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Emergence of aggregation in the swarm sphere model with adaptive coupling laws
Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria |
$ \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. $ |
$n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$ |
$ε(p) = -\sum_{i = 1}^d$ |
$\cos(2π p_i)$ |
$\mathcal{F}_f$ |
$f$ |
$. = 0$ |
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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[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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