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Article Dans Une Revue Philosophical Magazine Année : 2010

On the $p,q$-binomial distribution and the Ising model

Per Håkan Lundow
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A Rosengren
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Résumé

We employ $p,q$-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case $p=q$. We take our investigation to the simple $d$-dimensional lattices for $d=1,2,3,4,5$ and fit $p,q$-binomial distributions to our data, some of which are exact but most are sampled. For $d=1$ and $d=5$ the magnetisation distributions are remarkably well-fitted by $p,q$-binomial distributions. For $d=4$ we are only slightly less successful, while for $d=2,3$ we see some deviations (with exceptions!) between the $p,q$-binomial and the Ising distribution. However, at certain temperatures near $T_c$ the statistical moments of the fitted distribution agree with the moments of the sampled data within the precision of sampling. We begin the paper by giving results on the behaviour of the $p,q$-distribution and its moment growth exponents given a certain parameterization of $p,q$. Since the moment exponents are known for the Ising model (or at least approximately for $d=3$) we can predict how $p,q$ should behave and compare this to our measured $p,q$. The results speak in favour of the $p,q$-binomial distribution's correctness regarding their general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution is not settled though.

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Dates et versions

hal-00605633 , version 1 (03-07-2011)

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Per Håkan Lundow, A Rosengren. On the $p,q$-binomial distribution and the Ising model. Philosophical Magazine, 2010, 90 (24), pp.3313-3353. ⟨10.1080/14786435.2010.484406⟩. ⟨hal-00605633⟩

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