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 alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}) 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q=v/\alpha $$\end{document}. For a total density rho\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} of particles, in the limit Q -> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \rightarrow 0$$\end{document}, the cluster size distribution is shown to be P(m)=1/rho-1e-m/ln rho\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(m) = \left( 1/\rho - 1\right) e<^>{-m/\ln \rho }$$\end{document} and the mean cluster size ⟨m⟩=1/(1-rho)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle m \rangle = 1/(1-\rho )$$\end{document}, results which are independent of Q and are identical to those for the simple exclusion process. By contrast, in the opposite limit, Q >> 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\gg 1$$\end{document}, we find the average cluster size, ⟨m⟩proportional to Q1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle m \rangle \propto Q<^>{1/2}$$\end{document}-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 ms=rho L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\rm{s}} = \rho L$$\end{document}. However, this peak vanishes in the thermodynamic limit L ->infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L \rightarrow \infty $$\end{document}. Interestingly, the probability of this largest size cluster, P(ms)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(m_{\rm{s}})$$\end{document}, for different L,rho\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L, \rho $$\end{document} and Q exhibits data collapse in terms of the scaled variable Qs equivalent to Q/L2 rho(1-rho)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{\rm{s}}\equiv Q/L<^>2 \rho (1-\rho )$$\end{document}. The statistical features of the clustering observed for this minimal model may be relevant for active particle systems in 1D.