Multi-Scaling of Correlation Functions in Single Species Reaction-Diffusion Systems

الأصل · EN

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