ABSTRACT

Brownian motion is not merely random motion of a very fine particle; in gen-

eral it is randommotion of a physical quantity to be observed in a macrosystem

M. Toda, R. Kubo, and N. Saito ∗.

In the previous chapter the traditional method of the Boltzmann equa-

tion was introduced and it was shown that the Boltzmann equation leads to

the Drude formula under RTA. The full Boltzmann equation or the “beyond

RTA” Boltzmann equation was also introduced. The Bloch-Boltzmann the-

ory and the case of electron-phonon scattering in DC resistivity was explicitly

considered. One has to carefully analyze the premises on which the Boltzmann

equation, as applied to the electrons in metals, is based. We heuristically jus-

tified that using the uncertainty principle and recognizing that electrons can

be treated as wave packets with size much greater than a lattice constant†

and thus a semiclassical distribution function g(r,k, t) was written. The wave

vector k remains a good quantum number as an uncertainty in it (arising due

to finite mean free path that is further due to scattering) remains small as

Now compare and contrast the above situation with a situation in which

strong electron-electron interaction or any other strong interaction is present

such that momentum is no longer a good quantum number. Consider a sit-

uation in which the mean free path is of the order of lattice constant (a).

Then the uncertainty in momentum will be of the order of the reciprocal wave

vector (∆k ∼ 1a ) which is of the order of k itself. In such a situation g(r,k, t) itself cannot be defined and the Boltzmann equation has no justification for

such a case.