Leírás
We study a system of N particles with some energies E_2,E_2,...,E_N. These energies are constant for some random time (which depends also on the energy values), but after this random time, two randomly chosen particles "collide", meaning that they redistribute their total energy between themselves in a random way, while all other energies remain unchanged. Then energies are constant again for some random time until the next "collision", and so on - in a Markov process. The process we are looking at is very specific, and the precise form of the evolution rules is dictated by the deterministic mechanical heat conduction model, from which it arises as a kind of "rare interaction limit" (not to be discussed in this talk). We show that this Markov chain has an L^2 spectral gap, and discuss the dependence of the spectral gap on the system size N. The main motivation is a potential application in the study of the hydrodynamic limit. This problem was believed to be solved for almost 10 years, until a serious gap in the proof was found. The present solution uses a substantially different approach. I present a few key steps and difficulties.
Joint work with Eric Carlen and Gustavo Posta.