Quantum particle in a washboard potential. I. Linear mobility and the Einstein relation

The Einstein-Kubo relation between the diffusion constant D and the linear mobility {ν} is investigated for a dissipative quantum particle in a potential V(x)=-Fx+V₀cos(k₀x). This system is directly related to a current-biased Josephson junction. It is shown that D=kT{ν} holds to all order in V₀ for a general dissipation spectrum J({ω}), as long as J({ω}) is linear near {ω}{→}0 and is not pathological. A generalized version of the Einstein relation is shown to hold also for sub-Ohmic dissipation, J({ω}){~}ωˢ, 0s1 at {ω}{→}0 where the motion is subdiffusive. The super-Ohmic case 1s2 is also discussed.