Videos

Colloids, Lattice Gasses, and Other Models with Hard Constraints

Presenter
February 22, 2012
Keywords:
  • mathematical statistical mechanics
  • percolation
  • probability theory
  • phase transitions
  • scaling laws
  • SLE
  • Ising model
  • colloids
  • clustering
MSC:
  • 82C20
  • 82C26
  • 82C27
  • 82C05
  • 82C44
  • 82C43
  • 82Cxx
  • 82-xx
Abstract
Colloids are binary mixtures of molecules with one type of molecule suspended in another where all non-overlapping arrangements are equally likely. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters, suggesting a phase transition occurs as the density of both types of molecules increases. In this talk we will discuss various strategies for sampling configurations of non-overlapping particles including lattice gasses and colloids in order to gain empirical evidence of this conjecture. Then we will show how to characterize the two phases for a general family of "interfering colloid models" by demonstrating that they exhibit a "clustering property" at high density and not at low density. The clustering property holds when there is a region with very high area to perimeter ratio and very high density of one type of molecule. A special case of interfering colloids is mixtures of squares and diamonds on Z^2, which correspond to the Ising model at fixed magnetization.