American Institute of Mathematical Sciences

Advanced
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.   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

[1]

Paolo Antonelli, Pierangelo Marcati. Quantum hydrodynamics with nonlinear interactions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 1-13. doi: 10.3934/dcdss.2016.9.1
[2]

Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020063
[3]

Nicola Zamponi. Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization. Kinetic & Related Models, 2012, 5 (1) : 203-221. doi: 10.3934/krm.2012.5.203
[4]

José Luis López, Jesús Montejo-Gámez. On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models. Kinetic & Related Models, 2012, 5 (3) : 517-536. doi: 10.3934/krm.2012.5.517
[5]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[6]

Bernard Bonnard, Jean-Baptiste Caillau, Olivier Cots. Energy minimization in two-level dissipative quantum control: Th e integrable case. Conference Publications, 2011, 2011 (Special) : 198-208. doi: 10.3934/proc.2011.2011.198
[7]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
[8]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[9]

Dmitry Jakobson. On quantum limits on flat tori. Electronic Research Announcements, 1995, 1: 80-86.

[10]

Jin-Cheng Jiang, Chi-Kun Lin, Shuanglin Shao. On one dimensional quantum Zakharov system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5445-5475. doi: 10.3934/dcds.2016040
[11]

Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287
[12]

Mason A. Porter, Richard L. Liboff. The radially vibrating spherical quantum billiard. Conference Publications, 2001, 2001 (Special) : 310-318. doi: 10.3934/proc.2001.2001.310
[13]

Jianhong (Jackie) Shen, Sung Ha Kang. Quantum TV and applications in image processing. Inverse Problems & Imaging, 2007, 1 (3) : 557-575. doi: 10.3934/ipi.2007.1.557
[14]

Huai-Dong Cao and Jian Zhou. On quantum de Rham cohomology theory. Electronic Research Announcements, 1999, 5: 24-34.

[15]

Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375
[16]

Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang. Thermodynamical potentials of classical and quantum systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1411-1448. doi: 10.3934/dcdsb.2018214
[17]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420
[18]

Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77
[19]

Yu. Dabaghian, R. V. Jensen, R. Blümel. Integrability in 1D quantum chaos. Conference Publications, 2003, 2003 (Special) : 206-212. doi: 10.3934/proc.2003.2003.206
[20]

Roderick Melnik, B. Lassen, L. C Lew Yan Voon, M. Willatzen, C. Galeriu. Accounting for nonlinearities in mathematical modelling of quantum dot molecules. Conference Publications, 2005, 2005 (Special) : 642-651. doi: 10.3934/proc.2005.2005.642

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences