5
$\begingroup$

Soft question: I am a mathematician self-learning statistical mechanics. The (mathematical) literature is concentrated on lattice models like the Ising model and the lattice-gas model. I understand the value of these models. However, what are some of the main obstacles to extending the results to more realistic models?

For a concrete example, why is it so hard to prove the existence of solid-liquid phase-transition for a collection of particles interacting with Lennard-Jones potential?

Improve this question
$\endgroup$
10
  • 2
    $\begingroup$ Elementary excitations of a Bose liquid is an interesting realistic application of quantum statistical physics. See. e.g., Ch. III Superfluidity (beginning on p. 85) of "Statistical Physics, Part 2, Theory of condensed states" by Lifshitz and Pitaevskii, Volume 9 of Course of Theoretical Physics (haidinh89.wordpress.com/wp-content/uploads/2015/08/…). $\endgroup$
    – Tom Copeland
    Commented Sep 12 at 1:14
  • $\begingroup$ I have not studied quantum mechanics yet. But quantum makes models even more complicated. Let's focus on classical statistical mechanics. $\endgroup$
    – Plemath
    Commented Sep 12 at 1:22
  • $\begingroup$ The Ising model would usually be considered a quantum system, so you've studied quantum statistical mechanics already! $\endgroup$
    – Michael Engelhardt
    Commented Sep 12 at 1:55
  • 2
    $\begingroup$ As far as I know the transition has not been proved for other simpler potentials. The difficulty is not the form of the potential but going from the lattice to the continuum, and in particular showing the breaking of translational symmetry, i.e., the spontaneous formation of a crystal. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Sep 13 at 1:17
  • 2
    $\begingroup$ Was thinking of something related to hard core, e.g., $V(r)=\infty$ for $r<r_1$ then $V(r)=-\epsilon$ for $r_1<r<r_2$, and then zero for $r>r_2$. I am not an expert though. A better source would be, e.g., arxiv.org/abs/1504.01153 $\endgroup$
    – Abdelmalek Abdesselam
    Commented Sep 13 at 2:32

1 Answer 1

Reset to default
1
$\begingroup$

General remark on why one would study simple models:
In the statistical mechanics of phase transitions one distinguishes relevant and irrelevant variables. A phase transition is associated with a divergent correlation length, irrelevant variables drop out of the model in that limit. The lattice constant is one irrelevant variable, which is why one studies lattice models rather than continuum models. The precise form of a short-range interaction potential is irrelevant, which is why one restricts oneself to a nearest-neighbor interaction.

Q: Why is it so hard to prove the existence of solid-liquid phase-transition for a collection of particles interacting with Lennard-Jones potential?

The phase transition is signaled by the appearance of long-range order in the pair correlation function $g(r)$, so you seek to evaluate a high-dimensional integral of the form $$g(|\vec{x}_1-\vec{x}_2|)\propto\int_V d\vec{x}_3\int_V d\vec{x}_4\cdots\int_V\vec{x}_N\,e^{-\beta H(\vec{x}_1,\vec{x}_2,\ldots \vec{x}_N)}$$ $$H(\vec{x}_1,\vec{x}_2,\ldots \vec{x}_N)=\sum_{i<j}\left(|\vec{x}_i-\vec{x}_j|^{-12}-|\vec{x}_i-\vec{x}_j|^{-6}\right),$$ in the thermodynamic limit $N,V\rightarrow\infty$ at fixed density $N/V$. Such a calculation can only be done numerically, which introduces errors and does not allow for a rigorous proof.

From universality arguments one would expect the Lennard-Jones liquid to have the same critical exponents as the Ising model, and computer calculations (on a system with $N\approx 180,000$) are consistent with that expectation.

Improve this answer
$\endgroup$
13
  • 1
    $\begingroup$ 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?" $\endgroup$
    – Sam Hopkins
    Commented Sep 12 at 16:54
  • 1
    $\begingroup$ Why the downvote? $\endgroup$
    – Abdelmalek Abdesselam
    Commented Sep 13 at 1:08
  • 1
    $\begingroup$ 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. $\endgroup$
    – Yvan Velenik
    Commented Sep 13 at 12:05
  • 1
    $\begingroup$ 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). $\endgroup$
    – Yvan Velenik
    Commented Sep 13 at 12:05
  • 1
    $\begingroup$ 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). $\endgroup$
    – Yvan Velenik
    Commented Sep 13 at 12:05

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .