Phase Transitions in Cellular Automata for Cargo Transport and Kinetically Constrained Traffic

Abstract

A probabilistic cellular automaton for cargo transport is presented that generalizes the totally asymmetric exclusion process with a defect from continuous time to parallel dynamics. It appears as an underlying principle in cellular automata for traffic flow with non-local jumps for the kinetic constraint to drive as fast as possible. The exactly solvable model shows a discontinuous phase transition between two regions with different cargo velocities.

The parallel TASEP, fixed particle number and weighted Motzkin paths

“…For parallel update such a process is not as straightforward to define [13], though. Here we presented one possible choice of second-class particle dynamics in terms of a cargo process [16] for which the generating function shares the thermodynamics with the generating function that has been derived here for the TASEP. Since the parallel TASEP is equivalent to the Nagel-Schreckenberg model with maximal velocity 1, the results also apply to a bottleneck situation in traffic flow.…”

“…In (B9), x and λ are the thermodynamic particle and lattice fugacities that were first found in [13]. They can further be simplified to [16] x = (ρ − J)(pρ − J)…”

“…In [13] it was argued that second-class particle dynamics in the parallel TASEP are not as natural to define as for the generic random update. In this section a model system is presented that mimics second-class particle dynamics and for which preliminary results were published in [16]. We change the boundary conditions and consider a closed periodic chain with one of the particles carrying a cargo.…”

“…This is due to the fact that for random update the weights for N particles on a lattice with L sites are the same for open boundaries and a second-class particle on the ring. Reference [16] shows the equivalence of this process and the one studied in [13]. It focuses on the thermodynamic limit and simplifies some results of [13].…”

“…In [16] it was argued that the second-class particle process can be mapped onto the (single species) traffic model in [13] which is defined by the transitions 100 → 001 and 101 → 011. We repeat shortly the arguments here.…”