Journal of Statistical Physics

, Volume 134, Issue 5–6, pp 1133–1172 | Cite as

Not to Normal Order—Notes on the Kinetic Limit for Weakly Interacting Quantum Fluids

  • Jani Lukkarinen
  • Herbert Spohn


The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrödinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on normal order in the Duhamel expansion. We discuss the term by term convergence of the expansion and the equilibrium time correlation 〈a(t)* a(0)〉.


Boltzmann-Nordheim equation Uehling-Uhlenbeck equation Kinetic theory Weakly interacting fermions Weakly interacting bosons Quantum BBGKY hierarchy Time-dependent perturbation theory 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsingin yliopistoFinland
  2. 2.Zentrum MathematikTU MünchenGarchingGermany

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