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.

18 pages, 8 figures

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Cited by in corpus (1)

• Nonstationary generalized TASEP in KPZ and jamming regimes