Modified Landau Theory of the Second Order Phase Transition

Abstract

1:he difficulty near the critical.point encountered by the Landau theory or the classical theory of the second order phase transition is removed with the least modification of the original framework. The essential point of our idea is to note that the temperature region in which the Landau theoryvalids depends sensitively on the way of defining the local order parameter. It is shown that the relations among the singularities of various thermodynamic quantities predicted by this modified form of the Landau theory are in good agreement with those obtained by nonclassical theories such as the static scaling theory. It is often pointed out that the Landau theory of the second order phase transi­ tion suffers from an internal inconsistency.1) In fact under the assumption of small ' fluctuation this theory predicts the large fluctuation at the critical point. The failure of this theory in the vicinity of the critical point, especially in the system with short range interaction, seems to be the consequence of this inconsistency. When we work with free energy density, there seems to be at least two possible ways to avoid this difficulty. One way is to take into account the higher order terms with respect to the local fluctuations and their spatial derivatives. **) The other way, which is of our main concern in this paper, is to change the defini­ tion of the local order parameter with temperature without taking into account the higher order terms. In the next section the definition of the local order parameter is carefully examIned, and this is indispensable for our whole discus­ sion. The procedure of removing the inconsistency is then presented. In § 3 the asymptotic behaviors of various thermodynamic quantities are studied and all the independent relations among the critical indices are derived which turn out to be indentical with those obtain~d by the static scaling theory.3),4) Through­ out the p:resent paper the language is used appropriate for the Ising ferromagnets though the same kind of arguments may apply to the classical gas or liquid system as well.