Coagulation processes with Gibbsian time evolution

TL;DR

This paper characterizes the conditions under which the time evolution of a pure coagulation process follows a Gibbs distribution, providing explicit solutions and analyzing the probability of forming a giant cluster.

Contribution

It establishes the necessary and sufficient conditions for Gibbsian time evolution in coagulation processes and derives explicit solutions for specific cases.

Findings

Gibbsian time evolution occurs iff rates have a specific form.

Explicit solutions for three particular models.

Probability analysis of giant cluster formation.

Abstract

We prove that time dynamics of a stochastic process of pure coagulation is given by a time dependent Gibbs distribution if and only if rates of single coagulations have the form $\psi(i,j)=if(j)+jf(i)$, where $f$ is an arbitrary nonnegative function on the set of integers $\ge 1$. We also obtained a recurrence relation for weights of these Gibbs distributions, that allowed explicit solutions in three particular cases of the function $f$. For the three corresponding models, we study the probability of coagulation into one giant cluster, at time $t>0.$