Homogeneity of the Correlation Length in the Critical Region

Abstract

We examine a variant of Widom's homogeneity that permits us to consider critical indices that have different values above and below the critical temperature $T_c$. It is based on the observation that if homogeneity holds at all, it can only be expected to hold locally (that is, in a limiting sense as one approaches the critical point) rather than globally.. The inequality ${\nu }^{\prime }>\nu$ is considered in particular, where $\nu$ is the exponent that measures the temperature variation of the correlation length ${\kappa }^{-1\mathrm{}}$, and the prime refers to temperature below $T_c$. It is shown that this inequality is consistent with the local one-phase homogeneity of $\kappa$ in Widom's variables $x$ and $y$.