February  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  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.

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:
[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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.

[52]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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.

[52]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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