关键词:
Interacting particle systems
Duality
Markov processes
Hydrodynamics
Random walks
摘要:
Within the mathematical statistical physics program of rigorously explain- ing the emergence of macroscopic phenomena in terms of the underlying mi- croscopic dynamics, stochastic interacting particle systems (IPS) play an em- inent role. Introduced in the early '70s as simplified stochastic "cartoons" of more realistic and complex microscopic Newtonian deterministic systems, IPS, on the one side, enable an extensive modeling flexibility – the possibility of describing particle interactions and, in turn, macroscopic systems of vari- ous nature, e. g. attraction as well as repulsion, independence as well as "non- linear" dependence. On the other side, they bear a significant reduction of the set of assumptions needed to investigate scaling limits, especially if compared to those required when studying more realistic Hamiltonian systems. Interacting particle systems and, in particular, a subclass which we call conservative factorized symmetric IPS are the main objects of this thesis. More specifically, the first part is entirely dedicated to the derivation of the solution to a linear heat (or diffusion) equation from an underlying microscopic system modeled as a symmetric simple exclusion process in presence of dynamic ran- dom conductances. The second part focuses on duality and self-duality, useful mathematical tools in the context of Markov processes and, in particular, of IPS, as they typically reduce the study of observables of complicated processes to that of quantities of more tractable processes. Duality plays a crucial role already in Chapter 2, in which we prove the hydrodynamic behavior in path space for the symmetric simple exclusion pro- cess in Z d evolving on uniformly bounded time-dependent conductances. To achieve this result – the precise statement may be found in Theorem 2.3 – we require essentially two assumptions: firstly, the association of the initial conditions to a macroscopic profile, namely that the initial empirical density of particles "ap