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 topolo gical defect in a superfluid from the microscopic to the semiclassical leve l: Starting from the microscopic kinetic equations for a clean superconduct or, we derive a Boltzmann equation for the quasiparticle distribution funct ion in and around the defect. The velocity (q) over dot and force (p) over dot appearing in this Boltzmann equation are given through the Hamiltonian equations (q) over dot = partial derivative(p)E(n)(p,q) and (p) over dot = -partial derivative(q)E(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 defec t in terms of the total momentum transfer of the quasiparticles from the he at 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.