Multiscaling of correlation functions in single species reaction-diffusion systems

We derive the multi-scaling of probability distributions of multi-particle configurations for the binary reaction-diffusion system A+A ->subset of in d <= 2 and for the ternary system 3A ->subset of in d=1. For the binary reaction we find that the probability P-t(N,Delta V) of finding N particles in a fixed volume element Delta V at time t decays in the limit of large time as (ln t/t)(N)(ln t)(-N(N-1)/2) for d=2 and t(-Nd/2)t(-N(N-1)epsilon/4+)O((epsilon 2)) for d < 2. Here epsilon=2-d. For the ternary reaction in one dimension we find that P-t(N,Delta V)similar to(Lnt/t)(N/2)(Ln t)(-N(N-1)(N-2)/6). The principal tool of our study is the dynamical renormalization group. We compare predictions of epsilon expansions for P-t(N,Delta V) for a binary reaction in one dimension against the exact known results. We conclude that the epsilon corrections of order two and higher are absent in the previous answer for P-t(N,Delta V) for N=1,2,3,4. Furthermore, we conjecture the absence of epsilon(2) corrections for all values of N.