Calculation of exchange second virial coefficient of a hard-sphere gas by path integrals

Elliott H. Lieb

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52 Scopus citations


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., Bexch/B 0exch = exp {-1/2π3(a/Λ)2 + O[(a/Λ)2/3]}, where Λ is the thermal wavelength and a is the hard-sphere radius. While it was known before that Bexch/ B0exch is exponentially small for high temperatures, this is the first time that a precise asymptotic formula is both proposed and proved to be correct.

Original languageEnglish (US)
Pages (from-to)43-52
Number of pages10
JournalJournal of Mathematical Physics
Issue number1
StatePublished - 1967

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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