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

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

Received  February 2020 Revised  July 2020 Published  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.

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:
[1]

P. AndriesK. 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

[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.  Google Scholar

[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.  Google Scholar

[5]

M. BennouneM. 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.  Google Scholar

[6]

F. BernardA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. V. BobylevM. BisiM. GroppiG. 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

[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.  Google Scholar

[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]

S. BrullV. 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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[16]

G. DimarcoL. 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. EscobedoS. 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. FilbetJ. 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. GarzóA. 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. HaS. 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. HaackC. 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

[36] I. M. Khalatnikov, An Introduction to the Theory of Superfluidity, CRC Press, 2018.  doi: 10.1201/9780429502897.  Google Scholar
[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.  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. KlingenbergM. 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]

C. KlingenbergM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

P.-G. Reinhard and E. Suraud, A quantum relaxation-time approximation for finite fermion systems, Ann. Physics, 354 (2015), 183-202.  doi: 10.1016/j.aop.2014.12.011.  Google Scholar

[52]

C. Ringhofer, Computational methods for semiclassical and quantum transport in semiconductor devices, Acta numerica, 6 (1997), 485-521.  doi: 10.1017/S0962492900002762.  Google Scholar

[53]

G. RussoP. Santagati and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 50 (2012), 1111-1135.  doi: 10.1137/100800348.  Google Scholar

[54]

G. Russo and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 56 (2018), 3580-3610.  doi: 10.1137/17M1163360.  Google Scholar

[55]

U. SchneiderL. HackermüllerJ. P. RonzheimerS. WillS. BraunT. BestI. BlochE. DemlerS. MandtD. 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.  Google Scholar

[56]

Y.-H. Shi and J. Y. Yang, A gas-kinetic BGK scheme for semiclassical Boltzmann hydrodynamic transport, J. Comput. Phys., 227 (2008), 9389-9407.  doi: 10.1016/j.jcp.2008.06.036.  Google Scholar

[57]

V. Sofonea and R. F. Sekerka, BGK models for diffusion in isothermal binary fluid systems, Physica A: Statistical Mechanics and its Applications, 299 (2001), 494-520.  doi: 10.1016/S0378-4371(01)00246-1.  Google Scholar

[58]

A. C. Sparavigna, The Boltzmann equation of phonon thermal transport solved in the relaxation time approximation-I-Theory, Mechanics, Materials Science & Engineering Journal, 2016 (2016), 34-35.   Google Scholar

[59]

N.-D. SuhM. R. Feix and P. Bertrand, Numerical simulation of the quantum Liouville-Poisson system, Journal of Computational Physics, 94 (1991), 403-418.  doi: 10.1016/0021-9991(91)90227-C.  Google Scholar

[60]

B. N. Todorova and R. Steijl, Derivation and numerical comparison of Shakhov and ellipsoidal statistical kinetic models for a monoatomic gas mixture, Eur. J. Mech. B Fluids, 76 (2019), 390-402.  doi: 10.1016/j.euromechflu.2019.04.001.  Google Scholar

[61]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. Ⅰ, Physical Review, 43 (1933), 552-561.  doi: 10.1103/PhysRev.43.552.  Google Scholar

[62]

E. A. Uehling, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. Ⅱ, Physical Review, 46 (1934), 917-929.  doi: 10.1103/PhysRev.46.917.  Google Scholar

[63]

L. WuJ. Meng and Y. Zhang, Kinetic modelling of the quantum gases in the normal phase, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 1799-1823.  doi: 10.1098/rspa.2011.0673.  Google Scholar

[64]

J.-Y. Yang and L.-H. Hung, Lattice Uehling-Uhlenbeck Boltzmann-Bhatnagar-Gross-Krook hydrodynamics of quantum gases, Physical Review E, 79 (2009), 056708. doi: 10.1103/PhysRevE.79.056708.  Google Scholar

[65]

R. Yano, Fast and accurate calculation of dilute quantum gas using Uehling-Uhlenbeck model equation, J. Comput. Phys., 330 (2017), 1010-1021.  doi: 10.1016/j.jcp.2016.10.071.  Google Scholar

show all references

References:
[1]

