Percolation-like phase transition in a nonequilibrium steady state
We study the Gierer-Meinhardt model of reaction-diffusion on a site-disordered square lattice. Let p be the site occupation probability of the square lattice. For p greater than a critical value p(c), the steady state consists of stripe-like patterns with long-range connectivity. For p < p(c), the connectivity is lost. The value of p(c) is found to be much greater than that of the site percolation threshold for the square lattice. In the vicinity of p(c), the cluster-related quantities exhibit power-law scaling behavior. The method of finite-size scaling is used to determine the values of the fractal dimension d(f), the ratio, gamma /v, of the average cluster size exponent gamma and the correlation length exponent v. The values appear to indicate that the disordered GM model belongs to the universality class of ordinary percolation.
percolation phase transition nonequilibrium reaction-diffusion turing patterns