New Approaches to Modern Physics/DEFINITIONS


118093 - New Approaches to Modern Physics
WINTER 2005

Advanced Statistical Mechanics


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[JOAN ADLER's lectures]

DEFINITIONS

  1. Thermodynamics - unlike the phase transitions in spin models such as the Ising model, which have the underpinnings of thermodynamics, percolation is an order-disorder transition which does not come with this baggage. Actually, in some ways because of recent work by mathematicians, the basis for percolation is somewhat better because there is an exact proof of hyperscaling in two dimensions.

  2. However there are analogies with magnetic transitions - formalized either via a dilute Ising model or thru the Potts mapping leading to an order parameter that we call P(p), the ``Percolation Probability'' which has the critical exponent beta like the magnetization. It is zero at Pc and normalised to one at p=1. (Mirror image of a magnetization graph.) The ``Mean Cluster Size'', S(p), is the analog of the susceptibility which diverges with critical exponent gamma.

  3. In site (bond) percolation each site (bond) is randomly occupied with probability p and is empty with probability q=1-p. Our goal is to determine the threshold value pc at which an infinite (spanning) connected cluster of occupied nearest-neighbor sites (bonds) occurs, and to describe the nature of the phase transition into the connected phase.

  4. The percolation order parameter P(p) is the probability that an occupied site (bond) belongs to the infinite cluster. The quantity P(p) is zero for p < pc and behaves

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