation principle for the empirical processes and the empirical measures. We derive different representations of the rate functions. By one of these representations it becomes obvious that it is exponentially unlikely that empirical processes deviate from the deterministic flow. In this chapter we allow the system to be more general, e.g. it can contain a random environment and a more general diffusion coefficient.<br /> The main distinctive features of the spin system considered in this dissertation are the relevance of the spatially fixed positions of the spins and the possibility of unbounded spins. The spatial positions of the spins affect the interaction and the initial distributions. Therefore new approaches in the proofs are necessary, in particular compared to mean field models. All these results can be used in the future to study long time phenomena like tunnelling and metastability.