American Institute of Mathematical Sciences

Advanced
February  2021, 14(1): 25-44. doi: 10.3934/krm.2020047

BGK model of the multi-species Uehling-Uhlenbeck equation

Gi-Chan Bae 1, , Christian Klingenberg 2, , Marlies Pirner 3, and  Seok-Bae Yun 1,

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

Received  February 2020 Revised  July 2020 Published  February 2021 Early access  September 2020

Fund Project: Christian Klingenberg acknowledges support by the DFG grant KL-566/20-2. Marlies Pirner is supported by the Austrian Science Fund (FWF) project F65 and the Humboldt foundation. Seok-Bae Yun is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02

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.

 

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Keywords: BGK models, boltzmann equation, Uehling-Uhlenbeck equation, relaxation time approximation, gas mixture.
Mathematics Subject Classification: 82C40, 76P05, 35Q20, 35Q40, 81-10, 80-10.
Citation: Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047
References:
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show all references

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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[13]

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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.  Google Scholar

[17]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[18]

P. Drude, Zur elektronentheorie der metalle, Annalen der physik., 306 (1900), 566-613.  doi: 10.1002/andp.19003060312.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

B. B. Hamel, Kinetic model for binary gas mixtures, The Physics of Fluids, 8 (1965), 418-425.  doi: 10.1063/1.1761239.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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