EXACT SOLUTION OF THE N-BODY INITIAL-VALUE PROBLEM FOR THE DIFFUSION-LIMITED LOGISTIC DIFFUSION-REACTION SYSTEM

We consider the many-body initial-value problem for the reversible single-species diffusion-controlled reaction X + X half-arrow-right-over-half-arrow-left X. A mapping to a dual process is identified which reduces the N-body problem to the dynamics of interfaces, which for general initial conditions is reduced to simple quadratures. The diffusion-controlled limit and the continuous space limit are taken as leading orders in power series expansions. The spectrum of the reaction operator is given explicitly and its use for the calculation of corrections to the diffusion-controlled limit is explained. A closed form is given for the time- and space-dependent concentration rho(R, t) in one dimension following an arbitrary initial state.