New Approaches to Modern Physics/ANTS


118093 - New Approaches to Modern Physics
WINTER 2005

Advanced Statistical Mechanics


[Course Summary][Links to Lecture Notes and Homework][Announcements] [For Lecturers]
[JOAN ADLER's lectures]

CLUSTERS, ANTS and TERMITES

  1. A percolation cluster, especially near pc is an object of interest independently of the percolation question. DeGennes termed the problem where some particle diffuses on a cluster as ``the ant in the labyrinth''. A diffusing particle selects one of its neighbours randomly and goes there. We call the average distance traveled by the ants R(t) .
    1. For p well above pc this is normal diffusion with R 2 proportional to t .

    2. For p well below the critical point the diffusion approaches a constant value.

    3. Near the threshold there is anomalous diffusion, which was of intense interest in the 80's. Time permitting I will return to this next week.

  2. A variant is to set the (until now) absent bonds to be normally conducting and the (until now) connected ones superconducting. The walkers here are called ``termites'' and a whole new set of phenomena can be studied.

  3. The ant concept has two applications of immediate interest to us:

    1. one is an easy way to introduce directed percolation - see the next page

    2. the second is that it was via ant diffusion simulations that Amnon Aharony and I stumbled over a set of models that we called diffusion percolation that turned out to be analogous to bootstrap percolation which has remained an area of interest both in applications and theory until today. (Bootstrap percolation was independently discovered by several groups, including Ilan Reiss at the Technion.) Some bootstrap applications even relate to Condensed Matter Physics although others are in Computer Science.