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Stanley [229]. According to him, the terminology refers only to the classical thermodynamic variables, disregarding the magnetic ones. , of a magnetic gas or fluid), and what we now call volume remains a particle number. Lattice gases and spin systems 36 without it, and the simple paradigmatic models it provoked (such as the Ising model), most of the progress of the last 80 years would not have been possible. Before we turn, in the next chapter, to the rigorous probabilistic setup of Gibbs measures for lattice spin systems, we will look at two singular situations that were studied in the early days of these models, and that gave rise to some confusion.

Therefore, no jump in the derivative of the free energy can occur. The problem with this argument is that, in general, it does not survive the thermodynamic limit. In any event, Ising drew the conclusion that something like a real phase transition, with a magnetization having a real jump-discontinuity at the values h = 0, cannot occur in his model. 5 The Curie–Weiss model Already in 1907, Weiss [259], following the discovery of the critical temperature (Curie temperature), above which ferromagnetism disappears, by Pierre Curie in 1895, had developed a theory of ferromagnetism based on a spin system analogue of the van der Waals theory.

When introducing statistical mechanics, we had been defining entropy or free energy in terms of logarithms of partition functions with a given number of particles. This was actually imprecise. If we claim that S(E, V, N ) = ln z E,V,N , even in the simplest example it is not strictly true that this function will be strictly a homogeneous function in N . Therefore, the true relation between statistical mechanics and thermodynamics stipulates such relations ‘to leading order in N ’, where N is now really the number of particles.

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