The Quantum-Statistical Theory of Transport Phenomena, I

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Abstract
By means of the N-representation of density operator, Irving and Zwanzig's equation of motion for the phase-space distribution function is rewritten in such a form that the influence of the symmetry effect appears more explicitly in the collision term of the equation for the singlet distribution function. By means of the customary theory of time-dependent perturbation, the equation of motion for the time averaged distribution function over an interval as long as the mean life of the unperturbed quantum states, is derived from the Irving-Zwanzig equation. This equation is compared with the Uehling-Uhlenbeck equation, which is obtained as the quantum mechanical modification of the Boltzmann integro-differential equation by means of a physical argument. (Then, it is concluded that the deviation from the Uehling-Uhlenbeck equation becomes important if that the local variation of the external field or of the distribution function in the range of the order of the magnitude of the de Broglie wave length is appreciable.) And it seems probable that the Uehling-Uhlenbeck equation becomes invalid at extremely low temperatures, such as the γ-point of liquid helium. We have, however, not succeeded in the derivation of an equation for the general case, because the postulate of random a priori phases destroys the dependence of the initial distribution on coordinates.

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