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

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Abstract

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)〉.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsingin yliopisto, Finland

    Jani Lukkarinen

  2. Zentrum Mathematik, TU München, 85747, Garching, Germany

    Herbert Spohn

Authors
  1. Jani Lukkarinen
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  2. Herbert Spohn
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Correspondence to Herbert Spohn.

Additional information

Jürg Fröhlich and Tom Spencer have profoundly shaped the interface between physics and mathematics. It is therefore a particular pleasure to dedicate to them this article on a chapter of mathematical physics which is far from being closed.

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Lukkarinen, J., Spohn, H. Not to Normal Order—Notes on the Kinetic Limit for Weakly Interacting Quantum Fluids. J Stat Phys 134, 1133–1172 (2009). https://doi.org/10.1007/s10955-009-9682-8

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  • DOI: https://doi.org/10.1007/s10955-009-9682-8

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