Abstract
In order to illustrate the power of the equilibrium statistical mechanical formalism, beyond ideal gases, we give here an outline of the theory of phase transitions. We start with the prototypical model of such transitions, the Ising model, discuss its mean field approximation, its behavior at high and low temperatures and the extension of these results to a variety of other models, including those covered by the Pirogov-Sinai theory. We end this chapter by a brief overview of critical phenomena.
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Notes
- 1.
The name comes from Ising who was student of Lenz. Ising solved the one-dimensional model in his thesis but his life in Germany was made impossible by the Nazi regime and he emigrated to the United States. For more information on Ising’s life, see Kobe [195] and for the history of the Ising model, see Brush [60].
- 2.
One can introduce a stochastic dynamics for that model, but we will not discuss that.
- 3.
Below we will use the same notation \(B(\Lambda )\) for the set of nearest neighbor pairs under consideration, although the latter may vary between different formulas.
- 4.
One may also ask whether, in systems where there is a genuine time evolution, and there are several equilibrium states, the system might jump from one state to another. In a finite system, this will eventually happen but, for the usual macroscopic systems, it will take a time much longer than the one of the universe.
- 5.
This follows from Griffiths’ correlation inequalities that prove that expectation values \(<s_A>_{\Lambda , \beta , h}\) increase with \(\Lambda \): \(<s_A>_{\Lambda , \beta , h}\le <s_A>_{\Lambda ', \beta , h} \) if \(\Lambda \subset \Lambda '\). This implies that the functions \(<s_A>_{\Lambda , \beta , h}\) converge as \(\Lambda \uparrow \mathbb Z^d\), for all finite subsets \(A\subset \mathbb Z^d\), and from this one can deduce that the measures \(\mu _{\Lambda , \beta , h}\) converge also, because linear combinations of functions of the form \(s_A\) form a dense set in \({L^1} (\Omega , \mu )\) or in \(\mathcal{C} (\Omega )\), in the corresponding topologies (a set E of functions is dense in \({L^1} (\Omega , \mu )\) if, \(\forall f \in {L^1} (\Omega , \mu )\), \(\forall \epsilon >0\), \(\exists g \in E\), with \(\int _\Omega |f(x)-g(x)| d\mu (x) \le \epsilon \). The same definition holds for dense sets in \(\mathcal{C} (\Omega )\), with the integral replaced by \(\sup _{x \in \Omega } |f(x)-g(x)|\)). See e.g. Griffitths [161, 162, Sect. VI], Kelley and Sherman [188], Glimm and Jaffe [148, p. 83], Friedli and Velenik [131, Sect. 3.6].
- 6.
See Sect. 9.7.6 for a discussion of non translation Gibbs states.
- 7.
The French mathematician Gustave Choquet extended the notion and theory of simplexes to infinite dimensional spaces.
- 8.
For example, an interval in one dimension, a triangle in two dimensions, a tetrahedron in three dimensions etc.
- 9.
From that point of view, non extremal states, namely states that are convex combinations of extremal ones are not physical although they are useful mathematically.
- 10.
David Wallace thinks that this non-uniqueness of the equilibrium state is a problem for the Botzmannian approach, see [326, Sect. 5], because it does not explain why one has the same probability to arrive at any given Gibbs state. But that can be answered, again, by a typicality argument, see [155, Sect. 7].
- 11.
If we had \(|s_i|\le C\) instead of \(|s_i|\le 1\), we would replace \(s_i s_j+1\) by \(s_i s_j+C^2\) and the arguments below would still be valid.
- 12.
Peierls’ argument was not completely rigorous. The first rigorous proofs of a phase transition in the Ising model were given by Dobrushin in [102] and by Griffiths in [160].
- 13.
Actually, using correlation inequalities, one can show that the limits \(\Lambda \uparrow \mathbb Z^d\) of the states with “plus” or “minus” boundary conditions exist, because expectation values \(<s_A>_{\Lambda , \beta , +}\) decrease with \(\Lambda \): \(<s_A>_{\Lambda , \beta , +}\ge <s_A>_{\Lambda ', \beta , +} \) if \(\Lambda \subset \Lambda '\) (see Griffitths [162, Sect. VI] or Friedli and Velenik [131, Sect. 3.6]).
- 14.
See Bricmont, Lebowitz and Pfister [47] for a simple and general proof.
- 15.
To be precise, the Euclidean version of such a theory, see Glimm and Jaffe [148].
- 16.
This is considered one of the hardest open problem in the field of lattice spin models.
- 17.
Each site has 2d nearest neighbor, but each pair contains 2 sites.
- 18.
Given a site i one has 2d choices of a site j with \(|i-j|=1\) and then \(2d-2\) choices of a site \(k \ne i\) with \(|j-k|=1\) and \(|i-k|=\sqrt{2}\). We divide by 3 because each triangles contains three sites.
- 19.
Strictly speaking, we should distinguish here between exponents \(\alpha \) and \(\alpha '\), depending on whether \(T\rightarrow T_c\) from above or from below, but we will not do that distinction, since in the examples discussed here both exponents are equal. The same remark holds for the exponent \(\nu \) defined in (9.9.5) below, and could have been made about the exponent \(\gamma \) in (9.9.1), where, for \(T<T_c\), it would be defined through the one sided derivative of \(m( \beta , h)\) with respect to h at \(h=0\).
- 20.
Here \(\alpha =0\) means a logarithmic singularity rather than a discontinuity as in the mean field model, see (9.9.7) The proof of \(\delta = 15\) is in [68].
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Bricmont, J. (2022). Phase Transitions. In: Making Sense of Statistical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-91794-4_9
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