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Coagulation processes with Gibbsian time evolution

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 $ψ(i,j)=if(j)+jf(i)$, 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 $t>0.$