Thermodynamics of liquidHe4

Using the equation-of-motion method we derive, based on the pair approximation [Hartree-Fock (HF), BCS] of the hard-sphere model for superfluid He, the Bogoliubov equation for the superfluid phase. At the temperature $T=T_{\lambda }$ the off-diagonal pairing factor (BCS) has to vanish, and the excitation energy must become that of the HF solution. This condition yields a transition temperature $T_{\lambda }=2.3\mathrm{}°$ K. The superfluid density ${\rho }_S$ as a function of temperature is also investigated. Furthermore assuming the net translational kinetic energy of the liquid to be $E_T=\left(\frac{1}{2}\right)\Omega ({\rho }_S{V}_S^2+{\rho }_N{V}_N^2)$ we investigate the response of our system to the corresponding perturbation $H_I$. We calculate the persistent current components $〈J^N\mathrm{}\left(0\right)\mathrm{}〉$ and $〈J^S\mathrm{}\left(0\right)\mathrm{}〉$ which exhibits the right behavior for $T\to 0$ and $T\to T_{\lambda }$.