The proposed project considers problems with respect to various scaling limits of a tagged, or distinguished particle, the space-time evolution of the mass density of particles, or hydrodynamics, and their underlying connections in certain systems of interacting random walks such as simple exclusion and zero- range processes. The proposal will concentrate on relevant but less studied situations when these systems are out of equilibrium, when particles interactions are not finite-range, and when the systems are disordered. Specific questions focused upon include (1) large deviations of a tagged particle in symmetric simple exclusion processes, (2) hydrodynamics, and stable process limits of a tagged particle in long range exclusion models, (3) hydrodynamics and tagged particle limits in zero-range systems with random rates, and (4) diffusive limits of a tagged particle in bounded rate zero- range processes.
Understanding the motion of a tagged interacting with others in large scale systems is important in many applied contexts such as traffic, queues, fluids, and granular and molecular flows, and is also a long standing concern in statistical physics with regard to formulating the physical basis of Brownian motion. One of the main difficulties in the analysis is that the bulk contribution from the other particles is not easily incorporated into the asymptotics of the tagged particle. In this context, a main thrust of the proposal is to develop mathematical techniques to handle complex microscopic particle interactions so that different types of macroscopic tagged particle behaviors may be captured.
