New Approaches to Modern Physics/NEIGHBOURS


118093 - New Approaches to Modern Physics
WINTER 2005

Advanced Statistical Mechanics


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[JOAN ADLER's lectures]

HIGHER NEIGHBOURS, etc

  1. Until now I only mentioned nearest neighbours while we were getting the basic concepts down. Unfortunately, except for examples such as the beaker of glass and metal balls, real systems are usually more complex.
  2. The simplest extension toward reality is the inclusion of more than nearest neighbour interactions. For example in 2D - square and triangular lattices

    what what

    The square has 4 nearest neighbours, another 4 second neighbours and ...... If sites at all these distances are connected, will pc rise or fall? The triangular lattice has 6 nearest neighbours..........

  3. 3D - simple cubic lattice - 6 nearest neighbours, 8 second nearest neighbours........

    what

  4. What about longer range interactions? At some point we have to address the following:

    Percolation is an excellent model for so many macroscopic systems, and for microscopic systems such as say a lead/germanium alloy at low temperatures where lead is a superconductor. However truly quantum systems and magnetic systems can not be modeled by simple percolation models. There are ``quantum percolation'' models and a rich variety of models that are relevant, but I am going to keep within classical systems. There is plenty to talk about here. The best reference I know for Quantum Percolation is http://www.edpsciences.org/articles/epjb/pdf/2000/14/b0108.pdf which describes the case where an electron propagates in the tight-binding approximation on a lattice with some bonds or sites missing, but there is more than one model with this name.