PHYSICAL REVIEW B VOLUME 36, NUMBER 7 1 SEPTEMBER 1987 Percolation in spatially disordered systems T. Odagaki Department of Physics, Brandeis Uniuersity, Waltham, Massachusetts 02254 M. Lax Department of Physics, City College of the City University of New York, New York, New York l0031 and AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received 11 June 1984; revised manuscript received 16 October 1986) The coherent-medium approximation is employed to study the percolation processes in a spa- tially disordered system. Short-range order in the distribution of percolating particles is intro- duced through the pair distribution function. The effect of a hard-core and a particle interaction on the percolation threshold are studied. The critical percolation density is shown to be an in- creasing function of the hard-core diameter when the distribution is completely random. When the distribution is not completely random, the critical percolation density can be a decreasing function of the hard-core diameter over a limited range. A density-dependent parameter is intro- duced in the approximation, in which the critical index of the static difT'usion constant at the per- colation threshold shows a significant improvement. The dynamic (frequency-dependent) diffusion constant for the percolation model is also obtained in the present approximation. I. INTRODUCTION The concept of percolation is quite useful in under- standing physical properties of disordered systems. ' In addition to percolation processes in crystal lattices, the percolation problem in spatially disordered systems has been studied extensively. ' Usually, a number of points are randomly distributed in three-dimensional space, and a sphere (a disc in two dimensions; a square or a cube has also been used) with radius ro is drawn centering at each point. Centers of two spheres are con- nected by a bond if one center is located inside the other sphere. The critical number density p, is determined by the occurrence of a channel of infinitely connected bonds. Currently available estimation of the critical per- colation density gives p, =4. 4 — 4. 6 in units of ~ro in two dimensions. ' ' ' '' ' and p, =2. 7 — 2. 8 in units of 4mro/3 in three dimensions. " ' It has been seen in computer simulation that the critical percolation density decreases when a hard core is introduced around each site. ' There have also been several works on percola- tion processes in continuum systems. ' In the studies mentioned above, the geometrical struc- ture of percolating channels or percolating clusters is of central interest. The percolation theory in lattices is con- cerned with it, and the main interest is the appearance of an infinite channel and the probability of finding such an infinite channel. Percolation of "particles" is assumed to occur when an infinite channel is opened. Recently, quantum percolation has been studied by many authors. In the quantum percolation problem, a quantum-mechanical "particle" is assumed to How along the channels and one studies if the particle can spread infinitely from its initial position. Dynamics of the particle becomes important even in the usual classical percolation problem when we are interested in its dynamic response to an oscillatory external field. In fact, we have recently studied hopping conduction for the bond-percolation model and predicted various critical be- haviors of the ac conductivity at the percolation thresh- ld 33 — 36 The purpose of the present paper is to study the classi- cal discrete percolation problem in spatially disordered systems from the dynamic point of view. To accomplish this, we first reformulate in Sec. II the percolation prob- lem in terms of a random-walk equation, which is as- sumed to govern the motion of the percolating classical particle. We use the coherent-medium approximation (CMA) to solve the random-walk equation. In this ap- proach we can study the dynamic (frequency-dependent) response of the particle as well as the usual static proper- ties. Section III treats the static property, where effects of hard-core and particle interactions are investigated with the use of an appropriate form of the pair distribution function. It is shown that a simple hard core always in- creases the percolation threshold and an attractive interac- tion can increase or decrease the threshold depending on the ratio of the interaction range to the jumping distance. In Sec. IV, we propose an improvement of the present theory by exploiting a density-dependent parameter. The critical exponent of the static diffusion constant shows a significant improvement over the usual effective- medium — type approximation. The frequency-dependent (dynamic) diffusion constant is also obtained in Sec. IV. Section V discusses results. II. PERCOLATION MODEL AND THE COHERENT-MEDIUM APPROXIMATION A. Random-walk model of percolation In order to discuss dynamics of the percolating particle in discrete space, we employ a random-walk master equa- tion for the probability P(s, t ~ so, O) that one finds the par- 36 3851