The parallel TASEP, fixed particle number and weighted Motzkin paths

Abstract

In this paper the totally asymmetric exclusion process (TASEP) with parallel update on an open lattice of size L is considered in the maximum-current region. A formal expression for the generating function for the weight of configurations with N particles is given. Further an interpretation in terms of (u, l, d)-colored weighted Motzkin paths is presented. Using previous results (Woelki and Schreckenberg 2009 J. Stat. Mech P05014, Woelki 2010 Cellular Automata, pp 637-645) the generating function is compared with the one for a possible 2nd-class particle dynamics for the parallel TASEP. It is shown that both become physically equivalent in the thermodynamic limit.

On the Blockage Problem and the Non-analyticity of the Current for Parallel TASEP on a Ring

“…All formulations of the stationary distribution of the parallel TASEP (both with open and periodic boundary conditions) are considered quite involved in [21], where the author studies parallel TASEP with open boundaries in terms of Motzkin paths. In this respect, our work offers a significant contribution, for it frames the steady state of a parallel TASEP in a simple yet fruitful formulation.…”

On the blockage problem and the non-analyticity of the current for the parallel TASEP on a ring

“…Conversely, if σ 1 = 1 then the exponents appearing in ( 23) may increase or decrease by a unit at most. This correction is negligible in the thermodynamic limit, thus we will use (23) to compute the current according to (21). For large L,…”

“…All formulations of the stationary distribution of the parallel TASEP (both with open and periodic boundary conditions) are considered quite involved in [21], where the author studies parallel TASEP with open boundaries in terms of Motzkin paths. In this respect, our work offers a significant contribution, for it frames the steady state of parallel TASEP in a simple yet fruitful formulation.…”