Timeline for Rigorous statistical mechanics: difficulty of realistic models
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 13 at 13:47 | comment | added | Plemath | Thank you very much!! Also, congratulations about your beautiful book!! | |
| Sep 13 at 12:05 | comment | added | Yvan Velenik | Of course, there are examples in which you can prove such a transition in a much simpler way (for instance the Widom Rownlinson model, see the original proof by Ruelle), but these are physically far less satisfactory. In both cases, the argument goes through discretization (although, in the Widom-Rowlinson case, you can avoid doing that). | |
| Sep 13 at 12:05 | comment | added | Yvan Velenik | The best result to date remains the work by Lebowitz, Mazel and Presutti. In the latter, they consider a model with a Kac-type interaction and perturb around the mean-field model. The paper implementing the full proof is 70 pages long and is very technical. Browsing through it might give you an idea of the difficulties involved. And this is only to establish the (presumably simpler) liquid-gas phase transition. And they cannot treat the case of 2-body interactions only (they need an additional repulsive 4-body interactions). | |
| Sep 13 at 12:05 | comment | added | Yvan Velenik | You need to be familiar, at least, with the Peierls argument for the Ising model (or its extensions, such as the Pirogov-Sinai theory). I'll discuss the case of the Ising model for simplicity. In the latter case, you prove that the ground state is stable under perturbation (at very low temperature). This requires that you understand the ground state and that you find a way to sum over all possible excitations. Off lattice, this leads to (at least) two major difficulties: determine around what you want to perturb, and then sum over all excitations. | |
| Sep 13 at 11:45 | comment | added | Plemath | @YvanVelenik thanks for answering! Can you sketch this "combinatorial" difficulty? It would be very helpful for incomers to the field! | |
| Sep 13 at 9:24 | comment | added | Yvan Velenik | Note that even the liquid-gas transition is not known for a model in the continuum with only 2-body interactions (as the Lennard-Jones example). | |
| Sep 13 at 9:24 | comment | added | Yvan Velenik | (The downvote is not mine, but) (1) a solid-liquid transition will be first-order, so the correlation length will not diverge; the motivation in the first paragraph is thus only valid for models with a continuous transition; (2) we can prove the existence of a phase transition in the Ising model (and in other much more complicated lattice models) without having recourse to exact computation (using, for instance, the Peierls argument or its generalizations). The main difficulty of leaving the lattice is of a "combinatorial" nature. | |
| Sep 13 at 6:21 | comment | added | Carlo Beenakker | @Plemath -- in the Ising model the integral is a finite lattice sum over nearest neigbors; it has a special structure on a 2D lattice which allows for an exact evaluation by means of transfer matrices; the Lennard-Jones integrals lack the simplifying structure of the Ising sum. | |
| Sep 13 at 1:08 | comment | added | Abdelmalek Abdesselam | Why the downvote? | |
| Sep 13 at 0:22 | comment | added | Plemath | Thank you for taking the time. However, the core of my question is this: this high dimensional integral can be handled in the case of the Ising model, to rigorously show the existence of phase transition. In Lennard-Jones, we don't know how to do it (rigorously). What are some of the main obstacles for the generalization? | |
| Sep 12 at 21:27 | comment | added | Carlo Beenakker | I've added a section on the Lennard-Jones case. | |
| Sep 12 at 21:27 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
Lennard-Jones potential
|
| Sep 12 at 17:30 | comment | added | Plemath | Exactly what @SamHopkins said. Let's stick to the question. If the general question is too broad, let's stick to the concrete example I gave. | |
| Sep 12 at 16:54 | comment | added | Sam Hopkins | This seems like more of an answer to "why is it okay to restrict to these simplified models?" than "what is hard about extending to more sophisticated (and hence, more realistic) models?" | |
| Sep 12 at 11:32 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |