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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

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), 473-506.  Google Scholar

[2]

L. G. Brown and H. Kosaki, Jensen's inequality in semi-finite von Neumann algebras, J. Operator Theory, 23 (1990), 3-19.  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), 226-249. doi: 10.1016/j.jcp.2006.06.027.  Google Scholar

[4]

_____, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X.  Google Scholar

[5]

_____, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. Google Scholar

[6]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, in "Quantum Transport," Lecture Notes in Math., 1946, Springer, Berlin, (2008), 111-168.  Google Scholar

[7]

P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3.  Google Scholar

[8]

_____, Quantum hydrodynamic models derived from the entropy principle, in "Nonlinear Partial Differential Equations and Related Analysis," Contemp. Math., Amer. Math. Soc., 371, Providence, RI, (2005), 107-131. Google Scholar

[9]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Statist. Phys., 112 (2003), 587-628. 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), 193-220. 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), 347-367. 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), 806-814. 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), 46-68. doi: 10.1137/050644823.  Google Scholar

[14]

M. Junk, Domain of definition of Levermore's five-moment system, J. Statist. Phys., 93 (1998), 1143-1167. 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), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[16]

P.-L. Lions, Hartree-Fock and related equations, in "Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. IX" (Paris, 1985-1986), Pitman Res. Notes Math. Ser., 181, Longman Sci. Tech., Harlow, (1988), 304-333.  Google Scholar

[17]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565-602. 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, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980.  Google Scholar

[19]

B. Simon, "Trace Ideals and their Applications," Second edition, Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, RI, 2005.  Google Scholar

show all references

References:
[1]

A. Arnold, Self-consistent relaxation-time models in quantum mechanics, Comm. Partial Differential Equations, 21 (1996), 473-506.  Google Scholar

[2]

L. G. Brown and H. Kosaki, Jensen's inequality in semi-finite von Neumann algebras, J. Operator Theory, 23 (1990), 3-19.  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), 226-249. doi: 10.1016/j.jcp.2006.06.027.  Google Scholar

[4]

_____, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X.  Google Scholar

[5]

_____, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. Google Scholar

[6]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, in "Quantum Transport," Lecture Notes in Math., 1946, Springer, Berlin, (2008), 111-168.  Google Scholar

[7]

P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3.  Google Scholar

[8]

_____, Quantum hydrodynamic models derived from the entropy principle, in "Nonlinear Partial Differential Equations and Related Analysis," Contemp. Math., Amer. Math. Soc., 371, Providence, RI, (2005), 107-131. Google Scholar

[9]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Statist. Phys., 112 (2003), 587-628. 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), 193-220. 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), 347-367. 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), 806-814. 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), 46-68. doi: 10.1137/050644823.  Google Scholar

[14]

M. Junk, Domain of definition of Levermore's five-moment system, J. Statist. Phys., 93 (1998), 1143-1167. 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), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[16]

P.-L. Lions, Hartree-Fock and related equations, in "Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. IX" (Paris, 1985-1986), Pitman Res. Notes Math. Ser., 181, Longman Sci. Tech., Harlow, (1988), 304-333.  Google Scholar

[17]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565-602. 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, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980.  Google Scholar

[19]

B. Simon, "Trace Ideals and their Applications," Second edition, Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, RI, 2005.  Google Scholar

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