Riv. Mat. Univ. Parma, Vol. 3, No. 2, 2012

Ansgar Jüngel

Dissipative quantum fluid models

Pages: 217-290
Received: 29 November 2010   
Accepted in revised form: 11 February 2011
Mathematics Subject Classification (2000): 35A05, 35B40, 35B45, 35G25, 35K30, 35M10, 35Q30, 35Q40, 76Y05, 81S30.

Keywords: Quantum transport, Schrödinger equation, Wigner equation, Wigner-Fokker-Planck equation, quantum drift-diffusion equation, DLSS equation, quantum hydrodynamic equations, viscous quantum hydrodynamic equations, quantum Navier-Stokes equations.

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Abstract: Some macroscopic models for diffusive quantum systems are reviewed. The model equations are derived from Wigner-Boltzmann equations by a moment method. Depending on the properties of the collision operator and the scaling of the Wigner equation, three model classes are presented: the quantum drift-diffusion equations, inviscid and viscous quantum hydrodynamic equations, and quantum Navier-Stokes equations. For each of these models, the derivation is sketched and analytical results are reviewed.

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