Order of the Phase Transition Related to the Interaction and the Degeneracy

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Abstract
It is investigated how the order of phase transition in a system of many identical interacting subsystems depends on the degeneracies of the states of each subsystem and on the interaction between subsystems; in particular, on the symmetry and uniformity of the degeneracy and on the symmetry of the interaction. By denoting the state of the i-th subsystem by a spin-like variable σi (i = 1,2,…,N), the degeneracy is expressed as D(σi) as a function of σi and the interaction between the i-th and j-th subsystems as V(σi,σj). (In the text use is made of such a normalized value as ΣσiD(σi) = 1). Then the symmetry means the equalities D(σi) = D(-σi) and V(σi,σj) = V(-σi,-σj) and the uniformity of degeneracy means the independency of D(σi) upon σi. The system of magnetic spins with exchange interactions belongs to the uniform symmetric category, which is used as a reference standard system in the present paper. The thermostatistical average of σi which is denoted by σ represents the order parameter in the system. Dependence of σ upon temperature and some applied field are investigated to see the features of the phase transition. In the non-uniform symmetric case it is found that the phase transition of first order can occur for an adequate form of D(σi), and that it is necessarily of second order in the uniform symmetric case. The induced transition by the applied field is also possible in the former case. In the asymmetric case the phase transition of a peculiar type as well as the induced transition may occur. Under some respective conditions on the degeneracy and interaction parameters, transitions of first and second orders can occur. Analytic properties of the partition functions are also discussed. An exact problem is proposed for the simplest model which is expected to exhibit the first-order phase transition.

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