The parallel TASEP, fixed particle number and weighted Motzkin paths

In this paper the totally asymmetric simple 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 637-45) the generating function is compared with the one for a possible second-class particle dynamics for the parallel TASEP. It is shown that both become physically equivalent in the thermodynamic limit.