Critical Percolation in Systems of Mobile Disks

Abstract

Numerical results for critical diffusion in a percolating system of moving disks are reported. Critical percolation ensues as the disk density 8 approaches the percolation threshold density Q c and U + 0, where U is a measure of disk speed. Pure percolation behaviour is observed only for U << (ece)'/q, where q = 0.3. For U >>(ec -@)'/q, diffusion is dominated by kinetic effects; therein, Du q p , where D is the diffusion constant and , U 1.3.Introduction. -Percolation theory has provided a useful framework for the understanding of transport phenomena that take place on random matrices that do not change with time [l]. The matrices may, for instance, be random passageways (for instance, porous rocks [2]) or random impurity sites (as in semiconductors [31). Percolation theory has been generalized in order to be able to treat matrices that change with time [4]. The generalization is designed to treat phenomena such as: 1) electron hopping through mowing macromolecules that takes place in some systems of biological interest [51; 2) electrical conduction in microemulsions (such as conducting water spherules that move in oil)[61; 3) conduction in electroactive polymers above the glass transition temperature [7].In this letter we focus our attention on the following question. Let U be a measure of how fast hopping sites (the amacromoleeules*) move. As U + 0, ordinary (i.e. static) percolation ensues. Roughly speaking, up to what values of U do the well-known (U = 0) critical percolation effects carry on? Such crossover effect has been studied in a lattice model with bonds that change slowly in time at randomr81. However, the applicability of the results obtained to systems with continuous degrees of freedom is questionable, since it is not clear