Abstract
We consider a model of stochastically interacting particles on ℤ, where each site is assumed to be empty or occupied by at most one particle. Particles jump to each empty neighboring site with rateγ/2 and also create new particles with rate 1/2 at these sites. We show that as seen from the rightmost particle, this process has precisely one invariant distribution. The average velocity of this particle V(γ) then satisfiesγ −1/2V(γ)→\(\sqrt 2 \) asγ→∞. This limit corresponds to that of the macroscopic density obtained by rescaling lengths by a factorγ 1/2 and lettingγ→∞. This density solves the reaction-diffusion equation\(u_t = \tfrac{1}{2}u_{xx} + u(1 - u)\), and under Heaviside initial data converges to a traveling wave moving at the same rate\(\sqrt 2 \).
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Bramson, M., Calderoni, P., De Masi, A. et al. Microscopic selection principle for a diffusion-reaction equation. J Stat Phys 45, 905–920 (1986). https://doi.org/10.1007/BF01020581
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DOI: https://doi.org/10.1007/BF01020581