Anisotropic Percolation Analysis of Discharge

摘要

Exploiting a nonlinear resistor network on a square lattice in two dimensions, we investigate discharge when two opposite sides of the lattice are subjected to a constant voltage difference. Each site is ionized randomly with a probability in proportion to the square of the strength of the electric field, and the resistivity between two ionized sites is assumed to be 10~(-6) times smaller than the original resistivity. Using Monte Carlo simulation, we obtain the current and distribution of clusters of ionized sites as functions of the fraction of ionized sites. It is found that a wall of potential drop is formed as the fraction approaches a critical value, which is followed by discharge. The critical value is much smaller than the critical percolation probability of the standard site percolation on the square lattice. We also find that a singular behavior of the cluster distribution is expected at a critical fraction differently from that for the current, and that the critical exponents characterizing the cluster distribution satisfy the scaling relation known for two-dimensional, percolation, while the critical exponent of the percolation probability is close to the value reported for a directed percolation.