Kinetic theory of flux-line hydrodynamics: Liquid phase with disorder

We study the Langevin dynamics of flux lines of high-${\mathit{T}}_{\mathit{c}}$ superconductors in the presence of random quenched pinning. The hydrodynamic theory for the densities is derived by starting with the microscopic model for the flux-line liquid. The dynamic functional is expressed as an expansion in the dynamic order parameter and the corresponding response field. We treat the model within the Gaussian approximation and calculate the dynamic structure function in the presence of pinning disorder. The disorder leads to an additive static peak proportional to the disorder strength. On length scales larger than the line static transverse wandering length and at long times, we recover the hydrodynamic results of simple frictional diffusion, with interactions additively renormalizing the relaxational rate. On shorter length and time scales line internal degrees of freedom significantly modify the dynamics by generating wave-vector-dependent corrections to the density-relaxation rate.