Generalized Ising measures for one-dimensional lattice gases and their applications - Ngoc, Ngo Phuoc Nguyen (2022)
The processes studied are Interacting Particle Systems in which particles move on sites of a lattice with periodic boundary conditions. The particle interaction rule is the exclusion rule. Among the two classes of processes studied here, the first class is a generalization of the Exclusion Process. The generalization amounts to an extension of the dependence upon neighboring particles of a particle transition rate. In the other class of the processes studied, we substituted particles with rods that occupy several contiguous sites, defined that each rod may be in one of two possible states, and defined the rules for state switch. The transition rules for rod motion are similar to the rules for particles in our generalizations of the Exclusion Process. For each process constructed, we studied the conditions that ensure that its time-invariant distribution is an Ising type measure. The results relate the rates of motion and their ranges to the range and the interactions expressed in the potential function of Ising measure. We also studied the behavior of the process when they evolve in their respective invariant states. The study revealed novel phenomena in the behavior of particle flux as a function of particle density. One of the phenomena is non-monotonicity which has not been observed in the Classical Exclusion Process. For the interacting rod system, we revealed and explained the effect of cooperative pushing that had been observed previously in the RNA transcription process executed by RNA polymerase enzymes. Our results apply to understanding this biological process and to the traffic of cars that are traditionally modeled by the Exclusion Process.