Diffusion and vibrational properties of critically disordered systems

Sonali Mukherjee, Purdue University

Abstract

In this work we have studied diffusion in critically disordered system modeled by a fractal in the framework of Markov chain analysis. Within this framework the time evolution of the diffusing particle is governed by the transition probability matrix whose spectral analysis in turn leads to the the critical exponents which describe the power-law behavior of the mean-square displacement, velocity autocorrelation function, etc. of the particle diffusing in the fractal. The power-law behavior of the quantities characterizing the diffusing particle is the result of the translation of the spatial correlation of the sites of the underlying fractal substrate into temporal correlation in the trajectory of the diffusing particle. The precise extraction of the exponents demanded incorporation of numerical algorithms like Arnoldi-Saad and computational techniques like making the code compatible for parallel computations in the Markov chain analysis technique. In particular we were successful in obtaining very precise values of the walk exponent $d\sb w$ which describes the power-law behavior of the mean-square displacement of the diffusing particle and the spectral exponent $d\sb s$ which describes the number of distinct sites visited by the diffusing particle for the percolation cluster. Also the change of exponents was discernible with the change of the universality class when the site occupation probability for the percolation cluster p is increased beyond $p\sb c$ transforming the underlying substrate from a fractal to a non-fractal medium. Furthermore the increased accuracy of the exponents obtained from this method made it possible to question the validity of scaling relation between the dynamical exponents $d\sb w$ and $d\sb s$ and the static exponent $d\sb f$ given by $d\sb s=2d\sb f/d\sb w$ in loopless fractal structures. The outcome proved that the validity of the scaling relation is highly dependent on the intricate structure of the fractal rather than on the presence or absence of the loops which was the traditional belief. Finally by casting the scalar elasticity problem into the diffusion problem we have studied the vibrational problem of percolation cluster with absorbing boundaries. The fractal boundaries of percolation cluster had interesting effect on the vibrational spectrum when it was made absorbing, which was manifested in new scaling relations for the slowest non-trivial mode of the transition probability matrix and the peaks of the density of vibrational states.

Degree

Ph.D.

Advisors

Nakanishi, Purdue University.

Subject Area

Condensation

COinS