Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

the boundary between mathematics and physics 409defined critical temperature at which a phase transition takes place. If we turnthe external field off above the critical temperature, the material no longerexhibits magnetic properties. However, if we turn it off below the criticaltemperature, the material retains a residual magnetic field, a phenomenoncalled ‘spontaneous magnetization’.Spontaneous magnetization appears as a collective property of large numbersof ferromagnetic atoms. Although the physical property presumably arises fromthe interaction of individual atoms, Pierre Weiss (1907) proposed the simplifiedmodel of a ‘molecular field’ representing the magnetic force η produced by thecollection of atoms and postulated the self-consistent equation η = tanh(η/T ),where T is a variable representing the temperature. This simple and primitivepicture, the most elementary example of the class of mean field models, in factdoes exhibit some of the key features of the physical system it is intended todepict. It shows a phase transition at a critical temperature, and in general itsqualitative features show that we are mathematically on the right track, eventhough the numerical values we obtain from it are in fact not in accord withphysical measurements (Yeomans, 1992,Chapter4). In spite of the well-knownshortcomings of mean field theory, it is the usual starting point for workers instatistical mechanics investigating a new system (Fischer and Hertz, 1991,p.19).A partial explanation of the success of Weiss’s highly simplified model wasprovided by a Japanese physicist (Husimi, 1954), who pointed out that Weiss’smolecular field can be derived as the large N limit of a collection of classicalspin variables σ i =+1, −1, interacting through a Hamiltonian in which everyspin acts on every other spin—in the large N limit the interaction strengthis infinitesimal, that is to say, its strength is of order 1/N. The mathematicaldetails of Husimi’s observation were worked out later in (Kac, 1968)—see also(Thompson, 1972, §4.5).Physicists use the phrase ‘mean field model’ to refer to two distinct categoriesof model. The first is a model of the Weiss type, in which the collective actionof a large set of microscopic variables is replaced by a macroscopic quantityrepresenting an average or mean; the second is an interaction model of thetype described by Husimi and Kac, in which microscopic variables (such asspins) interact with an unbounded number of other microsopic variables. Thetwo types of model, however, are not completely equivalent mathematically,even in the paradigmatic case of the Curie–Weiss model.It is clear in the case of mean field models that we are dealing withmathematical caricatures of physical systems, that nevertheless exhibit some ofthe most important broad features of critical phenomena and phase transitions.It is tempting to say that these models are simply mathematical objects in theirown right that are related in only a very rough and ready way to the real