Analytic Free Energy: A Basis for Scaling Laws

Abstract

Using the idea that the thermodynamic potential, whose natural variables are intensive, is analytically continuable in all these variables from the liquid state to the vapor state, we have arrived at a picture of a phase transition which is in accord with all currently observed behavior in the neighborhood of a critical point. Using functions which are algebraic in both the applied field and the temperature, we have shown for a magnetic system that (a) Rushbrooke's inequality holds with the equality sign as a consequence of the symmetry of the free energy in the applied field, and (b) there is a set of relations between the critical exponents of derivatives of the free energy higher than the second. Although they only lead to rational critical exponents, algebraic functions are uniquely suited to this problem since they automatically satisfy the requirements of analytic continuation. The homogeneous functions of Widom with rational critical exponents are shown to be a special case of these algebraic functions, and several examples are given which illustrate some apparently anomalous behavior, such as (a) the critical exponent $\ensuremath{\beta}$ may have an odd denominator, (b) the critical exponent $\ensuremath{\delta}$ may have an even numerator, and (c) the primed and unprimed critical exponents may be unequal.

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