## Rapid Stirring Limits

Stirring is a mechanism that exchanges the values of neighboring sites. When stirring happens at fast rate, it makes adjacent sites independent. A result of DeMasi, Ferrari, and Lebowitz (1986) when applied to the quadratic contact process, implies that if we use scale the square lattice by multiplying by epsilon and stir at rate epsilon^{-2}, i.e., epsilon to the minus 2 power, then in the limit as epsilon tends to 0 the density of occupied sites satisfies In words, the Laplacian Lu results from diffusion of particles, while rapid stirring makes adjacent sites independent, so densities evolve according to the mean field ODE.

The mean field ODE for the contact process has a two nontrivial fixed points p1 < p2 when delta < 1/4. This might suggest that as stirring becomes more rapid the critical value delta_c approaches 1/4. However, results of Noble (1992) and Durrett and Neuhauser (1994) show

Theorem. As epsilon to 0, the critical value converges to 2/9. Furthermore, the density of the stationary distribution which is the limit starting from all 1's converges to p2, the larger fixed point of the mean field ODE.

To explain the value 2/9's we recall that in one dimension the limiting reaction diffusion equation has travelling wave solutions u(x,t) = h (x-vt) that keep their shape but move at velocity v. Further, if adopt the convention that h tends to p2 at minus infinity and 0 at plus infinity then v > 0, i.e., the region where the density is near p2 expands, if and only if delta < 2/9.

This strategy has been used to facilitate the study of other models. See the pages on predator prey systems, predator mediated coexistence and catalyst surfaces.

s3 Exercise. Consider the quadratic contact process, setting the fraction of stirring steps = 0.5 and delta = 0.15. The process an equilibrium distribution in which about 70% of the sites are occupied. This is not quite as large as the 81.6% predicted by solving u(1-u) = delta, however it is a drastic improvement over the rapid extinction that occurs in the process without stirring.

s3 Footnote. To increase the effect of single stirring steps we exchange the value at x with a site y chosen at random from the square of radius 3 centered at x.

DeMasi, A., Ferrari, P. and Lebowitz, J. (1986) Reaction diffusion equations for interacting particle systems. J. Stat. Phys. 44, 589-644

Noble, C. (1992) Equilibrium behavior of the sexual reproduction process with rapid diffusion. Ann. Probab. 20, 724-745

Durrett, R. and Neuhauser, C. (1994) Particle systems and reaction diffusion equations. Ann. Probab. 22, 289-333