December  2021, 14(6): 949-959. doi: 10.3934/krm.2021031

Relativistic BGK model for massless particles in the FLRW spacetime

1. 

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

2. 

Department of Mathematics and Research Institute for Basic Science, Kyung Hee University, Seoul, 02447, Republic of Korea

Received  April 2021 Revised  August 2021 Published  December 2021 Early access  September 2021

In this paper, we address the Cauchy problem for the relativistic BGK model proposed by Anderson and Witting for massless particles in the Friedmann-Lemaȋtre-Robertson-Walker (FLRW) spacetime. We first derive the explicit form of the Jüttner distribution in the FLRW spacetime, together with a set of nonlinear relations that leads to the conservation laws of particle number, momentum, and energy for both Maxwell-Boltzmann particles and Bose-Einstein particles. Then, we find sufficient conditions that guarantee the existence of equilibrium coefficients satisfying the nonlinear relations and we show that the condition is satisfied through all the induction steps once it is verified for the initial step. Using this observation, we construct explicit solutions of the relativistic BGK model of Anderson-Witting type for massless particles in the FLRW spacetime.

Citation: Byung-Hoon Hwang, Ho Lee, Seok-Bae Yun. Relativistic BGK model for massless particles in the FLRW spacetime. Kinetic and Related Models, 2021, 14 (6) : 949-959. doi: 10.3934/krm.2021031
References:
[1]

J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica, 74 (1974), 466–488. doi: 10.1016/0031-8914(74)90355-3.

[2]

H. BarzegarD. Fajman and G. Heißel, Isotropization of slowly expanding spacetimes, Phys. Rev. D, 101 (2020), 044046.  doi: 10.1103/PhysRevD.101.044046.

[3]

D. BazowG. S. DenicolU. HeinzM. Martinez and J. Noronha, Analytic solution of the Boltzmann equation in an expanding system, Phys. Rev. Lett., 116 (2016), 022301.  doi: 10.1103/PhysRevLett.116.022301.

[4]

D. BazowG. S. DenicolU. HeinzM. Martinez and J. Noronha, Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaȋtre-Robertson-Walker spacetime, Phys. Rev. D, 94 (2016), 125006.  doi: 10.1103/physrevd.94.125006.

[5]

A. BellouquidJ. CalvoJ. Nieto and J. Soler, On the relativistic BGK-Boltzmann model: Asymptotics and hydrodynamics, J. Stat. Phys., 149 (2012), 284-316.  doi: 10.1007/s10955-012-0600-0.

[6]

A. BellouquidJ. Nieto and L. Urrutia, Global existence and asymptotic stability near equilibrium for the relativistic BGK model, Nonlinear Anal., 114 (2015), 87-104.  doi: 10.1016/j.na.2014.10.020.

[7]

P. L. BhatnagarE. P. Gross and M. L. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[8]

J. Calvo, P.-E. Jabin and J. Soler, Global weak solutions to the relativistic BGK equation, Comm. Partial Differential Equations, 45 (2020), 191–229. doi: 10.1080/03605302.2019.1669642.

[9]

W. Florkowski, R. Ryblewski and M. Strickland, Anisotropic hydrodynamics for rapidly expanding systems, Nucl. Phys. A, 916 (2013), 249–259. doi: 10.1016/j.nuclphysa.2013.08.004.

[10]

W. Florkowski, R. Ryblewski and M. Strickland, Testing viscous and anisotropic hydrodynamics in an exactly solvable case, Phys. Rev. C., 88 (2013), 024903. doi: 10.1103/PhysRevC.88.024903.

[11]

B.-H. Hwang, T. Ruggeri and S.-B. Yun, On a relativistic BGK model for polyatomic gases near equilibrium, Preprint; arXiv: 2102.00462.

[12]

B.-H. Hwang and S.-B. Yun, Anderson-Witting model of the relativistic Boltzmann equation near equilibrium, J. Stat. Phys., 176 (2019), 1009–1045. doi: 10.1007/s10955-019-02330-3.

