Fluctuations of the Phase Difference Across an Array of Josephson Junctions in Superfluid He-4

We present a formal thermodynamic treatment of superfluid flow in a Josephson junction. We show that the current and the phase difference are thermodynamic conjugate variables. We derive quantitative expressions for the rms fluctuations of these variables. Also, we discuss the thermodynamic stability and the thermal activation to the phase slip region. We apply the developed formalism to show why an array of apertures in He-4 can exhibit the Josephson effect near the Lambda transition despite strong thermal fluctuations.