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BGK model of the multi-species Uehling-Uhlenbeck equation
1. | Department of mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea |
2. | Department of mathematics, Würzburg University, Emil Fischer Str. 40, 97074 Würzburg, Germany |
3. | Department of mathematics, Vienna University, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria |
We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and $ H $-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficients of the local equilibriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under the condition, leading to the well-definedness of our model.
References:
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P. Andries, K. Aoki and B. Perthame,
A consistent BGK-type model for gas mixtures, J. Statist. Phys., 106 (2002), 993-1018.
doi: 10.1023/A:1014033703134. |
[2] |
N. W. Ashcroft and N. D. Mermin, Solid State Physic Holt, Rinehart and Winston, New York, USA. 1976. Google Scholar |
[3] |
G.-C. Bae and S.-B. Yun,
Stationary quantum BGK model for bosons and fermions in a bounded interval, J. Stat. Phys., 178 (2020), 845-868.
doi: 10.1007/s10955-019-02466-2. |
[4] |
G.-C. Bae and S.-B. Yun,
Quantum BGK model near a global Fermi-Dirac distribution, SIAM J. Math. Anal., 52 (2020), 2313-2352.
doi: 10.1137/19M1270021. |
[5] |
M. Bennoune, M. Lemou and L. Mieussens,
Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics., J. Comput. Phys., 227 (2008), 3781-3803.
doi: 10.1016/j.jcp.2007.11.032. |
[6] |
F. Bernard, A. Iollo and G. Puppo,
Accurate asymptotic preserving boundary conditions for kinetic equations on Cartesian grids, J. Sci. Comput., 65 (2015), 735-766.
doi: 10.1007/s10915-015-9984-8. |
[7] |
P. L. Bhathnagor, E. P. Gross and M. Krook, A model for collision processes in gases, Physical Review, 94 (1954), 511.
doi: 10.1103/PhysRev.94.511. |
[8] |
M. Bisi and M. J. Cáceres,
A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325.
doi: 10.4310/CMS.2016.v14.n2.a1. |
[9] |
M. Bisi, M. Groppi and G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamic limit, Physical Review E, 81 (2010), 036327.
doi: 10.1103/PhysRevE.81.036327. |
[10] |
A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga and I. F. Potapenko,
A general consistent BGK model for gas mixtures, Kinet. Relat. Models, 11 (2018), 1377-1393.
doi: 10.3934/krm.2018054. |
[11] |
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.
doi: 10.3934/krm.2019019. |
[12] |
M. Braukhoff, Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation, Kinet. Relat. Models, 13 (2020), 187–210. preprint, arXiv: 1803.00379.
doi: 10.3934/krm.2020007. |
[13] |
S. Brull, V. Pavan and J. Schneider,
Derivation of a BGK model for mixtures, Eur. J. Mech. B Fluids, 33 (2012), 74-86.
doi: 10.1016/j.euromechflu.2011.12.003. |
[14] |
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge university press, 1990.
![]() |
[15] |
N. Crouseilles and G. Manfredi,
Asymptotic preserving schemes for the Wigner-Poisson-BGK equations in the diffusion limit, Comput. Phys. Commun., 185 (2014), 448-458.
doi: 10.1016/j.cpc.2013.06.002. |
[16] |
G. Dimarco, L. Mieussens and V. Rispoli,
An asymptotic preserving automatic domain decomposition method for the Vlasov-Poisson-BGK system with applications to plasmas, J. Comput. Phys., 274 (2014), 122-139.
doi: 10.1016/j.jcp.2014.06.002. |
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G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.
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Entropy maximisation problem for quantum relativistic particles, Bull. Soc. Math. France, 133 (2005), 87-120.
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[22] |
F. Filbet, J. Hu and S. Jin, A numerical scheme for the quantum Boltzmann equation efficient in the fluid regime, preprint, arXiv: 1009.3352. Google Scholar |
[23] |
F. Filbet, J. Hu and S. Jin,
A numerical scheme for the quantum Boltzmann equation with stiff collision terms, ESAIM Math. Model. Numer. Anal., 46 (2012), 443-463.
doi: 10.1051/m2an/2011051. |
[24] |
F. Filbet and S. Jin,
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.
doi: 10.1016/j.jcp.2010.06.017. |
[25] |
R. H. Fowler and L. Nordheim,
Electron emission in intense electric fields, Proceedings of the Royal Society A, 119 (1928), 173-181.
doi: 10.1098/rspa.1928.0091. |
[26] |
V. Garzó, A. Santos and J. J. Brey,
A kinetic model for a multicomponent gas, Physics of Fluids A: Fluid Dynamics, 1 (1989), 380-383.
