From microscopic theory to Boltzmann kinetic equation: Application to vortex dynamics

We show how to lift the problem of calculating the force acting on a topological defect in a superfluid from the microscopic to the semiclassical level: Starting from the microscopic kinetic equations for a clean superconductor, we derive a Boltzmann equation for the quasiparticle distribution function in and around the defect. The velocity q˙ and force p˙ appearing in this Boltzmann equation are given through the Hamiltonian equations q˙=∂_pE_n(p,q) and p˙=-∂_qE_n(p,q), where E_n(p,q) denotes the (nth branch in the) spectrum of the quasiparticles in the vicinity of the defect. Second, we reformulate the microscopic expression for the force acting on the defect in terms of the total momentum transfer of the quasiparticles from the heat bath to the vortex core. We illustrate our result with an application to vortices in s-wave superconductors, where we derive the vortex equation of motion and identify the Magnus, Hall, and dissipative forces.