### Abstract

In these notes, we review the recent theory of quantum hydrodynamic and diffusion models derived from the entropy minimization principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subject to local constraints of given moments. We provide a framework to develop this minimization approach and successively apply it to quantum hydrodynamic models and quantum diffusion models. The results of numerical simulations show that these models capture well the various features of quantum transport.

Original language | English (US) |
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Title of host publication | Lecture Notes in Mathematics |

Pages | 111-168 |

Number of pages | 58 |

Volume | 1946 |

DOIs | |

State | Published - 2008 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 1946 |

ISSN (Print) | 00758434 |

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### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*Lecture Notes in Mathematics*(Vol. 1946, pp. 111-168). (Lecture Notes in Mathematics; Vol. 1946). https://doi.org/10.1007/978-3-540-79574-2_3

**Quantum hydrodynamic and diffusion models derived from the entropy principle.** / Degond, Pierre; Gallego, Samy; Méhats, Florian; Ringhofer, Christian.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Lecture Notes in Mathematics.*vol. 1946, Lecture Notes in Mathematics, vol. 1946, pp. 111-168. https://doi.org/10.1007/978-3-540-79574-2_3

}

TY - CHAP

T1 - Quantum hydrodynamic and diffusion models derived from the entropy principle

AU - Degond, Pierre

AU - Gallego, Samy

AU - Méhats, Florian

AU - Ringhofer, Christian

PY - 2008

Y1 - 2008

N2 - In these notes, we review the recent theory of quantum hydrodynamic and diffusion models derived from the entropy minimization principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subject to local constraints of given moments. We provide a framework to develop this minimization approach and successively apply it to quantum hydrodynamic models and quantum diffusion models. The results of numerical simulations show that these models capture well the various features of quantum transport.

AB - In these notes, we review the recent theory of quantum hydrodynamic and diffusion models derived from the entropy minimization principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subject to local constraints of given moments. We provide a framework to develop this minimization approach and successively apply it to quantum hydrodynamic models and quantum diffusion models. The results of numerical simulations show that these models capture well the various features of quantum transport.

UR - http://www.scopus.com/inward/record.url?scp=46949093671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46949093671&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-79574-2_3

DO - 10.1007/978-3-540-79574-2_3

M3 - Chapter

AN - SCOPUS:46949093671

SN - 9783540795735

VL - 1946

T3 - Lecture Notes in Mathematics

SP - 111

EP - 168

BT - Lecture Notes in Mathematics

ER -