Effective Stochastic Generator with Site-dependent Interactions

TL;DR

This paper explores conditions under which the stochastic generators of effective processes in particle systems can be local and site-dependent, focusing on one-dimensional lattice models with local interactions.

Contribution

It demonstrates that, under specific constraints, local and site-dependent stochastic generators can be derived from systems with initially local interactions.

Findings

Local stochastic generators are possible under certain microscopic constraints.

Examples include the asymmetric zero-temperature Glauber model and the A-model with diffusion.

Site-dependent generators can be obtained from local, site-independent original interactions.

Abstract

It is known that the stochastic generators of effective processes associated with the unconditioned dynamics of rare events might consist of non-local interactions; however, it can be shown that there are special cases for which these generators can include local interactions. In this paper we investigate this possibility by considering systems of classical particles moving on a one-dimensional lattice with open boundaries. The particles might have hard-core interactions similar to the particles in an exclusion process or there can be arbitrary many particles at single site as in a zero-range process. Assuming that the interactions in the original process are local and site-independent, we will show that under certain constraints on the microscopic reaction rules, the stochastic generator of unconditioned process can be local but site-dependent. As two examples, the asymmetric zero-temperature Glauber model and the A-model with diffusion are presented and studied under the above mentioned constraints.