Superfluid helium in fully saturated porous media

Abstract

The flow of superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ through spongelike media at full saturation is modeled by the flow of current through an Ohmic network with random resistors. Solving Kirchhoff’s equations leads to the conclusion that the superfluid critical point is a percolation threshold, with critical exponent 1.7. The fractal dimension of the percolating cluster is 2.6. These lead to a specific-heat exponent α=-5.4, by the Josephson hyperscaling relation. Existing experiments apparently do not cover the critical region. Instead, they measure ‘‘mean-field’’ exponents, whose values for Vycor, aerogel, and xerogel can all be reproduced by choosing appropriate distribution functions for the resistors.