A possibility to obtain equalities for the critical indices by thermodynamic way

Abstract

Tisza's theory (~) (TT) is, so far as we know, the sole most general and rigorous thermodynamic theory of second-kind phase transitions tha t justifies completely the divergences that some thermodynamic quantities, such as Cv, K z , Ce, K s for pure fluids and Cn, Xz, C,,,, Xs ~or simple uniaxial ferromagnetic systems, have at critical point. The key-point of the theory, based on a suitable reformulation of the conditions of thermodynamic stability, consists in considering the critical point as the one at which the usual thermodynamic formalism becomes singular and in characterizing it rigorously according to the mathematical point of view. Of course, because the stabil i ty conditions are involved in TT, it has to be possible to derive, from it, relations among the critical indices (~). As we shall see, neverthless, making definite the rigorous mathematical condition that characterizes a critical point, i t is possible to establish, besides, criteria on the ground of which we can say when we must expect inequalities and when certainly strict equalities for the critical indices. Such a possibility seemed, till now, not to be included in no thermodynamic theory of critical phenomena. For simplicity, let us consider a simple uniaxial ferromagnetic system; according to TT, the impor tant quantit ies (expressed in terms of T and H , intensive parameters conjugate to s and m) are then