# Talk:Potts model

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In two dimensions, phase transition is first order for q >4, not q>3. (exact solution by Rodney Baxter, Journal of Physcs A, 1973).

This has been fixed.

In three (or more) dimensions, phase transitions first order for q >2. (strong numerical evidence).

(All this is for J >0). For J < 0, existence of phase transitions depends on which lattice.

(This is in discussion, because I need more practice before I start editing articles directly).

You should feel free to make it better. Mistakes can always be reverted.

## modular group symmetry

One of the more exciting things to me is, a certain variant of the 1-D Potts model, with a long-range interaction, called the Kac model, has a canonical ensemble that is fractally self-similar, under the usual fractal semi-group/groupoid of the modular group. In particular, the integral over the space of states is nothing other than the Minkowski question mark function. The force in the Kac model is exponentially decaying, can can thus be visualized to correspond to a massive particle mdeiating the force. Why is it that such a simple model has hyperbolic aspects, with all those number theoretic implications, connections to elliptic curves, and also all these particle physics aspects? I love this stuff. linas 05:36, 14 September 2005 (UTC)

## "The partition function can be used to obtain the thermodynamic properties in the usual way."

Needs more explanation, or at least a link to somewhere. GangofOne 21:56, 16 September 2005 (UTC)

Hm, I'll try to reword this; what I was trying to say is that the all the stuff discussed in the article on the partition function can be applied directly to this case. linas 23:42, 16 September 2005 (UTC)

## To Do from the article

I am moving this list here. It shouldn't be in the article. Deepak 20:15, 12 May 2006 (UTC)

### ToDo: (article under development)

• Show graph of energy states of the Ising model.
• Show the general form of the solution for a finite-range interaction.
• Show that the infinite-range force (Kac model) is the trace of a transfer operator.
• Mark Kac (1956); Foundations of Kinetic Theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, Vol. 3, pp. 171-197
• Show that the largest eigenvalue of the transfer operator, per the Ruelle-Perron-Frobenius theorem, is the state giving the thermodynamic equilibrium of the system.