Journal of Statistical Physics

, Volume 112, Issue 3–4, pp 587–628 | Cite as

Quantum Moment Hydrodynamics and the Entropy Principle

  • P. Degond
  • C. Ringhofer


This paper presents how a non-commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models. Our starting point is the moment method, which consists in integrating the quantum Liouville equation with respect to momentum p against a given vector of monomials of p. Like in the classical case, the so-obtained moment system is not closed. Inspired from Levermore's procedure in the classical case,(26) we propose to close the moment system by a quantum (Wigner) distribution function which minimizes the entropy subject to the constraint that its moments are given. In contrast to the classical case, the quantum entropy is defined globally (and not locally) as the trace of an operator. Therefore, the relation between the moments and the Lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators (instead of purely local ones in the classical case). In the present paper, we discuss some practical aspects and consequences of this nonlocal feature.

density matrix quantum entropy quantum moments local quantum equilibria quantum BGK models quantum hydrodynamics 


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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • P. Degond
    • 1
  • C. Ringhofer
    • 2
  1. 1.MIP, UMR 5640 (CNRS-UPS-INSA) Université Paul SabatierToulouse cedexFrance
  2. 2.Department of MathematicsArizona State UniversityTempe

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