Preprint C56/2007
Central Limit Theorem for a Tagged Particle in Asymmetric Simple Exclusion and Hydrodynamic Limit for a Particle System with Degenerate Rates
Patrícia Gonçalves
Keywords: Hydrodynamic limit | equilibrium fluctuations | Tagged Particle | Asymmetric Simple Exclusion
This work has to do with the study of two different models. The first is the Asymmetric Simple Exclusion in $\mathbb{Z}$. In this process, each particle, independently from the others, waits a mean one exponential time, at the end of which being at $x$ it jumps to $x+1$ at rate $p$ or $x-1$ at rate $1-p$. If the site is occupied the jump is suppressed to respect the exclusion rule. The Bernoulli product measures $\nu_{\alpha}$ are invariant for this process. We prove a Functional Central Limit Theorem for the position of a Tagged Particle in the one-dimensional setting and in the hyperbolic scaling, for the process starting from a Bernoulli product measure conditioned to have a particle at the origin. We also show that the position of the Tagged Particle at time $t$ depends on the initial configuration, by the number of empty sites in the interval $[0,(p-q)\alpha t]$ divided by $\alpha$ in the hyperbolic and in a longer time scale, namely $N^{4/3}$. In the second part of the work, we study a conservative particle system with degenerate rates, namely with nearest neighbor exchange rates which vanish for some configurations. Due to this degeneracy the hyperplanes with a fixed number of particles can be decomposed into some irreducible sets of configurations plus isolated configurations that do not evolve under the dynamics. We show that, for initial profiles smooth enough and bounded away from zero and one, under the diffusive time scaling, the macroscopic density profile evolves according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed microscopic dynamics: we add jumps of the Symmetric Simple Exclusion which remove the degeneracy of rates and are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained.