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

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 α ) 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/α . For a total density ρ of particles, in the limit Q → 0 , the cluster size distribution is shown to be P(m) = ( 1/ρ - 1) e^-m/lnρ and the mean cluster size ⟨ m ⟩ = 1/(1-ρ ) , results which are independent of Q and are identical to those for the simple exclusion process. By contrast, in the opposite limit, Q≫ 1 , we find the average cluster size, ⟨ m ⟩∝ 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_s = ρ L . However, this peak vanishes in the thermodynamic limit L →∞ . Interestingly, the probability of this largest size cluster, P(m_s) , for different L, ρ and Q exhibits data collapse in terms of the scaled variable Q_s≡ Q/L^2 ρ (1-ρ ) . The statistical features of the clustering observed for this minimal model may be relevant for active particle systems in 1D.