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April  2019, 12(2): 445-482. doi: 10.3934/krm.2019019

Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness

Marcel Braukhoff

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Received  June 2018 Published  November 2018

Fund Project: The author was partially funded by the Austrian Science Fund (FWF) project F 65

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice:
$ \partial _t f + \nabla _pε(p)·\nabla _x f - \nabla _x n_f·\nabla _p f = n_f(1- n_f)(\mathcal{F}_f-f),\;\;\;\; x∈\mathbb{R}^d, p∈\mathbb{T}^d, t>0. $
This system contains an interaction potential
$n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$
 being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore,
$ε(p) = -\sum_{i = 1}^d$
$\cos(2π p_i)$
 is the dispersion relation and
$\mathcal{F}_f$
 denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on
$f$
 in this context.
In a dilute plasma—without collisions (r.h.s
$. = 0$
)—this system is closely related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial data that the semiconductor Boltzmann equation possesses a local, analytic solution. Here, we exploit the techniques of Mouhout and Villani by using Gevrey-type norms which vary over time. In addition, it is proved that this equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac equilibrium distribution functions.
Keywords: Vlasov-Dirac-Benney equation, BGK collision operator, Boltzmann equation, optical lattice, ill-posedness.
Mathematics Subject Classification: Primary: 35F25, 35F20, 35Q20; Secondary: 35Q83.
Citation: Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019
References:
[1]

N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3308-3333. doi: 10.1063/1.531567. Google Scholar

[2]

A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr and C. Lisdat, Noise and instability of an optical lattice clock, Phys. Rev. A, 92 (2015), 063814, 7 pages.Google Scholar

[3]

N. W. Ashcroft and N. D. Mermin, Solid state physics, Physics Today, 30 (1977), 61. doi: 10.1063/1.3037370. Google Scholar

[4]

C. Bardos and N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits, Kinet. Relat. Models, 6 (2013), 893-917. doi: 10.3934/krm.2013.6.893. Google Scholar

[5]

C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications. Selected Papers Based on the Presentations at the Conference on Hamiltonian PDEs: Analysis, Computations and applications, Toronto, Canada, January 10–12, 2014, pages 1– 30. Toronto: The Fields Institute for Research in the Mathematical Sciences; New York, NY: Springer, 2015. doi: 10.1007/978-1-4939-2950-4. Google Scholar

[6]

C. Bardos and N. Besse, Semi-classical limit of an infinite dimensional system of nonlinear Schrödinger equations, Bull. Inst. Math., Acad. Sin. (N.S.), 11 (2016), 43-61. Google Scholar

[7]

C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys., 53 (2012), 115621, 16pp. doi: 10.1063/1.4765338. Google Scholar

[8]

E. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30. Google Scholar

[9]

M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017.Google Scholar

[10]

M. Braukhoff and A. Jüngel, Energy-transport systems for optical lattices: Derivation, analysis, simulation, Mathematical Models and Methods in Applied Sciences, 28 (2018), 579-614. doi: 10.1142/S021820251850015X. Google Scholar

[11]

O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. Malomed, T. Sowinski and J. Zakrzewski, Non-standard Hubbard models in optical lattices: A review, Rep. Prog. Phys., 78 (2015), 066001, 47 pages.Google Scholar

[12]

A. Griffin, T. Nikuni and E. Zaremba, Bose-Condensed Gases at Finite Temperatures, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511575150. Google Scholar

[13]

D. Han-Kwan and T. T. Nguyen, Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Rational Mech. Anal., 221 (2016), 1317-1344. doi: 10.1007/s00205-016-0985-z. Google Scholar

[14]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. cole Norm. Sup., 49 (2016), 1445-1495. doi: 10.24033/asens.2313. Google Scholar

[15]

P.-E. Jabin and A. Nouri, Analytic solutions to a strongly nonlinear Vlasov equation, C. R., Math., Acad. Sci. Paris, 349 (2011), 541-546. doi: 10.1016/j.crma.2011.03.024. Google Scholar