[13]

B.-H. Hwang and S.-B. Yun, Stationary solutions to the Anderson–Witting model of the relativistic Boltzmann equation in a bounded interval, SIAM J. Math. Anal., 53 (2021), 730-753.  doi: 10.1137/20M1331378.

[14]

B.-H. Hwang and S.-B. Yun, Stationary solutions to the boundary value problem for the relativistic BGK model in a slab, Kinet. Relat. Models, 12 (2019), 749–764. doi: 10.3934/krm.2019029.

[15]

A. Jaiswal, R. Ryblewski and M. Strickland, Transport coefficients for bulk viscous evolution in the relaxation time approximation, Phys. Rev. C., 90 (2014), 044908. doi: 10.1103/PhysRevC.90.044908.

[16]

F. Jüttner, Das Maxwellsche gesetz der geschwindigkeitsverteilung in der relativtheorie, Ann. Physik, 339 (1911), 856–882. doi: 10.1002/andp.19113390503.

[17]

F. Jüttner, Die relativistische Quantentheorie des idealen Gases, Zeitschr. Physik, 47 (1928), 542–566. doi: 10.1007/BF01340339.

[18]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press., 1959.

[19]

H. Lee, The spatially homogeneous Boltzmann equation for massless particles in an FLRW background, J. Math. Phys., 62 (2021), 031502, 15 pp. doi: 10.1063/5.0037951.

[20]

H. LeeE. Nungesser and P. Tod, The massless Einstein-Boltzmann system with a conformal-gauge singularity in an FLRW background, Classical Quantum Gravity, 37 (2020), 035005.  doi: 10.1088/1361-6382/ab5f41.

[21]

H. Lee, E. Nungesser and P. Tod, On the future of solutions to the massless Einstein-Vlasov system in a Bianchi I cosmology, Gen. Relativity Gravitation, 52 (2020), no. 48. doi: 10.1007/s10714-020-02699-7.

[22]

R. Maartens and F. P. Wolvaardt, Exact non-equilibrium solutions of the Einstein-Boltzmann equations, Classical Quantum Gravity, 11 (1994), 203–225. doi: 10.1088/0264-9381/11/1/021.

[23]

C. Marle, Modele cinétique pour l'établissement des lois de la conduction de la chaleur et de la viscosité en théorie de la relativité, C. R. Acad. Sci. Paris, 260 (1965), 6539–6541.

[24]

C. Marle, Sur l'établissement des equations de l'hydrodynamique des fluides relativistes dissipatifs, I. L'equation de Boltzmann relativiste, Ann. Inst. Henri Poincaré, 10 (1969), 67–127.

[25]

M. Mendoza, I. Karlin, S. Succi and H. J. Herrmann, Relativistic lattice Boltzmann model with improved dissipation, Phys. Rev. D., 87 (2013), 065027. doi: 10.1103/PhysRevD.87.065027.

[26]

E. Molnár, H. Niemi and D. H. Rischke, Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation, Phys. Rev. D., 93 (2016), 114025. doi: 10.1103/PhysRevD.93.114025.

[27]

S. Pennisi and T. Ruggeri, A new BGK model for relativistic kinetic theory of monatomic and polyatomic gases, J. Phys. Conf. Ser., 1035 (2018), 012005. doi: 10.1088/1742-6596/1035/1/012005.

[28]

K. P. Tod, Isotropic cosmological singularities: Other matter models, Class. Quantum Grav., 20 (2003), 521-534.  doi: 10.1088/0264-9381/20/3/309.

[29]

R. M. Wald, General Relativity, University of Chicago Press, Chicago, IL, 1984. doi: 10.7208/chicago/9780226870373.001.0001.

[30]

P. Walender, On the temperature jump in a rarefied gas, Ark, Fys., 7 (1954), 507-553. 

show all references

References:
[1]

J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica, 74 (1974), 466–488. doi: 10.1016/0031-8914(74)90355-3.

