Conductivity of superconductors in the flux flow regime

We develop a theory of conductivity of type-II superconductors in the flux flow regime taking into account random spatial fluctuations of the system parameters, such as the gap magnitude $\mathrm{\Delta }\left(\mathbf{r}\right)$ and the diffusion coefficient $D\left(\mathbf{r}\right)$. We find a contribution to the conductivity that is proportional to the inelastic relaxation time ${\tau }_{\mathrm{in}}$, which is much longer than the elastic relaxation time. This contribution is due to Debye-type relaxation, and it can be much larger than the conventional flux flow conductivity due to Bardeen and Stephen. The new contribution is expected to dominate in clean superconductors at low temperatures and in magnetic fields much smaller than $H_{\mathrm{c}2}$.