Exactly solvable reaction diffusion models on a Bethe Lattice through the empty-interval method

Laleh Farhang Matin

doi:10.1007/s40094-015-0163-y

Abstract

The most general reaction-diffusion model on a Bethe Lattice with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models are discussed. For some special choice of reaction rates the dynamics of the system is also studied.

Highlights

Reaction–diffusion systems have been studied using various methods including analytical techniques, approximation methods, and simulation
The Cayley tree or Bethe lattice is a tree where every site is connected to n nearest neighbor sites
In [24, 25], all the one-dimensional reaction–diffusion models with nearest neighbor interactions which can be exactly solved by empty interval method (EIM) have been found and studied

Summary

Introduction

Reaction–diffusion systems have been studied using various methods including analytical techniques, approximation methods, and simulation. The empty interval method (EIM) has been used to analyze the one dimensional of diffusion-limited coalescence [18,19,20,21] Using this method, the probability that n consecutive sites are empty has been calculated. In [24, 25], all the one-dimensional reaction–diffusion models with nearest neighbor interactions which can be exactly solved by EIM have been found and studied. The most general single-species reaction– diffusion model with nearest-neighbor interactions on a Cayley tree is investigated, which can be solved exactly through the empty interval method. 2, the most general reaction– diffusion model with nearest neighbor interactions on a Cayley tree is studied, which can be solved exactly through EIM.

Models solvable through the empty interval method on a Cayley tree

The stationary solution

Dynamical solutions

Concluding remarks