TY - JOUR

T1 - Quantum Moment Hydrodynamics and the Entropy Principle

AU - Degond, P.

AU - Ringhofer, Christian

N1 - Funding Information:
P.D. was supported by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. P. Degond wishes to express his gratitude to F. Méhats for very valuable remarks and comments. C.R. was supported by NSF award Nrs. DMS0204543 and DECS0218008.

PY - 2003/8

Y1 - 2003/8

N2 - 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.

AB - 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.

KW - Density matrix

KW - Local quantum equilibria

KW - Quantum BGK models

KW - Quantum entropy

KW - Quantum hydrodynamics

KW - Quantum moments

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U2 - 10.1023/A:1023824008525

DO - 10.1023/A:1023824008525

M3 - Article

AN - SCOPUS:0037837827

VL - 112

SP - 587

EP - 628

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -