Lattice Statistical Physics
Lattice statistical physics analyses complex statistics of sytems on regular lattices and includes
models of cooperative phase transitions (Potts-Ising models) as well as statistical models such as
percolation and self-avoiding walks.
There are five main techniques for analysing lattice statistics problems:
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Exact solutions: only possible for special cases.
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Closed-form approximations: based on approximating higher order statistics.
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Power series expansions: expanding about simple limits, eg. fully ordered or total disorder.
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Renormalisation group analyses: transforming the generators of random fields.
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Monte Carlo simulations: based on sampling using the statistical definitions.
Statistical physics has provided concrete examples of a number of concepts
that are believed to be important in the study of complex systems.
(MORE).
Finite Lattice Method (FLM) of series expansion
The FLM has proved to be a very powerful technique for obtaining power
series expansions, particularly in two dimensions.
A review of the method is given by Enting, 1996, Nuclear Physics
B (proceedings supplement) 47, pp180-187.
My work in the FLM involves a long-standing collaboration with
present and former staff
and students at The University of Melbourne, including
(most notably)
Tony Guttmann,
Iwan Jensen,
Aleks Owczarek,
Richard Brak, Nick Wormald, Doochal Kim,
Andrew Conway, Debbie Bennett-Wood, Keith Briggs and most recently Nathan Clisby
Application areas have included:
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Standard Potts model
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Self-avoiding walks and polygons
- Planar Potts models and related systems showing Kosterlitz-Thouless transitions.
Recent achievements
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With Iwan Jensen, wrote chapter "Exact Enumerations" for book
Polygons, Polyominoes and Polycubes. (ed. A. J. Guttmann, Springer 2009).
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Series for surface contributions on square lattice 3-state Potts model. (Order parameters with
free and fixed boundary conditions). Presented at `Counting Complexity' conference, July 2005, and published in conference volume.
- New finite lattice weights for more efficient calculation of surface series with free boundary
conditions. Published in J_Stat. Calculations ongoing.
- Investigation of Kosterlitz-Thouless transitions on square lattice systems:
* low-temperature series for 5-state model -- exploring phase diagram;
* low-temperature series for 6-state planar Potts model;
Preliminary results presented at Statistical Mechanics meeting, Melbourne, Decemeber 2005,
paper at Complex II conference, paper submitted 2011 to special issue to mark
90th birthday of Prof. C. Domb..
External links
Stochastic Cellular Automata
The objectives of this project are:
understanding the behaviour of deterministic cellular automata by considering
them as limits of stochastic cellular automata which can be investigated
using the standard techniques of lattice statistical physics listed
above.
using stochastic cellular automata to model real-world systems.
searching for further insight into the behaviour of the corner-transfer-matrix
method for series expansion and variational approximation.
investigation general statistical problems, including inverse problems,
in systems with complex behaviour.
Disclaimer
This page, its contents and style, are the responsibility of the author and do not represent the
views, policies or opinions of The University of Melbourne.
Ian Enting: Last update 10/7/11