One-dimensional discrete aggregation-fragmentation model

Abstract

We study here one-dimensional model of aggregation and fragmentation of clusters of particles obeying the stochastic discrete-time kinetics of the generalized Totally Asymmetric Simple Exclusion Process (gTASEP) on open chains. Isolated particles and the first particle of a cluster of particles hop one site forward with probability $p$; when the first particle of a cluster hops, the remaining particles of the same cluster may hop with a modified probability $p_m$, modelling a special kinematic interaction between neighboring particles, or remain in place with probability $1-p_m$. The model contains as special cases the TASEP with parallel update ($p_m =0$) and with sequential backward-ordered update ($p_m =p$). These cases have been exactly solved for the stationary states and their properties thoroughly studied. The limiting case of $p_m =1$, which corresponds to irreversible aggregation, has been recently studied too. Its phase diagram in the plane of injection ($\alpha$) and ejection ($\beta$) probabilities was found to have a different topology. Here we focus on the stationary properties of the gTASEP in the generic case of attraction $p<p_m<1$ when aggregation-fragmentation of clusters occurs. We find that the topology of the phase diagram at $p_m =1$ changes sharply to the one corresponding to $p_m =p$ as soon as $p_m$ becomes less than $1$. Then a maximum current phase appears in the square domain $\alpha_c(p,p_m)\le\alpha\le 1$ and $\beta_c(p,p_m) \le \beta \le 1$, where $\alpha_c(p,p_m)= \beta_c(p,p_m)\equiv \sigma_c(p,p_m)$ are parameter-dependent injection/ejection critical values. The properties of the phase transitions between the three stationary phases at $p< p_m <1$ are assessed by computer simulations and random walk theory.