Class of solvable reaction-diffusion processes on Cayley tree

TL;DR

This paper identifies two integrable reaction-diffusion models on Cayley trees, providing exact solutions for shell probabilities and demonstrating their uniqueness under certain restrictions.

Contribution

It introduces and solves two specific reaction-diffusion models on Cayley trees, expanding understanding of integrable processes in such structures.

Findings

Exact probabilities for particle distributions on Cayley tree shells

Identification of only two integrable reaction-diffusion models under given constraints

Demonstration of the models' uniqueness within certain probability restrictions

Abstract

Considering the most general one-species reaction-diffusion processes on a Cayley tree, it has been shown that there exist two integrable models. In the first model, the reactions are the various creation processes, i.e. $\circ\circ\to\bullet\circ$, $\circ\circ\to\bullet\bullet$ and $\circ\bullet\to\bullet\bullet$, and in the second model, only the diffusion process $\bullet\circ\to\circ\bullet$ exists. For the first model, the probabilities $P_l(m;t)$, of finding $m$ particles on $l$-th shell of Cayley tree, have been found exactly, and for the second model, the functions $P_l(1;t)$ have been calculated. It has been shown that these are the only integrable models, if one restricts himself to $L+1$-shell probabilities $P(m_0,m_1,...,m_L;t)$s.