Relation between positional specific heat and static relaxation length: Application to supercooled liquids

A general identification of the {\em positional specific heat} as the thermodynamic response function associated with the {\em static relaxation length} is proposed, and a phenomenological description for the thermal dependence of the static relaxation length in supercooled liquids is presented. Accordingly, through a phenomenological determination of positional specific heat of supercooled liquids, we arrive at the thermal variation of the static relaxation length $\xi$, which is found to vary in accordance with $\xi \sim (T-T_0)^{-\nu}$ in the quasi-equilibrium supercooled temperature regime, where $T_0$ is the Vogel-Fulcher temperature and exponent $\nu$ equals unity. This result to a certain degree agrees with that obtained from mean field theory of random-first-order transition, which suggests a power law temperature variation for $\xi$ with an apparent divergence at $T_0$. However, the phenomenological exponent $\nu = 1$, is higher than the corresponding mean field estimate (becoming exact in infinite dimensions), and in perfect agreement with the relaxation length exponent as obtained from the numerical simulations of the same models of structural glass in three spatial dimensions.