[16]

A. Jaksch, Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381. Google Scholar

[17]

A. Jüngel, Transport Equations for Semiconductors, Lect. Notes Phys., 773. Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. Google Scholar

[18]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9. Google Scholar

[19]

N. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28. Google Scholar

[20]

A. Rapp, S. Mandt and A. Rosch, Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices, Phys. Rev. Lett., 105 (2010), 220405, 4 pages.Google Scholar

[21]

U. Schneider, L. Hackermüller, J. Ph. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch and A. Rosch, Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, Nature Physics, 8 (2012), 213-218. Google Scholar

show all references

References:
[1]

N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3308-3333. doi: 10.1063/1.531567. Google Scholar

[2]

A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr and C. Lisdat, Noise and instability of an optical lattice clock, Phys. Rev. A, 92 (2015), 063814, 7 pages.Google Scholar

[3]

N. W. Ashcroft and N. D. Mermin, Solid state physics, Physics Today, 30 (1977), 61. doi: 10.1063/1.3037370. Google Scholar

[4]

C. Bardos and N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits, Kinet. Relat. Models, 6 (2013), 893-917. doi: 10.3934/krm.2013.6.893. Google Scholar

[5]

C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications. Selected Papers Based on the Presentations at the Conference on Hamiltonian PDEs: Analysis, Computations and applications, Toronto, Canada, January 10–12, 2014, pages 1– 30. Toronto: The Fields Institute for Research in the Mathematical Sciences; New York, NY: Springer, 2015. doi: 10.1007/978-1-4939-2950-4. Google Scholar

[6]

C. Bardos and N. Besse, Semi-classical limit of an infinite dimensional system of nonlinear Schrödinger equations, Bull. Inst. Math., Acad. Sin. (N.S.), 11 (2016), 43-61. Google Scholar

[7]

C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys., 53 (2012), 115621, 16pp. doi: 10.1063/1.4765338. Google Scholar

[8]

E. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30. Google Scholar

[9]

M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017.Google Scholar

[10]

M. Braukhoff and A. Jüngel, Energy-transport systems for optical lattices: Derivation, analysis, simulation, Mathematical Models and Methods in Applied Sciences, 28 (2018), 579-614. doi: 10.1142/S021820251850015X. Google Scholar

[11]

O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. Malomed, T. Sowinski and J. Zakrzewski, Non-standard Hubbard models in optical lattices: A review, Rep. Prog. Phys., 78 (2015), 066001, 47 pages.Google Scholar

[12]

A. Griffin, T. Nikuni and E. Zaremba, Bose-Condensed Gases at Finite Temperatures, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511575150. Google Scholar

[13]

D. Han-Kwan and T. T. Nguyen, Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Rational Mech. Anal., 221 (2016), 1317-1344. doi: 10.1007/s00205-016-0985-z. Google Scholar

[14]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. cole Norm. Sup., 49 (2016), 1445-1495. doi: 10.24033/asens.2313. Google Scholar

[15]

P.-E. Jabin and A. Nouri, Analytic solutions to a strongly nonlinear Vlasov equation, C. R., Math., Acad. Sci. Paris, 349 (2011), 541-546. doi: 10.1016/j.crma.2011.03.024. Google Scholar

[16]

A. Jaksch, Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381. Google Scholar

[17]

A. Jüngel, Transport Equations for Semiconductors, Lect. Notes Phys., 773. Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. Google Scholar

[18]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9. Google Scholar

[19]

N. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28. Google Scholar

[20]

A. Rapp, S. Mandt and A. Rosch, Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices, Phys. Rev. Lett., 105 (2010), 220405, 4 pages.Google Scholar

[21]

U. Schneider, L. Hackermüller, J. Ph. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch and A. Rosch, Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, Nature Physics, 8 (2012), 213-218. Google Scholar

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