Curie-Weiss model

Treating a two-level system in an individual way is, of course, a delicate matter and can be quite misleading, as will be seen later in our discussion of the Curie-Weiss model. This point cannot be stressed often enough. It is only for reasons of simplicity that two-level systems are used here for purposes of illustration. 

In summary, statistical quantum mechanics permits us to derive strictly classical observables (such as the classical specific magnetization operator) by appropriate limit considerations (such as a limit of infinitely many spins in case of the Curie-Weiss model). However, statistical quantum mechanics cannot cope with fuzzy classical observables (for finitely many degrees of freedom) since different decompositions of a thermal state Dp are considered to be equivalent. The introduction of a canonical decomposition of Dp into pure states will give rise to an individual formalism of quantum mechanics in which fuzzy classical observables can be treated in a natural way. 

It is, unfortunately, not simple to compute the maximum entropy decomposition in molecular situations. We shall therefore consider again the simpler example of the (quantum-mechanical) Curie-Weiss model with the Hamiltonian ... 

FIGURE 11 An entropy function in the sense of fluctuation (i.e., large-deviation) theory, describing how fast the mean magnetization of a spin system gets classical with an increasing number of spins. The figure is based on an approximate calculation for the Curie-Weiss model. The temperature is fixed and has been taken here as one third of the critical (Curie) temperature. Above the Curie temperature the respective entropy Sn ,an would only have one minimum, nameiy, at m = 0. 

Bearing in mind the large-deviation considerations for the Curie-Weiss model, one could try to characterize molecules by some large-deviation entropy that describes how fast a nuclear molecular structure appears with increasing molecular nuclear masses. Such a large-deviation entropy would describe the decrease in fuzziness of the molecular nuclear structure when the nuclear masses increase. In the limit of infinite nuclear masses one expects a strictly classical nuclear framework, this not being fuzzy anymore at all. Such a large-deviation entropy would also nicely describe the quantum fluctuations round the strictly classical nuclear structure. 

Order-disorder transitions are examples of a second-order transformation. An order-parameter can be assigned that goes from one for a perfectly ordered state to zero for a completely random state, i.e., a solid solution. Using a technique similar to the Curie-Weiss model for ferromagnetism (see Chapter 25), it can be shown that the order parameter goes from 1 at low temperatures to 0 at the transition temperature and the system then becomes a solid solution again. 

Polarization versus temperature predicted using the Curie-Weiss model for spontaneous polarization. 

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