A problem of moment realizability in quantum statistical physics

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December 2011, 4(4): 1143-1158. doi: 10.3934/krm.2011.4.1143

A problem of moment realizability in quantum statistical physics

Florian Méhats ^1, and Olivier Pinaud ^2,

1.	IRMAR, Université de Rennes 1 and IPSO, INRIA Rennes, 35042 Rennes Cedex, France
2.	Université de Lyon, Université de Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, F - 69622 Villeurbanne, France

Received June 2011 Published November 2011

Abstract
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This work is a generalization of the results previously obtained in [17] in a one-dimensional setting: we revisit the problem of the minimization of the quantum free energy (entropy + energy) under local constraints (moments) and prove the existence of minimizers in various configurations. While [17] addressed the 1D case on bounded domains, we treat in the present paper the multi-dimensional case as well as unbounded domains and non-linear interactions as Hartree/Hartree-Fock. Moreover, whereas [17] dealt with the first moment only, namely the charge density, we extend the results to the second moment, the current density.

Keywords: moment problem, Quantum hydrodynamics, quantum entropy minimization under contraints..

Mathematics Subject Classification: Primary: 35Q405; Secondary: 82B1.

Citation: Florian Méhats, Olivier Pinaud. A problem of moment realizability in quantum statistical physics. Kinetic & Related Models, 2011, 4 (4) : 1143-1158. doi: 10.3934/krm.2011.4.1143

References:

[1]	A. Arnold, Self-consistent relaxation-time models in quantum mechanics,, Comm. Partial Differential Equations, 21 (1996), 473. Google Scholar
[2]	L. G. Brown and H. Kosaki, Jensen's inequality in semi-finite von Neumann algebras,, J. Operator Theory, 23 (1990), 3. Google Scholar
[3]	P. Degond, S. Gallego and F. Méhats, An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes,, J. Comput. Phys., 221 (2007), 226. doi: 10.1016/j.jcp.2006.06.027. Google Scholar
[4]	_____, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation,, Multiscale Model. Simul., 6 (2007), 246. doi: 10.1137/06067153X. Google Scholar
[5]	_____, On quantum hydrodynamic and quantum energy transport models,, Commun. Math. Sci., 5 (2007), 887. Google Scholar
[6]	P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle,, in, 1946 (2008), 111. Google Scholar
[7]	P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625. doi: 10.1007/s10955-004-8823-3. Google Scholar
[8]	_____, Quantum hydrodynamic models derived from the entropy principle,, in, 371 (2005), 107. Google Scholar
[9]	P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle,, J. Statist. Phys., 112 (2003), 587. doi: 10.1023/A:1023824008525. Google Scholar
[10]	J. Dolbeault, P. Felmer, M. Loss and E. Paturel, Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems,, J. Funct. Anal., 238 (2006), 193. doi: 10.1016/j.jfa.2005.11.008. Google Scholar
[11]	J. Dolbeault, P. Felmer and M. Lewin, Orbitally stable states in generalized Hartree-Fock theory,, Math. Models Methods Appl. Sci., 19 (2009), 347. doi: 10.1142/S0218202509003450. Google Scholar
[12]	A. Jüngel and D. Matthes, A derivation of the isothermal quantum hydrodynamic equations using entropy minimization,, ZAMM Z. Angew. Math. Mech., 85 (2005), 806. doi: 10.1002/zamm.200510232. Google Scholar
[13]	A. Jüngel, D. Matthes and J. P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization,, SIAM J. Appl. Math., 67 (2006), 46. doi: 10.1137/050644823. Google Scholar
[14]	M. Junk, Domain of definition of Levermore's five-moment system,, J. Statist. Phys., 93 (1998), 1143. doi: 10.1023/B:JOSS.0000033155.07331.d9. Google Scholar
[15]	C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Statist. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar
[16]	P.-L. Lions, Hartree-Fock and related equations,, in, 181 (1988), 1985. Google Scholar
[17]	F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics,, J. Stat. Phys., 140 (2010), 565. doi: 10.1007/s10955-010-0003-z. Google Scholar
[18]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics. I. Functional Analysis,", Second edition, (1980). Google Scholar
[19]	B. Simon, "Trace Ideals and their Applications,", Second edition, 120 (2005). Google Scholar

show all references

References:

[1]	A. Arnold, Self-consistent relaxation-time models in quantum mechanics,, Comm. Partial Differential Equations, 21 (1996), 473. Google Scholar
[2]	L. G. Brown and H. Kosaki, Jensen's inequality in semi-finite von Neumann algebras,, J. Operator Theory, 23 (1990), 3. Google Scholar
[3]	P. Degond, S. Gallego and F. Méhats, An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes,, J. Comput. Phys., 221 (2007), 226. doi: 10.1016/j.jcp.2006.06.027. Google Scholar
[4]	_____, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation,, Multiscale Model. Simul., 6 (2007), 246. doi: 10.1137/06067153X. Google Scholar
[5]	_____, On quantum hydrodynamic and quantum energy transport models,, Commun. Math. Sci., 5 (2007), 887. Google Scholar
[6]	P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle,, in, 1946 (2008), 111. Google Scholar
[7]	P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625. doi: 10.1007/s10955-004-8823-3. Google Scholar
[8]	_____, Quantum hydrodynamic models derived from the entropy principle,, in, 371 (2005), 107. Google Scholar
[9]	P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle,, J. Statist. Phys., 112 (2003), 587. doi: 10.1023/A:1023824008525. Google Scholar
[10]	J. Dolbeault, P. Felmer, M. Loss and E. Paturel, Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems,, J. Funct. Anal., 238 (2006), 193. doi: 10.1016/j.jfa.2005.11.008. Google Scholar
[11]	J. Dolbeault, P. Felmer and M. Lewin, Orbitally stable states in generalized Hartree-Fock theory,, Math. Models Methods Appl. Sci., 19 (2009), 347. doi: 10.1142/S0218202509003450. Google Scholar
[12]	A. Jüngel and D. Matthes, A derivation of the isothermal quantum hydrodynamic equations using entropy minimization,, ZAMM Z. Angew. Math. Mech., 85 (2005), 806. doi: 10.1002/zamm.200510232. Google Scholar
[13]	A. Jüngel, D. Matthes and J. P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization,, SIAM J. Appl. Math., 67 (2006), 46. doi: 10.1137/050644823. Google Scholar
[14]	M. Junk, Domain of definition of Levermore's five-moment system,, J. Statist. Phys., 93 (1998), 1143. doi: 10.1023/B:JOSS.0000033155.07331.d9. Google Scholar
[15]	C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Statist. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar
[16]	P.-L. Lions, Hartree-Fock and related equations,, in, 181 (1988), 1985. Google Scholar
[17]	F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics,, J. Stat. Phys., 140 (2010), 565. doi: 10.1007/s10955-010-0003-z. Google Scholar
[18]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics. I. Functional Analysis,", Second edition, (1980). Google Scholar
[19]	B. Simon, "Trace Ideals and their Applications,", Second edition, 120 (2005). Google Scholar

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