Here is a snapshot of a simulation of the diffusionlimited reaction
A+B>0 in two dimensions. The two species are indicated by the different
colors. Note the existence of three length scales to describe the spatial
distribution of reactants: (a) the typical distance between individual
reactants, which grows with time as t^{1/4}; (ii) the typical domain
size, which grows as t^{1/2}; and (iii) the "gap" distance between
domains, which grows at t^{1/3}. The latter is the white space
between domains and it is perhaps best visualized by squinting at the
picture.

Here is the "microcanonical" density profile of reactants in one
dimension when particles move by isotropic diffusion. This is defined by
taking each domain and stretching or shrinking its length so that it lies
between [1,1], and then superposing the resulting density profiles. These
profiles exhibit excellent data collapse when they are rescaled by
a factor t^{1/4}.


Microcanonical density profile in one dimension when all particles move
by driven diffusion. That is, all particles have the same overall drift
superimposed on the diffusion. Naively, one would expect that this drift
would have no effect on the density profile at large times. However, the
drift is asymptotically relevant. Surprisingly, the density decays as
t^{1/3} in one dimension, in contrast to the t^{1/4} decay
in the absence of drift. Moreover, there is a longtime asymmetry in the
density profile, as well as small deviations from scaling when the profiles
are all rescaled by a factor of t^{1/3}.
