Effects of bound pairs on the low-density transport properties of gases

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Abstract
In some derivations of the Boltzmann equation, the equation which describes the time evolution of the distribution function in the phase space of the atoms of a low-density gas, it has been assumed that all of the trajectories in the phase space of the pairs of atoms are such that transforming back sufficiently far in time leads to separated atoms. In a recent development, it has been shown that if this is not true the ‘‘bound-state’’ trajectories lead to an addition to the collision integral which, in turn, leads to corrections to the values of the low-density transport coefficients. In the present alternate development, it is shown that the ‘‘bound-state’’ trajectories may be joined to trajectories in the ‘‘unbound region’’ through the introduction of a complex time and analytic continuation. With this generalization of the concept of trajectories, one may introduce the ‘‘molecular chaos assumption’’ in the usual manner, and the ‘‘collision integral’’ may be generalized to include the bound-state region of the phase space. This leads to ‘‘corrections’’ to the low-density limiting values of the transport coefficients. These effects of the bound pairs of atoms on the low-density transport coefficients of a gas of atoms which interact according to a Lennard-Jones potential are investigated numerically. It is found that the effects become important at temperatures, scaled in the usual manner by ε/k, below about unity. Specifically, it is found that at a scaled temperature of unity, corresponding to a temperature in the range of about 100–200 K, the effect is to lower the self-diffusion coefficient by about 3.7% and the viscosity and thermal conductivity by about 0.7%. The effects become significantly greater at lower temperatures.

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