Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2005-06-16
Physical Review E Vol. 73, 051103 (2006).
Physics
Condensed Matter
Statistical Mechanics
10 pages, 6 figures
Scientific paper
10.1103/PhysRevE.73.051103
We derive the multi-fractal scaling of probability distributions of multi-particle configurations for the binary reaction-diffusion system $A+A \to \emptyset$ in $d \leq 2$ and for the ternary system $3A \to \emptyset$ 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 $(\frac{\ln t}{t})^{N}(\ln t)^{-\frac{N(N-1)}{2}}$ for $d=2$ and $t^{-Nd/2}t^{-\frac{N(N-1)\epsilon}{4}+\mathcal{O}(\ep^2)}$ for $d<2$. Here $\ep=2-d$. For the ternary reaction in one dimension we find that $P_{t}(N,\Delta V) \sim (\frac{\ln t}{t})^{N/2}(\ln t)^{-\frac{N(N-1)(N-2)}{6}}$. The principal tool of our study is the dynamical renormalization group. We compare predictions of $\ep$-expansions for $P_{t}(N,\Delta V)$ for binary reaction in one dimension against exact known results. We conclude that the $\ep$-corrections of order two and higher are absent in the above answer for $P_{t}(N, \Delta V)$ for $N=1,2,3,4$. Furthermore we conjecture the absence of $\ep^2$-corrections for all values of $N$.
Munasinghe Ranjiva M.
Rajesh R.
Zaboronski Oleg V.
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