Proof of the Kawabata formula

In this section we give a proof of the Kawabata formula (52), following a method due to Kaveh (1984) and Mott and Kaveh (1985a, b). We assume that an electron undergoes a random walk, which determines an elastic mean free path l and diffusion coefficient D. If an electron starts at time t=0 at the point r0 then the probability per unit volume of finding it at a distance r, at time U denoted by n(r, t) obeys a diffusion equation 

Here d is the dimensionality of the system d=3 in our case. For large values of t quantum mechanics must give the same result, so if j/(ry t) is the wave function describing this system then 

Taking the time and space Fourier transforms of 2, multiplying by e iq fc and integrating over r0, we obtain the following key relationship between the diffusion coefficient and the eigenstates  

Here the average is over all states separated by a fixed energy difference hcom and N0(Ef) is the density of states per spin. This formula has been derived by various methods (MacMilliah 1981, Kaveh and Mott 198 la, b, Imry et a/. 1982, Lee 1982). For non-zero q, we find, making o - 0, as in the derivation of the Kubo-Greenwood formula (34), 

The functions j are formed by the superposition of plane waves of given wavenumber k and with equal amplitudes but random phases for all directions of k. For these plane waves, denoted in the absence of scattering by 

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