Spreading and annihilating particles: surprises from mean-field theory

We study the pairwise annihilation process $A+A\to$ inert of a number of random walkers, which originally are localized in a small region in space. The size of the colony and the typical distance between particles increases with time and, consequently, the reaction rate goes down. In the long time limit the spatial density profile becomes scale invariant. The mean-field approximation of this scenario bears some surprises. It predicts an upper critical dimension $d_c=2$, with logarithmic corrections at the critical dimension and nontrivial scaling behavior for d<2. Based on an exact solution of the one dimensional system we conjecture that the mean-field {\it exponents} are in fact correct even below the upper critical dimension, down to d=1, while the corresponding scaling function that describes the spatial density profile, changes for d<2 due to the stochastic fluctuations.