|WikiProject Physics||(Rated Start-class, Mid-importance)|
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)
Transfer Operators, Subshifts of Finite Type, Topology, Measures, Etc
There is nothing unique to the Potts model in using transfer "operators". In statistical mechanics the terminology is a transfer matrix and it is defined for any model. In 1D problems transfer matrices allow one to solve models exactly. In 2D it's much more difficult, but Onsager solved the 2D Ising model by diagonalizing its transfer matrix. It is incredibly easy to show a particular example in 1D of a transfer matrix rather than going through an overly complicated discussion of topology and measures and subshifts of finite type, geez! Transfer matrices are standard in statistical mechanics literature and are easy to refer to. Any discussion of subshifts, etc, needs to not only be a minor part of this article - but they require references and should not supersede a typical transfer matrix discussion. They are not standard in discussions of 1D models in statistical mechanics. — Preceding unsigned comment added by 2001:8003:4400:EE01:31ED:FDF0:9829:338D (talk) 17:29, 14 May 2015 (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.
- Add inline links to the References (currently it is not clear which reference one should look at to verify and expand on a given section). --Natematic (talk) 23:23, 6 March 2016 (UTC)