Classical and quantum continuum percolation with hard core interactions

Abstract

We study the classical and quantum percolation of spheres in a three-dimensional continuum. Each sphere has an impenetrable hard core of diameter (J, and two spheres are considered to be directly connected if the distance between their centers is less than d. We calculate the critical percolation density as a function of (J/d. In the classical problem this is the density P at which a~ i~finite cluster of connected spheres first forms. In the quantum problem, we stud~ a tight-bmdl~g model where the hopping matrix element between two spheres is nonzero only if they are dlrectly.con~ected. In this case the critical density P q is the density at which the eigenstates of the Hamlltoman first become extended. Our method uses Monte Carlo simulation and finite-size scaling techniques, and for the quantum problem, the concept of quantum connectivity. We find that both Pc and P q exhibit nonmonotonic behavior as a function of (J/d. We also find that for all values of (J/d, P q >Pc> although the ratio of the thresholds decreases with increasing (J/d. We argue that a better understanding of this ratio is obtained by considering the average coordination number. We speculate about the nature of both classical and quantum percol~tion as (J/d approaches 1.

Monte carlo study of the percolation in two-dimensional polymer systems

“…The second method of the determination of the percolation threshold for an infinite system is based on the finding that percolation probability curves P ( φ ) for systems of different size intersect in one point [11, 21, 36]. Figure 4 shows the percolation probabilities P as functions of the polymer concentrations φ for the chain N  = 10 calculated for some sizes of the Monte Carlo box L .…”

Sol–gel transition in solutions of FG-Nups of the nuclear pore complex

“…This method relies on the percolation probability P, defined as the probability of forming an infinite cluster in the system. The percolation probability versus concentration curves P(c) for systems with different sizes are obtained and the intersection of these curves has been shown to determine c crit [42][43][44]. The percolation probabilities for different concentrations can be fitted to the function…”

Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution

“…Since the curve is approximately linear in an extended region, the value at the mid-point of the linear portion is typically used in rougher estimations. An alternative and more accurate approach is to obtain p-f curves from a pair of independent simulations with different cell sizes (or equivalently, different particle sizes); the value of f at which the two curves intersect is a good estimate of the percolation threshold f c (Stauffer 1979;Saven et al . 1991).…”