Abstract
Considering a system ofN identical interacting particles, which obey Fermi-Dirac or Bose-Einstein statistics, we derive new formulas for correlation functions of the type\(C(t) = \langle \Sigma _{i = 1}^N A_i (t) \Sigma _{j = 1}^N B_j \rangle \) (whereB j is diagonal in the free-particle states) in the thermodynamic limit. Thereby we apply and extend a superoperator formalism, recently developed for the derivation of long-time tails in semiclassical systems. As an illustrative application, the Boltzmann equation value of the time-integrated correlation functionC(t) is derived in a straightforward manner. Due to exchange effects, the obtained t-matrix and the resulting scattering cross section, which occurs in the Boltzmann collision operator, are now functionals of the Fermi-Dirac or Bose-Einstein distribution.
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Loss, D., Schoeller, H. A new quantum statistical evaluation method for time correlation functions. J Stat Phys 54, 765–795 (1989). https://doi.org/10.1007/BF01019775
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DOI: https://doi.org/10.1007/BF01019775