An exclusion process on a tree with constant aggregate hopping rate

TL;DR

This paper introduces a TASEP model on a tree with constant aggregate hopping rate, showing that mean field theory accurately describes the system, especially at large branching ratios, with applications in hierarchical distribution networks.

Contribution

The paper develops an exact solution for TASEP on a tree with constant aggregate hopping rate and demonstrates the validity of mean field theory in the large branching ratio limit.

Findings

Mean field theory matches the model's behavior.

Exact solution for a two-level tree enables correlation function computation.

Rapid convergence of simulations to mean field results at large branching ratios.

Abstract

We introduce a model of a totally asymmetric simple exclusion process (TASEP) on a tree network where the aggregate hopping rate is constant from level to level. With this choice for hopping rates the model shows the same phase diagram as the one-dimensional case. The potential applications of our model are in the area of distribution networks; where a single large source supplies material to a large number of small sinks via a hierarchical network. We show that mean field theory (MFT) for our model is identical to that of the one-dimensional TASEP and that this mean field theory is exact for the TASEP on a tree in the limit of large branching ratio, $b$(or equivalently large coordination number). We then present an exact solution for the two level tree (or star network) that allows the computation of any correlation function and confirm how mean field results are recovered as $b\rightarrow\infty$. As an example we compute the steady-state current as a function of branching ratio. We present simulation results that confirm these results and indicate that the convergence to MFT with large branching ratio is quite rapid.