Modulated solutions and superfluid fraction for the Gross-Pitaevskii equation with a nonlocal potential at T≠0

Modulated solutions of the nonlocal Gross-Pitaevskii equation are studied at T≠0. Stationary states are computed by constructing a stochastic process consisting of a noisy Ginzburg-Landau equation. An order parameter is introduced to quantify the superfluid fraction as a function of the temperature. When the temperature increases the superfluid fraction is shown to vanish. This is explained qualitatively by the thermal appearance of defects that disconnect the system wave function. We also deduce an explicit formula for the introduced order parameter in terms of an Arrhenius law. This allow us to estimate the "energy of activation" to create a disconnection in the wave function.