Percolation of interacting diffusing particles

We explore the connectivity properties of diffusing particles with short-range interactions for dimensions d=1,2. We consider both ‘‘blind’’ and ‘‘myopic’’ diffusion rules (for the blind case, the walker chooses the next step from among all neighbor sites while in the myopic case the walker chooses from among only the unblocked sites). We show that—for all d—the equilibrium state of a system of particles diffusing according to the blind rule, at density ρ, is equivalent to the lattice gas with interaction parameter J=0 and chemical potential μ=2 ${\sinh }^{\mathrm{-}1}${[(2ρ-1${)}^2$/4ρ(1-ρ)${]}^{1/2}$}. The connectivity properties of the blind diffusion system are described by random site percolation in all dimensions. The myopic diffusion system is more complicated. For d=1, the equilibrium state of a system of particles diffusing according to the myopic rule, with particle density ρ, is equivalent to a lattice gas with J=-ln(2) and μ=ln(2)+2 ${\sinh }^{\mathrm{-}1}${[(2ρ-1${)}^2$/2ρ(1-ρ)${]}^{1/2}$}. Also, for d=1, the number of clusters of size s is approximately ${\mathit{n}}_{\mathit{s}}$=ρ${\mathit{p}}_{\mathrm{eff}}^{\mathit{s}\mathrm{-}1}$(1-${\mathit{p}}_{\mathrm{eff}}$ ${)}^2$, where ${\mathit{p}}_{\mathrm{eff}}$≤ρ. An approximation for ${\mathit{p}}_{\mathrm{eff}}$ is given that agrees closely with Monte Carlo simulations. For d=2, the myopic diffusion system has no mapping to the lattice-gas model. Rather, it undergoes a percolation transition at a threshold density ${\mathrm{\rho }}_{\mathit{c}}$. On the square lattice, ${\mathrm{\rho }}_{\mathit{c}}$=0.617±0.004, a value that is higher than the threshold for random site percolation. However, percolation and myopic diffusion appear to be in the same universality class.