We consider in this work the problem of minimizing the von Neumann entropy under the constraints that the density of particles, the current, and the kinetic energy of the system is fixed at each point of space. The unique minimizer is a self-adjoint positive trace class operator, and our objective is to characterize its form. We will show that this minimizer is solution to a self-consistent nonlinear eigenvalue problem. One of the main difficulties in the proof is to parametrize the feasible set in order to derive the Euler–Lagrange equation, and we will proceed by constructing an appropriate form of perturbations of the minimizer. The question of deriving quantum statistical equilibria is at the heart of the quantum hydrodynamical models introduced by Degond and Ringhofer (J Statist Phys 112:(3–4), 587-628, 2003). An original feature of the problem is the local nature of constraints, i.e. they depend on position, while more classical models consider the total number of particles, the total current and the total energy in the system to be fixed.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Degond, P., Gallego, S., Méhats, F.: An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes. J. Comput. Phys. 221(1), 226–249 (2007)
Degond, P., Gallego, S., Méhats, F.: Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation. Multiscale Model. Simul. 6(1), 246–272 (2007)
Degond, P., Gallego, S., Méhats, F.: On quantum hydrodynamic and quantum energy transport models. Commun. Math. Sci. 5(4), 887–908 (2007)
Degond, P., Gallego, S., Méhats, F., Ringhofer, C.: Quantum hydrodynamic and diffusion models derived from the entropy principle. In: Quantum Transport, Volume 1946 of Lecture Notes in Mathematics, pp. 111–168. Springer, Berlin (2008)
Degond, P., Ringhofer, C.: Quantum moment hydrodynamics and the entropy principle. J. Statist. Phys. 112(3–4), 587–628 (2003)
Dolbeault, J., Felmer, P., Loss, M., Paturel, E.: Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems. J. Funct. Anal. 238(1), 193–220 (2006)
Duboscq, R., Olivier P.: A constrained optimization problem in quantum statistical physics. arXiv preprint arXiv:1904.00600 (2019)
Duboscq, R., Pinaud, O.: On the minimization of quantum entropies under local constraints. J. Math. Pure Appl. 128, 87–118 (2019)
Jüngel, A., Matthes, D.: A derivation of the isothermal quantum hydrodynamic equations using entropy minimization. ZAMM Z. Angew. Math. Mech. 85(11), 806–814 (2005)
Jüngel, A., Matthes, D., Milišić, J.P.: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67(1), 46–68 (2006)
Junk, M.: Domain of definition of Levermore’s five-moment system. J. Statist. Phys. 93(5–6), 1143–1167 (1998)
Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Statist. Phys. 83(5–6), 1021–1065 (1996)
Lions, P.-L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9, 553–618 (1993)
Méhats, F., Pinaud, O.: An inverse problem in quantum statistical physics. J. Stat. Phys. 140(3), 565–602 (2010)
Méhats, F., Pinaud, O.: A problem of moment realizability in quantum statistical physics. Kinet. Relat. Models 4(4), 1143–1158 (2011)
Méhats, F., Pinaud, O.: The quantum Liouville-BGK equation and the moment problem. J. Differ. Eq. 263(7), 3737–3787 (2017)
Nachtergaele, B., Yau, H.-T.: Derivation of the Euler equations from quantum dynamics. Commun. Math. Phys. 243(3), 485–540 (2003)
Pinaud, O.: The quantum drift-diffusion model: existence and exponential convergence to the equilibrium. Annales de l’Institut Henri Poincaré C, Analyse non linéaire 36(3), 811–836 (2019)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. Academic Press Inc, New York (1980)
von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955). Translated by Robert T. Beyer
OP is supported by NSF CAREER Grant DMS-1452349.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by J. Ball.
About this article
Cite this article
Duboscq, R., Pinaud, O. Constrained minimizers of the von Neumann entropy and their characterization. Calc. Var. 59, 105 (2020). https://doi.org/10.1007/s00526-020-01753-1
Mathematics Subject Classification
- Primary 35Q40
- Secondary 82B10