Diffusion noise and the high-frequency extension of the Einstein relation

Whereas the low frequency diffusion constant is uniquely related to the velocity auto correlation function, the extension to very high frequencies is not unambiguous. In this article we introduce a quantum mechanical Kubo-like form. This form satisfies the standard Einstein relation at all frequencies. The diffusion noise source now contains the quantum correction factor in a similar fashion as Nyquist's formula for thermal noise. The internal consistency is more evident than in the formulation of a previous paper1).