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 $p=11 - 29\,\mbox{ bar}$. The analysis supports a conclusion that superfluid $^3$He-B infused into high-porosity silica aerogels is a gapless superfluid at all pressures.
Categories: cond-mat.supr-con