Research Interests: Theoretical and Computational Condensed Matter Physics
My research is oriented towards those systems which exhibit some type of growth phenomena, such as domains in temperature-quenched magnetic systems, clustering and thin films on surfaces, dielectric breakdown, and viscous flow in porous media. In each of these systems, there exist two media separated by an interface, and one of the media grows at the expense of the other. Physical quantities such as the range of correlation or the roughness of the interface will increase with time, and a structure will develop which is often fractal in character. This is an area of physics which has seen great activity in the past few years, as numerous models of equilibrium and nonequilibrium growth have been introduced and studied. The techniques that can be employed range from renormalization group calculations to Monte Carlo simulations to numerical solution of differential equations.
The formation of many disordered systems can be described by a growth process in which individual particles join together to form an interconnected, percolating network. A simple model for such systems is the percolation model, in which sites on a lattice are randomly chosen and occupied. Nearest-neighbor particles are then considered to be connected, resulting in clusters of varying sizes and geometries (as shown with different colors in the picture on the left). When the concentration is larger than some critical value, it is found that one cluster will percolate through the entire lattice (the white one shown on the left). Its formation signifies a geometric phase transition, and is accompanied by the divergence of the mean cluster size and the range of connectivity. The percolating cluster is not solid, but rather has holes of many different sizes, making it a fractal object. If the cluster is made up of N particles and has a linear size L, then the fractal dimension D of the cluster is given by the relation N ~ LD. In two dimensions, a solid cluster has D = 2, but a percolating cluster is found to have D =1.9.
In the interactive percolation model we introduced, the occupation of a lattice site is dependent on the presence or absence of neighboring particles, which is a more realistic description of physical systems such as adsorption of gas atoms on metallic surfaces. This can dramatically effect the structure of the percolating cluster (note the background of this page). If the particles prefer to adsorb in the presence of other particles, solid islands with rough edges can grow for some time before joining together with other islands to form the percolating cluster. This is the situation shown in the picture at the right.
Interestingly, the fractal dimension of this cluster is the same as in ordinary percolation; this is because, on the large length scales which characterize a percolating cluster, there is no distinction between the individual particles and the larger, solid islands. In recent years, however, there have been several studies which indicate that the short-range details of a percolating cluster may have some effect on the ability of a particle to diffuse through the percolating cluster. In the picture above, the diffusion probability (starting from near the center) is indicated by the shade of gray, with white being the most-visited locations. The time necessary to travel a distance R through a cluster can be expressed as t ~ Ra, where a = 2 for a solid cluster. For an ordinary percolation cluster, however, a ~ 2.9. I am currently investigating how this diffusion exponent might depend on the strength of the interactions introduced into the percolation model.
The fractal dimension and the diffusion exponent are not the only exponents which characterize a percolation cluster. Diffusion is simply related to the ability of a current to pass through the cluster. It has recently been shown that the distribution of currents in the different branches of the cluster can be used to define an infinite number of independent dimensions which provide detailed information about its structure. In this sense, the percolation cluster is a "multifractal". In future research, therefore, I am interested in examining the multifractal characteristics of interactive percolation clusters.
©1996-2005 Scott R. Anderson, Ph.D.
Last Update: 2005 May 21