New England Quarterly Complex Fluids Conference
September 22, 2000
M. I. T.
Title: Renormalization of the Order-Parameter Distribution in Percolation
Speaker: Martin Z. Bazant
Dept. of Mathematics
M. I. T.
Time: 2:00-2:300pm
Place: Wang Auditorium, E51
Abstract:
Perocolation is the canonical model for quenched spatial disorder,
which has relevance for flow in porous media as well as the
thermodynamical and mechanical properties of complex fluids (in
certain limits). The order parameter for the percolation transition is
the fraction of all sites occupied by the largest cluster, which has
has a singularity at p=p_c in the infinite-system limit. Although the
finite-size scaling of the mean is well-known, fluctuations in the
largest cluster size have rarely been considered. In this work, the
entire probability distribution of the largest cluster size is studied
analytically using novel real-space renormalization-group methods and
supported by computer simulations. The analysis employs classical
ideas from probability theory away from the critical point, where
rigorous results are possible (due to the finite range of
correlations). At the critical point (where the correlation length
diverges), modern renormalization-group ideas are applied and
extended. Although the resulting approximations are not rigorous, many
highly nontrivial quantities can be predicted in reasonable agreement
with simulations.
For a discussion of the subcritical case, see M. Z. Bazant,
Phys. Rev. E 62, 1660 (2000), http://arXiv.org/abs/cond-mat/9905191.