P. AndriesK. 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

[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.  Google Scholar

[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.  Google Scholar

[5]

M. BennouneM. 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.  Google Scholar

[6]

F. BernardA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. V. BobylevM. BisiM. GroppiG. 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

[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.  Google Scholar

[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]

S. BrullV. 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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[16]

G. DimarcoL. 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. EscobedoS. 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. FilbetJ. 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. GarzóA. 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. HaS. 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. HaackC. 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

[36] I. M. Khalatnikov, An Introduction to the Theory of Superfluidity, CRC Press, 2018.  doi: 10.1201/9780429502897.  Google Scholar
[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.  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. KlingenbergM. 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]

C. KlingenbergM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

P.-G. Reinhard and E. Suraud, A quantum relaxation-time approximation for finite fermion systems, Ann. Physics, 354 (2015), 183-202.  doi: 10.1016/j.aop.2014.12.011.  Google Scholar

[52]

C. Ringhofer, Computational methods for semiclassical and quantum transport in semiconductor devices, Acta numerica, 6 (1997), 485-521.  doi: 10.1017/S0962492900002762.  Google Scholar

[53]

G. RussoP. Santagati and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 50 (2012), 1111-1135.  doi: 10.1137/100800348.  Google Scholar

[54]

G. Russo and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 56 (2018), 3580-3610.  doi: 10.1137/17M1163360.  Google Scholar

[55]

U. SchneiderL. HackermüllerJ. P. RonzheimerS. WillS. BraunT. BestI. BlochE. DemlerS. MandtD. 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.  Google Scholar

[56]

Y.-H. Shi and J. Y. Yang, A gas-kinetic BGK scheme for semiclassical Boltzmann hydrodynamic transport, J. Comput. Phys., 227 (2008), 9389-9407.  doi: 10.1016/j.jcp.2008.06.036.  Google Scholar

[57]

V. Sofonea and R. F. Sekerka, BGK models for diffusion in isothermal binary fluid systems, Physica A: Statistical Mechanics and its Applications, 299 (2001), 494-520.  doi: 10.1016/S0378-4371(01)00246-1.  Google Scholar

[58]

A. C. Sparavigna, The Boltzmann equation of phonon thermal transport solved in the relaxation time approximation-I-Theory, Mechanics, Materials Science & Engineering Journal, 2016 (2016), 34-35.   Google Scholar

[59]

N.-D. SuhM. R. Feix and P. Bertrand, Numerical simulation of the quantum Liouville-Poisson system, Journal of Computational Physics, 94 (1991), 403-418.  doi: 10.1016/0021-9991(91)90227-C.  Google Scholar

[60]

B. N. Todorova and R. Steijl, Derivation and numerical comparison of Shakhov and ellipsoidal statistical kinetic models for a monoatomic gas mixture, Eur. J. Mech. B Fluids, 76 (2019), 390-402.  doi: 10.1016/j.euromechflu.2019.04.001.  Google Scholar

[61]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. Ⅰ, Physical Review, 43 (1933), 552-561.  doi: 10.1103/PhysRev.43.552.  Google Scholar

[62]

E. A. Uehling, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. Ⅱ, Physical Review, 46 (1934), 917-929.  doi: 10.1103/PhysRev.46.917.  Google Scholar

[63]

L. WuJ. Meng and Y. Zhang, Kinetic modelling of the quantum gases in the normal phase, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 1799-1823.  doi: 10.1098/rspa.2011.0673.  Google Scholar

[64]

J.-Y. Yang and L.-H. Hung, Lattice Uehling-Uhlenbeck Boltzmann-Bhatnagar-Gross-Krook hydrodynamics of quantum gases, Physical Review E, 79 (2009), 056708. doi: 10.1103/PhysRevE.79.056708.  Google Scholar

[65]

R. Yano, Fast and accurate calculation of dilute quantum gas using Uehling-Uhlenbeck model equation, J. Comput. Phys., 330 (2017), 1010-1021.  doi: 10.1016/j.jcp.2016.10.071.  Google Scholar

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