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

We study the cluster size distribution of particles for a two-species exclusion process which involves totally asymmetric transport process of two oppositely directed species with stochastic directional switching of the species on a 1D lattice. As a function of $Q$ - the ratio of the translation rate and directional switching rate of particles, in the limit of $Q \rightarrow 0$, the probability distribution of the cluster size is an exponentially decaying function of cluster size $m$ and is exactly similar to the cluster size distribution of a TASEP. For $Q>>1$, the model can be mapped to persistent exclusion process (PEP) and the average cluster size, $\langle m \rangle \propto Q^{1/2}$. We obtain an approximate expression for the average cluster size in this limit. For finite system size system of $L$ lattice sites, for a particle number density $ρ$, the probability distribution of cluster sizes exhibits a distinct peak which corresponds to the formation of a single cluster of size $m_s = ρL$. However this peak vanishes in the thermodynamic limit $L \rightarrow \infty$. Interestingly, the probability of this largest size cluster, $P(m_s)$, exhibits scaling behaviour such that in terms of scaled variable $Q_1 \equiv Q/L^2 ρ(1-ρ)$, data collapse is observed for the probability of this cluster. The statistical features related to clustering observed for this minimal model may also be relevant for understanding clustering characteristics in {\it active } particles systems in confined 1D geometry.

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