BGK model of the multi-species Uehling-Uhlenbeck equation

Gi-Chan Bae; Christian Klingenberg; Marlies Pirner; Seok-Bae Yun

doi:10.3934/krm.2020047

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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 and Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047

References:

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[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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[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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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.
[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.
[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.
[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.
[47]	A. Nouri, An existence result for a quantum BGK model, Math. Comput. Modelling, 47 (2008), 515-529. doi: 10.1016/j.mcm.2007.05.002.
[48]	S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations, J. Sci. Comput., 32 (2007), 1-28. doi: 10.1007/s10915-006-9116-6.
[49]	M. Pirner, A BGK model for gas mixtures of polyatomic molecules allowing for slow and fast relaxation of the temperatures, J. Stat. Phys., 173 (2018), 1660-1687. doi: 10.1007/s10955-018-2158-y.
[50]	A. Rapp, S. Mandt and A. Rosch, Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices, Physical review letters., 105 (2010), 220405. doi: 10.1103/PhysRevLett.105.220405.
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