Mathematical Physics Seminar, October 25, 2011
Wojciech de Roeck
Spectral gap in interacting particle systems with locally conserved quantities
By interacting particle systems, I mean particle systems with a stochastic dynamics (for example: exclusion processes). By 'spectral gap' I mean the spectral gap of the generator of the time-evolution. The locally conserved quantity can be thought of as the number of particles. For such a system, one expects that particles will flow from regions with high density to regions with lower density, and 'blobs' of high density will disappear by diffusion. This diffusion is captured (to some extent) by the way the spectral gap depends on the linear size L of the system, namely as 1/L^2.
Concretely, I want to describe a method, introduced by P. Caputo (see 'Spectral gap inequalities in product spaces with conservation laws') to prove this scaling rigorously. A generalization of this result is at this moment an interesting open problem in mathematical non-equilibrium statistical mechanics.