Noise Theory for Hot Electrons

The subject of this note is the extension of the Nyquist theorem, which relates the electrical noise in a conductor in thermal equilibrium to the Ohmic conductivity,
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to the “hot electron” situation in which there is a steady electric field E strong enough to disturb the distribution of electrons among Bloch states and for which the steady current density J will not in general be proportional to E.
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For simplicity, we discuss the fluctuations of velocity, v, for a single electron. For a steady state the time average, equation, of a function ψ of the electron's state is the same as the ensemble average at any instant over all N electrons in the body: ψ̅ = ifψ ψ (ψ).