Microscopic models of traveling wave equations

Reaction-diffusion problems are often described at a macroscopic scale by partial derivative equations of the type of the Fisher or KolmogorovPetrovsky-Piscounov equation. These equations have a continuous family of front solutions, each of them corresponding to a different velocity of the front. By simulating systems of size up to N = 10 particles at the microscopic scale, where particles react and diffuse according to some stochastic rules, we show that a single velocity is selected for the front. This velocity converges logarithmically to the solution of the F-KPP equation with minimal velocity when the number N of particles increases. A simple calculation of the effect introduced by the cutoff due to the microscopic scale allows one to understand the origin of the logarithmic correction. PACS numbers: 02.50.Ey, 03.40.Kf, 47.20.Ky 1 The Fisher equation The Fisher equation[1], also called KPP equation (for Kolmogorov-PetrovskyPiscounov[2]) is widely used to describe front propagation in many problems of physics, chemistry and biology: ∂c ∂t = ∂c ∂x2 + c− c. (1) Fisher first introduced this equation to represent “The Wave of Advance of Advantageous Genes”[1] in a population. The concentration c(x, t) was the fraction of individuals in a population at position x and time t that exhibit some benefic genes, and (1) was used to describe how this favorable gene would spread in the population. Equation (1) can also model the dynamics of sick individuals in a population during a viral contagious infection, the proportion of burnt-out gases in a combustion[3], the concentration of some species produced in a chemical reaction, etc. It also appears in the mean field theory of directed