Static and dynamic exponents of disordered media

Mark Daniel Rintoul, Purdue University

Abstract

The primary focus of this work is to obtain precise values of critical exponents associated with randomly disordered media. First, using efficient new techniques, the backbone of critical percolation clusters in 2 and 3 dimensions is extracted, and the fractal dimension, $d\sb{f}$, is carefully measured. The larger sized samples in this work yield a slightly different value than previous works and this is explained on the basis of finite size effects. Next we study the behavior of diffusing particles on the backbones of three dimensional percolation clusters. The walk dimension, $d\sb{w},$ and the spectral dimension, $d\sb{s},$ are extracted using new numerical techniques which allow for a precise determination of their values. First, a transfer matrix which reflects the transitions probabilities between sites on the percolation cluster is created. The eigenvalue and eigenvector spectrum of the transition matrix are then mathematically related to the exponents in question. Finally the eigenvalues and eigenvectors themselves are extracted using a special mathematical technique developed by Arnoldi and Saad, which is based on the Lanczos method. This technique allows us to study very large matrices which correspond to large percolation clusters. The final part of this work consists of the study of polymer chains on disordered media. Percolation clusters are again used to represent the disordered media, and self-avoiding walks (SAW's) are used to represent the chains. The mean-square end-to-end distance exponent, $\nu$, and the free energy fluctuation exponent, $\chi$, are determined for a number of different conditions using exact enumeration methods. These conditions include all cluster vs. infinite cluster averages, chain averaging vs. kinetic averaging, and weak dilution vs. critical dilution. The value of $\nu$ is shown to be clearly different from its full lattice value, and inconsistencies with previous Monte Carlo results are explained.

Degree

Ph.D.

Advisors

Nakanishi, Purdue University.

Subject Area

Condensation

COinS