Theoretical Study of Discharge in a Nonlinear Medium

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 fixed voltage difference. We assume that each site can take one of three states; freshly-ionized (F-site), aged-ionized (A-site), and neutral (N-site) states. An F-site is rejuvenated with some probability, and when it is not rejuvenated, it becomes an A-site. An A-site can be rejuvenated back to an F-site or it can be neutralized. Otherwise, it stays as an A-site with older age. An N-site can be ionized with some probability to become an F-site. Each site is ionized with a probability which is an increasing function of the strength of the electric field. The resistivity between two ionized sites is assumed to be 10−6 times smaller than the original resistivity. We obtain the current between two electrodes at the opposite edges of the lattice and the fraction of ionized sites as functions of the potential difference by a Monte Carlo simulation. We find that as the potential difference is increased, the system changes its state discontinuously from the non-conductive to the conductive state, similar to the first-order phase transition. We show that non-conductive and conductive states can coexist at a given applied potential difference. We analyze the transition on the basis of a mean field approach and find that the local effect of a cluster structure of ionized sites on ionization and rejuvenation is important for the transition and that the discontinuous transition is induced by rejuvenation of ionized sites.