Clustering and finite size effects in a two-species exclusion process

Abstract

We study a two-species totally asymmetric exclusion process (TASEP) in 1D lattice in which the particles of both species move stochastically in opposite directions (with rate v) and switch directions stochastically (with rate $\alpha$) while adjacent a particle of either species. We focus on the cluster size distribution P(m), where a cluster is taken to be a contiguous set of sites occupied by either species, as a function of $Q=v/\alpha$. For a total density $\rho$ of particles, in the limit $Q \rightarrow 0$, the cluster size distribution is shown to be $P(m) = \left( 1/\rho - 1\right) e^{-m/\ln \rho }$ and the mean cluster size $\langle m \rangle = 1/(1-\rho )$, results which are independent of Q and are identical to those for the simple exclusion process. By contrast, in the opposite limit, $Q\gg 1$, we find the average cluster size, $\langle m \rangle \propto Q^{1/2}$—similar to the that for the persistent exclusion process (PEP), although the cluster size distributions are different in both limits. We further find that, for a finite system with L sites, the probability distribution of cluster sizes exhibits a distinct peak which corresponds to the formation of a single cluster of size $m_{\rm{s}} = \rho L$. However, this peak vanishes in the thermodynamic limit $L \rightarrow \infty$. Interestingly, the probability of this largest size cluster, $P(m_{\rm{s}})$, for different $L, \rho$ and Q exhibits data collapse in terms of the scaled variable $Q_{\rm{s}}\equiv Q/L^2 \rho (1-\rho )$. The statistical features of the clustering observed for this minimal model may be relevant for active particle systems in 1D.