Journal of Statistical Physics

, Volume 145, Issue 6, pp 1661–1673 | Cite as

A New Derivation of the Quantum Navier–Stokes Equations in the Wigner–Fokker–Planck Approach

  • Ansgar JüngelEmail author
  • José Luis López
  • Jesús Montejo–Gámez


A quantum Navier–Stokes system for the particle, momentum, and energy densities is formally derived from the Wigner–Fokker–Planck equation using a moment method. The viscosity term depends on the particle density with a shear viscosity coefficient which equals the quantum diffusion coefficient of the Fokker–Planck collision operator. The main idea of the derivation is the use of a so-called osmotic momentum operator, which is the sum of the phase-space momentum and the gradient operator. In this way, a Chapman–Enskog expansion of the Wigner function, which typically leads to viscous approximations, is avoided. Moreover, we show that the osmotic momentum emerges from local gauge theory.


Quantum Navier–Stokes model Wigner–Fokker–Planck equations Moment method Osmotic momentum Local gauge transformation 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Ansgar Jüngel
    • 1
    Email author
  • José Luis López
    • 2
  • Jesús Montejo–Gámez
    • 2
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  2. 2.Departamento de Matemática Aplicada, Facultad de CienciasUniversidad de GranadaGranadaSpain

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