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A problem of moment realizability in quantum statistical physics
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 |
References:
[1] |
A. Arnold, Self-consistent relaxation-time models in quantum mechanics, Comm. Partial Differential Equations, 21 (1996), 473-506. |
[2] |
L. G. Brown and H. Kosaki, Jensen's inequality in semi-finite von Neumann algebras, J. Operator Theory, 23 (1990), 3-19. |
[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. |
[4] |
_____, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272.
doi: 10.1137/06067153X. |
[5] |
_____, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[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. |
[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. |
[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. |
[19] |
B. Simon, "Trace Ideals and their Applications," Second edition, Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, RI, 2005. |
show all references
References:
[1] |
A. Arnold, Self-consistent relaxation-time models in quantum mechanics, Comm. Partial Differential Equations, 21 (1996), 473-506. |
[2] |
L. G. Brown and H. Kosaki, Jensen's inequality in semi-finite von Neumann algebras, J. Operator Theory, 23 (1990), 3-19. |
[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. |
[4] |
_____, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272.
doi: 10.1137/06067153X. |
[5] |
_____, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[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. |
[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. |
[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. |
[19] |
B. Simon, "Trace Ideals and their Applications," Second edition, Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, RI, 2005. |
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