doi: 10.1063/1.857458. |
[27] |
J. M. Greene,
Improved Bhatnagar–Gross–Krook model of electron–ion collisions, The Physics of Fluids, 16 (1973), 2022-2023.
doi: 10.1063/1.1694254. |
[28] |
M. Groppi, S. Monica and G. Spiga, A kinetic ellipsoidal BGK model for a binary gas mixture, EPL (Europhysics Letters), 96 (2011), 64002.
doi: 10.1209/0295-5075/96/64002. |
[29] |
E. P. Gross and M. Krook, Model for collision processes in gases: Small-amplitude oscillations of charged two-component systems, Physical Review, 102 (1956), 593.
doi: 10.1103/PhysRev.102.593. |
[30] |
S.-Y. Ha, S. E. Noh and and S. B. Yun,
Global existence and stability of mild solutions to the Boltzmann system for gas mixtures, Quart. Appl. Math., 65 (2007), 757-779.
doi: 10.1090/S0033-569X-07-01068-6. |
[31] |
J. R. Haack, C. D. Hauck and M. S. Murillo,
A conservative, entropic multispecies BGK model, J. Stat. Phys., 168 (2017), 826-856.
doi: 10.1007/s10955-017-1824-9. |
[32] |
B. B. Hamel,
Kinetic model for binary gas mixtures, The Physics of Fluids, 8 (1965), 418-425.
doi: 10.1063/1.1761239. |
[33] |
J. Hu and S. Jin,
On kinetic flux vector splitting schemes for quantum Euler equations, Kinet. Relat. Models, 4 (2011), 517-530.
doi: 10.3934/krm.2011.4.517. |
[34] |
T. Ihn, Electronic Quantum Transport in Mesoscopic Semiconductor Structures, Springer Tracts in Modern Physics, Vol. 192, Springer, New York, NY, 2004.
doi: 10.1007/b97630. |
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A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics, Vol. 773, Springer, Berlin, Heidelberg, 2009.
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Über die kinetische fundamentalgleichung in der quantenstatistik, Zeitschrift für Physik A Hadrons and nuclei, 60 (1930), 652-662.
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C. Klingenberg and M. Pirner,
Existence, uniqueness and positivity of solutions for BGK models for mixtures, J. Differential Equations, 264 (2018), 702-727.
doi: 10.1016/j.jde.2017.09.019. |
[39] |
C. Klingenberg, M. Pirner and G. Puppo,
A consistent kinetic model for a two-component mixture of polyatomic molecules, Commun. Math. Sci., 17 (2019), 149-173.
doi: 10.4310/CMS.2019.v17.n1.a6. |
[40] |
C. Klingenberg, M. Pirner and G. Puppo,
A consistent kinetic model for a two-component mixture with an application to plasma, Kinet. Relat. Models, 10 (2017), 445-465.
doi: 10.3934/krm.2017017. |
[41] |
C. Klingenberg, M. Pirner and G. Puppo, Kinetic ES-BGK models for a multi-component gas mixture, Theory, Numerics and Applications of Hyperbolic Problems. II, 195-208, Springer Proc. Math. Stat., Vol. 237, Springer, Cham, 2018.
doi: 10.1007/978-3-319-91548-7. |
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X. Lu,
A modified Boltzmann equation for Bose-Einstein particles: Isotropic solutions and long-time behavior, J. Statist. Phys., 98 (2000), 1335-1394.
doi: 10.1023/A:1018628031233. |
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X. Lu,
On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles, J. Statist. Phys., 105 (2001), 353-388.
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P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
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A. J. H. McGaughey and M. Kaviany, Quantitative validation of the Boltzmann transport equation phonon thermal conductivity model under the single-mode relaxation time approximation, Physical Review B, 69 (2004), 094303.
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Simulation of shock wave diffraction by a square cylinder in gases of arbitrary statistics using a semiclassical Boltzmann-Bhatnagar-Gross-Krook equation solver, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 651-670.
