Abstract
The Ising model2 plays a very special role in statistical mechanics and generates the simplest nontrivial example of a system undergoing phase transitions.
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References
This chapter is mostly taken from the paper Instabilities and phase transitions in the Ising model, La Rivista del Nuovo Cimento, 2, 133–169, 1972. Copyright by Societá Italiana di Fisica.
For a history of the Ising model see [Br69].
The original solution for the free energy of the Ising model in two dimensions can be found in [On44]. It was preceded by the proofs of existence of Peierls, [Pe36] , and van der Waerden, [VW41] , and by the exact location of the critical temperature by Kramers and Wannier, [KW41]. The spontaneous magnetization was found by Onsager, [KO49], but the details were never published; it was subsequently rediscovered by Yang, [Ya52]. A modern derivation of the solution is found in the review article by Schultz, Mattis and Lieb, [SML64]; the latter is reproduced in Sect. 7.4. Another interesting older review article is the paper [NM53]. A combinatorial solution has been found by Kac and Ward and can be found in [LL67], p. 538. Some aspects of this derivation were later clarified, and it has been discussed again in several papers, see [Be69]. Another approach to the solution (Kasteleyn’s pfaffian method) can be found in [Ka61].
In some cases the Ising model is a good phenomenological model for antiferromagnetic materials; this is the case of MnC12 • 4H2O, see [FS62], [Fi67a,b,c].
This term is usually omitted and in some sense its importance has only recently been recognized after the work of Dobrushin, Lanford and Ruelle, see [Do68a,b,c], [LR69]. In this chapter the main purpose is to emphasize the role of this term in the theory of phase transitions.
This definition is essentially in [LR69] where the equivalence of the above definition with a number of other possible definitions is shown. For instance the definition in question is equivalent to that based on the requirement that the correlation functions should be a solution of the equations for the correlation functions that can be derived for lattice gases or magnetic spin systems in analogy to those we discussed for the gases in Sect. 5.8. It is also equivalent to the other definitions of equilibrium state in terms of tangent planes (i.e. functional derivatives of a suitable functional: see [Ru69] , p. 184, [Ga81]).
The solution can also be found for instance in [NM53].
The expansion can be used as a starting point for the combinatorial solution mentioned above, see [LL67].
Of course we do not attach a deep physical meaning to the difference between these two approaches. Clearly they should be equivalent if one pretended to extract all possible information from them. What is really important is that the first questions raised by both approaches are very interesting and relevant from a physical point of view. One of the goals of the analytic theory of phase transitions is to understand the nature of the singularity at the critical point and at the “breaks” of the isotherms. A lot of interest has been devoted to this point and a number of enlightening phenomenological results are available. However the number of complete results on the matter is rather limited. An idea of the type of problems that are of interest can be obtained by reading the papers [Ka68] or the more detailed paper [Fi67].
This geometric picture of the spin configurations can be traced back at least as far as Peierls’ paper, [Pe36], and has been used, together with formula (6.4.12) to derive (6.6.8) (the “Kramers—Wannier duality” relation) and (6.6.9), [KW41]). A recent interesting generalization of the duality concept has been given in [We71], where some very interesting applications can be found as well as references to earlier works. The duality relation between (+) or (—) boundary conditions and open boundary conditions (which is used here) has been realized by several people. The reader can find other similar interesting relations in [BJS72] and further applications came in [BGJS73]. Duality has found many more applications, see for instance [GHM77] and, for a recent one, [BC94]. In particular a rigorous proof of the correctness of the Onsager—Yang value of the spontaneous magnetization is derived in [BGJS73].
The above proof is due to R.B. Griffiths and, independently, to R.L. Dobrushin and it is a mathematically rigorous version of [Pe36].
The number 333 is just an arbitrary constant and it is reported here because it appeared in the original literature, [MS67], as a joke referring to the contemporary papers on the KAM theorem (“Moser’s constant”). In fact it looks today somewhat confusing and quite strange: the modern generation does not seem to appreciate this kind of humour any more; they became more demanding and would rather ask here for the “best” constant; this is my case as well.
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© 1999 Springer-Verlag Berlin Heidelberg
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Gallavotti, G. (1999). Coexistence of Phases. In: Statistical Mechanics. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03952-6_6
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