Exactly solvable models through the generalized empty interval method, for multi-species interactions

TL;DR

This paper derives conditions under which multi-species reaction-diffusion systems on a one-dimensional lattice have exactly solvable dynamics using a generalized empty interval method, providing explicit solutions for certain cases.

Contribution

It establishes necessary and sufficient constraints on interaction rates for the closure of evolution equations in multi-species systems, extending the solvability framework.

Findings

Derived constraints for interaction rates ensuring closed evolution equations.

Solved the constraints for single-species symmetric systems.

Provided examples of multi-species systems with closed dynamics.

Abstract

Multi-species reaction-diffusion systems, with nearest-neighbor interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E^{\mathbf a}_n(t)$'s, the expectation value of the product of certain linear combination of the number operators on $n$ consecutive sites at time $t$. The constraints are solved for the single-species left-right-symmetric systems. Also, examples of multi-species system for which the evolution equations of $E^{\mathbf a}_n(t)$'s are closed, are given.