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show all references
References:
[1] |
P. Andries, K. Aoki and B. Perthame,
A consistent BGK-type model for gas mixtures, J. Statist. Phys., 106 (2002), 993-1018.
doi: 10.1023/A:1014033703134. |
[2] |
N. W. Ashcroft and N. D. Mermin, Solid State Physic Holt, Rinehart and Winston, New York, USA. 1976. Google Scholar |
[3] |
G.-C. Bae and S.-B. Yun,
Stationary quantum BGK model for bosons and fermions in a bounded interval, J. Stat. Phys., 178 (2020), 845-868.
doi: 10.1007/s10955-019-02466-2. |
[4] |
G.-C. Bae and S.-B. Yun,
Quantum BGK model near a global Fermi-Dirac distribution, SIAM J. Math. Anal., 52 (2020), 2313-2352.
doi: 10.1137/19M1270021. |
[5] |
M. Bennoune, M. Lemou and L. Mieussens,
Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics., J. Comput. Phys., 227 (2008), 3781-3803.
doi: 10.1016/j.jcp.2007.11.032. |
[6] |
F. Bernard, A. Iollo and G. Puppo,
Accurate asymptotic preserving boundary conditions for kinetic equations on Cartesian grids, J. Sci. Comput., 65 (2015), 735-766.
doi: 10.1007/s10915-015-9984-8. |
[7] |
P. L. Bhathnagor, E. P. Gross and M. Krook, A model for collision processes in gases, Physical Review, 94 (1954), 511.
doi: 10.1103/PhysRev.94.511. |
[8] |
M. Bisi and M. J. Cáceres,
A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325.
doi: 10.4310/CMS.2016.v14.n2.a1. |
[9] |
M. Bisi, M. Groppi and G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamic limit, Physical Review E, 81 (2010), 036327.
doi: 10.1103/PhysRevE.81.036327. |
[10] |
A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga and I. F. Potapenko,
A general consistent BGK model for gas mixtures, Kinet. Relat. Models, 11 (2018), 1377-1393.
doi: 10.3934/krm.2018054. |
[11] |
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.
doi: 10.3934/krm.2019019. |
[12] |
M. Braukhoff, Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation, Kinet. Relat. Models, 13 (2020), 187–210. preprint, arXiv: 1803.00379.
doi: 10.3934/krm.2020007. |
[13] |
S. Brull, V. Pavan and J. Schneider,
Derivation of a BGK model for mixtures, Eur. J. Mech. B Fluids, 33 (2012), 74-86.
doi: 10.1016/j.euromechflu.2011.12.003. |
[14] |
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge university press, 1990.
![]() |
[15] |
N. Crouseilles and G. Manfredi,
Asymptotic preserving schemes for the Wigner-Poisson-BGK equations in the diffusion limit, Comput. Phys. Commun., 185 (2014), 448-458.
doi: 10.1016/j.cpc.2013.06.002. |
[16] |
G. Dimarco, L. Mieussens and V. Rispoli,
An asymptotic preserving automatic domain decomposition method for the Vlasov-Poisson-BGK system with applications to plasmas, J. Comput. Phys., 274 (2014), 122-139.
doi: 10.1016/j.jcp.2014.06.002. |
[17] |
G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.
doi: 10.1017/S0962492914000063. |
[18] |
P. Drude,
Zur elektronentheorie der metalle, Annalen der physik., 306 (1900), 566-613.
doi: 10.1002/andp.19003060312. |
[19] |
P. Drude,
Zur elektronentheorie der metalle; Ⅱ. Teil. galvanomagnetische und thermomagnetische effecte, Annalen der Physik., 308 (1900), 369-402.
doi: 10.1002/andp.19003081102. |
[20] |
F. Duan and J. Guojun, Introduction to Condensed Matter Physics, Volume 1, World Scientific Publishing Company, 2005. Google Scholar |
[21] |
M. Escobedo, S. Mischler and M. A. Valle,
Entropy maximisation problem for quantum relativistic particles, Bull. Soc. Math. France, 133 (2005), 87-120.
doi: 10.24033/bsmf.2480. |
[22] |
F. Filbet, J. Hu and S. Jin, A numerical scheme for the quantum Boltzmann equation efficient in the fluid regime, preprint, arXiv: 1009.3352. Google Scholar |
[23] |
F. Filbet, J. Hu and S. Jin,
A numerical scheme for the quantum Boltzmann equation with stiff collision terms, ESAIM Math. Model. Numer. Anal., 46 (2012), 443-463.
doi: 10.1051/m2an/2011051. |
[24] |
F. Filbet and S. Jin,
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.
doi: 10.1016/j.jcp.2010.06.017. |
[25] |
R. H. Fowler and L. Nordheim,
Electron emission in intense electric fields, Proceedings of the Royal Society A, 119 (1928), 173-181.
doi: 10.1098/rspa.1928.0091. |
[26] |
V. Garzó, A. Santos and J. J. Brey,
A kinetic model for a multicomponent gas, Physics of Fluids A: Fluid Dynamics, 1 (1989), 380-383.
