Quantum particle in a washboard potential. I. Linear mobility and the Einstein relation
Yong-Cong Chen, Joel L. Lebowitz · Physical review. B, Condensed matter · 1992
The Einstein-Kubo relation between the diffusion constant D and the linear mobility \ensuremath{ u} is investigated for a dissipative quantum particle in a potential V(x)=-Fx+${\mathit{V}}_{0}$cos(${\mathit{k}}_{0}$x). This system is directly related to a current-biased Josephson junction. It is shown that D=kT\ensuremath{ u} holds to all order in ${\mathit{V}}_{0}$ for a general dissipation spectrum J(\ensuremath{\omega}), as long as J(\ensuremath{\omega}) is linear near \ensuremath{\omega}\ensuremath{\rightarrow}0 and is not pathological. A generalized version of the Einstein relation is shown to hold also for sub-Ohmic dissipation, J(\ensuremath{\omega})\ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}^{\mathit{s}}$, 0s1 at \ensuremath{\omega}\ensuremath{\rightarrow}0 where the motion is subdiffusive. The super-Ohmic case 1s2 is also discussed.