Hydrodynamic limit for two-species exclusion processes

We consider two-species exclusion processes on the d-dimensional discrete torus taking the effects of exchange, creation and annihilation into account. The model is, in general, of nongradient type. We prove that the (charged) particle density converges to the solution of a certain nonlinear diffusion equation under the diffusive rescaling in space and time. We also prove a lower bound on the spectral gap for the generator of the process confined in a finite volume.Comment: 28 page

Spectral gap for stochastic energy exchange model with nonuniformly positive rate function

We give a lower bound on the spectral gap for a class of stochastic energy exchange models. In 2011, Grigo et al. introduced the model and showed that, for a class of stochastic energy exchange models with a uniformly positive rate function, the spectral gap of an $N$-component system is bounded from below by a function of order $N^{-2}$. In this paper, we consider the case where the rate function is not uniformly positive. For this case, the spectral gap depends not only on $N$ but also on the averaged energy $\mathcal{E}$, which is the conserved quantity under the dynamics. Under some assumption, we obtain a lower bound of the spectral gap which is of order $C(\mathcal{E})N^{-2}$ where $C(\mathcal{E})$ is a positive constant depending on $\mathcal {E}$. As a corollary of the result, a lower bound of the spectral gap for the mesoscopic energy exchange process of billiard lattice studied by Gaspard and Gilbert [J. Stat. Mech. Theory Exp. 2008 (2008) p11021, J. Stat. Mech. Theory Exp. 2009 (2009) p08020] and the stick process studied by Feng et al. [Stochastic Process. Appl. 66 (1997) 147-182] are obtained.Comment: Published at http://dx.doi.org/10.1214/14-AOP916 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org