Civil Engineering Seminar
MIT, Friday, Feb. 11
Title: Finite-Size Scaling in Percolation
Speaker: Professor Martin Z. Bazant
Department of Mathematics, MIT
Abstract:
Percolation is used in almost every area of science as the simplest model
for spatial disorder. In civil engineering, for example, it has been
applied to transport in porous media, fracture of heterogenous rocks,
polymer gelation and the strength of composites. In spite over fifty
years of theoretical research, however, we still do not have a complete
understanding of the model itself, particularly finite-size effects.
In applications, the quantity most closely related to properties of
interest (e.g. conductivity, diffusion coefficent, correlation length) is
often the size of the largest cluster, or more precisely, its probability
distribution. Although the scaling of the mean is well-known, however, the
scaling of higher moments and the shape of the limiting distribution are
not. In this talk, these quantities are derived analytically and checked
numerically on 2d square lattices of up to 30 million sites. A
"renormalization-group" framework is proposed that draws on classical
ideas from probability theory (due to Levy, Fisher and Tippett) as well as
the modern theory of critical phenomena.