Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals

Abstract

By direct examination of the path (Wiener)‐integral representation of the diffusion Green's function in the presence of an opaque sphere, we are able to obtain upper and lower bounds for that Green's function. These bounds are asymptotically correct for short‐time, even in the shadow region. Essentially, we have succeeded in showing that diffusion probabilities for short‐time intervals are concentrated mainly on the optical path. By integrating the Green's function, we obtain upper‐ and lower‐bound estimates for the exchange part of the second virial coefficient of a hard‐sphere gas. We can show that, for high temperature, it is asymptotically very small compared to the corresponding quantity for an ideal gas, viz., $$B_{\mathrm{exch}}{\mathrm{/B}}_{\mathrm{exch}}^0=\exp \mathrm{\hspace{0.17em}\{-}\frac{1}{2}{\pi }^3{\mathrm{(a/\Lambda )}}^2{\mathrm{+O[(a/\Lambda )}}^{\text{2/3}}\mathrm{]\}}$$ , where Λ is the thermal wavelength and a is the hard‐sphere radius. While it was known before that B_{ex ch}/B⁰_{ex ch} is exponentially small for high temperatures, this is the first time that a precise asymptotic formula is both proposed and proved to be correct.