doi: 10.1063/1.857458. |
[27] |
J. M. Greene,
Improved Bhatnagar–Gross–Krook model of electron–ion collisions, The Physics of Fluids, 16 (1973), 2022-2023.
doi: 10.1063/1.1694254. |
[28] |
M. Groppi, S. Monica and G. Spiga, A kinetic ellipsoidal BGK model for a binary gas mixture, EPL (Europhysics Letters), 96 (2011), 64002.
doi: 10.1209/0295-5075/96/64002. |
[29] |
E. P. Gross and M. Krook, Model for collision processes in gases: Small-amplitude oscillations of charged two-component systems, Physical Review, 102 (1956), 593.
doi: 10.1103/PhysRev.102.593. |
[30] |
S.-Y. Ha, S. E. Noh and and S. B. Yun,
Global existence and stability of mild solutions to the Boltzmann system for gas mixtures, Quart. Appl. Math., 65 (2007), 757-779.
doi: 10.1090/S0033-569X-07-01068-6. |
[31] |
J. R. Haack, C. D. Hauck and M. S. Murillo,
A conservative, entropic multispecies BGK model, J. Stat. Phys., 168 (2017), 826-856.
doi: 10.1007/s10955-017-1824-9. |
[32] |
B. B. Hamel,
Kinetic model for binary gas mixtures, The Physics of Fluids, 8 (1965), 418-425.
doi: 10.1063/1.1761239. |
[33] |
J. Hu and S. Jin,
On kinetic flux vector splitting schemes for quantum Euler equations, Kinet. Relat. Models, 4 (2011), 517-530.
doi: 10.3934/krm.2011.4.517. |
[34] |
T. Ihn, Electronic Quantum Transport in Mesoscopic Semiconductor Structures, Springer Tracts in Modern Physics, Vol. 192, Springer, New York, NY, 2004.
doi: 10.1007/b97630. |
[35] |
A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics, Vol. 773, Springer, Berlin, Heidelberg, 2009.
doi: 10.1007/978-3-540-89526-8. |
[36] |
I. M. Khalatnikov, An Introduction to the Theory of Superfluidity, CRC Press, 2018.
doi: 10.1201/9780429502897.![]() |
[37] |
S. Kikuchi and L. Nordheim,
Über die kinetische fundamentalgleichung in der quantenstatistik, Zeitschrift für Physik A Hadrons and nuclei, 60 (1930), 652-662.
doi: 10.1007/BF01339761. |
[38] |
C. Klingenberg and M. Pirner,
Existence, uniqueness and positivity of solutions for BGK models for mixtures, J. Differential Equations, 264 (2018), 702-727.
doi: 10.1016/j.jde.2017.09.019. |
[39] |
C. Klingenberg, M. Pirner and G. Puppo,
A consistent kinetic model for a two-component mixture of polyatomic molecules, Commun. Math. Sci., 17 (2019), 149-173.
doi: 10.4310/CMS.2019.v17.n1.a6. |
[40] |
C. Klingenberg, M. Pirner and G. Puppo,
A consistent kinetic model for a two-component mixture with an application to plasma, Kinet. Relat. Models, 10 (2017), 445-465.
doi: 10.3934/krm.2017017. |
[41] |
C. Klingenberg, M. Pirner and G. Puppo, Kinetic ES-BGK models for a multi-component gas mixture, Theory, Numerics and Applications of Hyperbolic Problems. II, 195-208, Springer Proc. Math. Stat., Vol. 237, Springer, Cham, 2018.
doi: 10.1007/978-3-319-91548-7. |
[42] |
X. Lu,
A modified Boltzmann equation for Bose-Einstein particles: Isotropic solutions and long-time behavior, J. Statist. Phys., 98 (2000), 1335-1394.
doi: 10.1023/A:1018628031233. |
[43] |
X. Lu,
On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles, J. Statist. Phys., 105 (2001), 353-388.
doi: 10.1023/A:1012282516668. |
[44] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[45] |
A. J. H. McGaughey and M. Kaviany, Quantitative validation of the Boltzmann transport equation phonon thermal conductivity model under the single-mode relaxation time approximation, Physical Review B, 69 (2004), 094303.
doi: 10.1103/PhysRevB.69.094303. |
[46] |
B. P. Muljadi and J.-Y. Yang,
Simulation of shock wave diffraction by a square cylinder in gases of arbitrary statistics using a semiclassical Boltzmann-Bhatnagar-Gross-Krook equation solver, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 651-670.
doi: 10.1098/rspa.2011.0275. |
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