The Quantum-statistical Theory of Transport Phenomena, II : On the Theory of Metallic Conductivity

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The main purpose of the present paper is to derive the Bloch integro-differential equation, upon which the modern electron theories of metallic conductivity are based, analytically from the principles of quantum-statistical mechanics. In the first place, a generalization of the phase-space distribution function of Wigner is obtained by means of the integral transformations which may be, in general, different from the Fourier transformation. And the starting point of the present theory is the equation of motion for the singlet generalized phase-space distribution function, which is derived from the Schrodinger equation for the quantized wave function. The Irving-Zwanzig equation for the Wigner distribution function is immediately obtained from the above-mentioned equation of motion. The generalized phase-space distribution functions of electrons are defined so as to be suitable for Bloch's band scheme, and the equation of motion for the singlet electronic distribution function is obtained. The macroscopic electric current density is defined from the quantized density field with the help of the time-averaging procedure over a microscopic interval. If one uses the customary approximation method together with the postulate of random a priori phases, the Bloch integro-differential equation is obtained from the equation of motion for the time-averaged electronic distribution function.
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