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Kinetic models of conservative economies with need-based transfers as welfare
Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria |
$ \partial_t f + \nabla\epsilon(p)\cdot\nabla_x f - \nabla \rho_f(x,t)\cdot\nabla_p f = \frac{\mathcal F_\lambda(p)-f}\tau, \quad x\in\mathbb{R}^d,\ p\in B, \ t>0 $ |
$ \tau>0 $ |
$ \mathcal F_\lambda $ |
$ \rho_f: = \int_B fdp $ |
$ \partial_t f + Lf = Q(f), $ |
$ L $ |
$ C^0 $ |
$ \|e^{tL}\|\leq Ce^{\omega t} $ |
$ t\in\mathbb R $ |
$ \omega>0 $ |
$ L $ |
$ Q $ |
References:
[1] |
N. W. Ashcroft and N. D. Mermin,
Solid state physics, Physics Today, 30 (1977), P61.
doi: 10.1063/1.3037370. |
[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.
doi: 10.1103/PhysRevA.92.063814. |
[3] |
N. B. Abdallah and P. Degond,
On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333.
doi: 10.1063/1.531567. |
[4] |
E. Bloch,
Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30.
doi: 10.1038/nphys138. |
[5] |
M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017. |
[6] |
M. Braukhoff, Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness, Kinet. Relat. Models, 12 (2019), 445–482, arXiv 1711.06015 [math.AP].
doi: 10.3934/krm.2019019. |
[7] |
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. |
[8] |
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. |
[9] |
C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications, 1–30, Fields Inst. Commun., 75, Fields Inst. Res. Math. Sci., Toronto, ON, 2015.
doi: 10.1007/978-1-4939-2950-4_1. |
[10] |
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.
|
[11] |
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. |
[12] |
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.
doi: 10.1088/0034-4885/78/6/066001. |
[13] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, 2000. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
A. Jaksch,
Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381.
doi: 10.1080/00107510410001705486. |
[18] |
A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics, 773. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-89526-8. |
[19] |
T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. |
[20] |
S. Mandt, Transport and Non-Equilibrium Dynamics in Optical Lattices. From Expanding Atomic Clouds to Negative Absolute Temperatures, PhD thesis, University of Cologne, 2012. |
[21] |
G. Metivier, Remarks on the well-posedness of the nonlinear Cauchy problem, Geometric Analysis of PDE and Several Complex Variables, 337–356, Contemp. Math., 368, Amer. Math. Soc., Providence, RI, 2005.
doi: 10.1090/conm/368/06790. |
[22] |
C. Mouhot and C. Villani,
On Landau damping,, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[23] |
N. Ramsey,
Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28.
doi: 10.1103/PhysRev.103.20. |
[24] |
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.
doi: 10.1103/PhysRevLett.105.220405. |
[25] |
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.
doi: 10.1038/nphys2205. |
show all references
References:
[1] |
N. W. Ashcroft and N. D. Mermin,
Solid state physics, Physics Today, 30 (1977), P61.
doi: 10.1063/1.3037370. |
[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.
doi: 10.1103/PhysRevA.92.063814. |
[3] |
N. B. Abdallah and P. Degond,
On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333.
doi: 10.1063/1.531567. |
[4] |
E. Bloch,
Ultracold quantum gases in optical lattices, Nature Physics, 1 (2005), 23-30.
doi: 10.1038/nphys138. |
[5] |
M. Braukhoff, Effective Equations for a Cloud of Ultracold Atoms in an Optical Lattice, Ph.D thesis, University of Cologne, Germany, 2017. |
[6] |
M. Braukhoff, Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness, Kinet. Relat. Models, 12 (2019), 445–482, arXiv 1711.06015 [math.AP].
doi: 10.3934/krm.2019019. |
[7] |
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. |
[8] |
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. |
[9] |
C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-benney equation, In Hamiltonian Partial Differential Equations and Applications, 1–30, Fields Inst. Commun., 75, Fields Inst. Res. Math. Sci., Toronto, ON, 2015.
doi: 10.1007/978-1-4939-2950-4_1. |
[10] |
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.
|
[11] |
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. |
[12] |
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.
doi: 10.1088/0034-4885/78/6/066001. |
[13] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, 2000. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
A. Jaksch,
Optical lattices, ultracold atoms and quantum information processing, Contemp. Phys., 45 (2004), 367-381.
doi: 10.1080/00107510410001705486. |
[18] |
A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics, 773. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-89526-8. |
[19] |
T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. |
[20] |
S. Mandt, Transport and Non-Equilibrium Dynamics in Optical Lattices. From Expanding Atomic Clouds to Negative Absolute Temperatures, PhD thesis, University of Cologne, 2012. |
[21] |
G. Metivier, Remarks on the well-posedness of the nonlinear Cauchy problem, Geometric Analysis of PDE and Several Complex Variables, 337–356, Contemp. Math., 368, Amer. Math. Soc., Providence, RI, 2005.
doi: 10.1090/conm/368/06790. |
[22] |
C. Mouhot and C. Villani,
On Landau damping,, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[23] |
N. Ramsey,
Thermodynamics and statistical mechanics at negative absolute temperature, Phys. Rev., 103 (1956), 20-28.
doi: 10.1103/PhysRev.103.20. |
[24] |
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.
doi: 10.1103/PhysRevLett.105.220405. |
[25] |
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.
doi: 10.1038/nphys2205. |
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