Quantum Moment Hydrodynamics and the Entropy Principle

P. Degond, Christian Ringhofer

Research output: Contribution to journalArticlepeer-review

142 Scopus citations


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, 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.

Original languageEnglish (US)
Pages (from-to)587-628
Number of pages42
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - Aug 2003


  • Density matrix
  • Local quantum equilibria
  • Quantum BGK models
  • Quantum entropy
  • Quantum hydrodynamics
  • Quantum moments

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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