Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances

Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization

We consider continuous-time random walks on a random locally finite subset of $$\mathbb {R}^d$$ R d with random symmetric jump probability rates. The jump range can be unbounded. We assume some second-moment conditions and that the above randomness is left invariant by the action of the group $$\mathbb {G}=\mathbb {R}^d$$ G = R d or $$\mathbb {G}=\mathbb {Z}^d$$ G = Z d . We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space–time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix D of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on $$\mathbb {Z}^d$$ Z d and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.

Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension ≥ 2

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus Td. In the discrete torus N-1TdN, we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is ej, then the exchange rate is defined as N-1 times the absolute value of the inner product between ej and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.