A Continuous Ising Model Exhibiting Phase Transitions of First or Second Order

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A continuous Ising model is studied which is an analogue of the discrete Ising model with degeneracy presented recently. The state density D(σ_i) of each component system is given as a sum of Gaussian error functions of the continuous state variable σ_i (i=1, 2,・・・, N), N denoting the total number of component systems : D(σ_i)=A exp(-aσ_i^2) + B exp[-b(σ_i-σ_0)^2] + C exp[-c(σ_i+σ_0)^2], where A, B, C, a, b, c and σ_0 are constants only conditioned by the relation (B/C)^2 = b/c. The free energy is obtained as a function of order parameter equal to the thermal average of one σ_i in a mean field approximation, on the basis of which the phase transition is investigated. In the asymmetric case B≠C, the transition is necessarily of first order. In the symmetric case B=C, the transition depends on the parameters D = (A/B)√<b/a> and α=1-(2a)^<-1> +(2b)^<-1> . The transition of first order occurs, so far as D(3α^2 -2)> 4 and that of second order does, otherwise. Temperature dependences of the order parameter are also investigated.
理論物理学刊行会の論文
1976-09-25