Dispersion in Second Sound and Anomalous Heat Conduction at the Lambda Point of Liquid Helium

The absence of a characteristic length at the lambda temperature $T_{\lambda }$ of liquid helium is used to determine the wave-number dependence of the phase fluctuations and the second-sound dispersion relation $\omega =a{k}^{\frac{3}{2}}$ where $\omega$ and $\overrightarrow{\mathrm{k}}$ are the frequency and wave number and $a\approx 0.1 {\mathrm{cm}}^{\frac{3}{2}}$ ${\sec }^{-1\mathrm{}}$. Further predictions are ${|T-T_{\lambda }|}^{-\frac{1}{3}}$ singular temperature variation for second-sound damping ($T<\mathrm{}{T}_{\lambda }$) and the thermal conductivity ($T>\mathrm{}{T}_{\lambda }$).