The Heat Capacity of ${}^3$He-B in Silica Aerogel

The thermodynamic potential for superfluid ${}^3$He-B embedded in a homogeneously distributed random potential is calculated from a quasiclassical reduction of the Luttinger-Ward functional to leading order in k_{\mbox{\tiny B}} T_c/E_f. The resulting functional provides an extension of the Ginzburg-Landau free energy functional to all temperatures $0<T\le T_c$. Theoretical predictions based on this functional for the heat capacity of superfluid ${}^3$He-B embedded in homogeneous, isotropic silica aerogel are in good agreement with experimental reports for superfluid ${}^3$He-B infused into 98.2% porous silica aerogel over the pressure range $p=11-29bar$. The analysis supports a conclusion that superfluid ${}^3$He-B infused into high-porosity silica aerogels is a gapless superfluid at all pressures.