[2]

H. BarzegarD. Fajman and G. Heißel, Isotropization of slowly expanding spacetimes, Phys. Rev. D, 101 (2020), 044046.  doi: 10.1103/PhysRevD.101.044046.

[3]

D. BazowG. S. DenicolU. HeinzM. Martinez and J. Noronha, Analytic solution of the Boltzmann equation in an expanding system, Phys. Rev. Lett., 116 (2016), 022301.  doi: 10.1103/PhysRevLett.116.022301.

[4]

D. BazowG. S. DenicolU. HeinzM. Martinez and J. Noronha, Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaȋtre-Robertson-Walker spacetime, Phys. Rev. D, 94 (2016), 125006.  doi: 10.1103/physrevd.94.125006.

[5]

A. BellouquidJ. CalvoJ. Nieto and J. Soler, On the relativistic BGK-Boltzmann model: Asymptotics and hydrodynamics, J. Stat. Phys., 149 (2012), 284-316.  doi: 10.1007/s10955-012-0600-0.

[6]

A. BellouquidJ. Nieto and L. Urrutia, Global existence and asymptotic stability near equilibrium for the relativistic BGK model, Nonlinear Anal., 114 (2015), 87-104.  doi: 10.1016/j.na.2014.10.020.

[7]

P. L. BhatnagarE. P. Gross and M. L. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[8]

J. Calvo, P.-E. Jabin and J. Soler, Global weak solutions to the relativistic BGK equation, Comm. Partial Differential Equations, 45 (2020), 191–229. doi: 10.1080/03605302.2019.1669642.

[9]

W. Florkowski, R. Ryblewski and M. Strickland, Anisotropic hydrodynamics for rapidly expanding systems, Nucl. Phys. A, 916 (2013), 249–259. doi: 10.1016/j.nuclphysa.2013.08.004.

[10]

W. Florkowski, R. Ryblewski and M. Strickland, Testing viscous and anisotropic hydrodynamics in an exactly solvable case, Phys. Rev. C., 88 (2013), 024903. doi: 10.1103/PhysRevC.88.024903.

[11]

B.-H. Hwang, T. Ruggeri and S.-B. Yun, On a relativistic BGK model for polyatomic gases near equilibrium, Preprint; arXiv: 2102.00462.

[12]

B.-H. Hwang and S.-B. Yun, Anderson-Witting model of the relativistic Boltzmann equation near equilibrium, J. Stat. Phys., 176 (2019), 1009–1045. doi: 10.1007/s10955-019-02330-3.

[13]

B.-H. Hwang and S.-B. Yun, Stationary solutions to the Anderson–Witting model of the relativistic Boltzmann equation in a bounded interval, SIAM J. Math. Anal., 53 (2021), 730-753.  doi: 10.1137/20M1331378.

[14]

B.-H. Hwang and S.-B. Yun, Stationary solutions to the boundary value problem for the relativistic BGK model in a slab, Kinet. Relat. Models, 12 (2019), 749–764. doi: 10.3934/krm.2019029.

[15]

A. Jaiswal, R. Ryblewski and M. Strickland, Transport coefficients for bulk viscous evolution in the relaxation time approximation, Phys. Rev. C., 90 (2014), 044908. doi: 10.1103/PhysRevC.90.044908.

[16]

F. Jüttner, Das Maxwellsche gesetz der geschwindigkeitsverteilung in der relativtheorie, Ann. Physik, 339 (1911), 856–882. doi: 10.1002/andp.19113390503.

[17]

F. Jüttner, Die relativistische Quantentheorie des idealen Gases, Zeitschr. Physik, 47 (1928), 542–566. doi: 10.1007/BF01340339.

[18]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press., 1959.

[19]

H. Lee, The spatially homogeneous Boltzmann equation for massless particles in an FLRW background, J. Math. Phys., 62 (2021), 031502, 15 pp. doi: 10.1063/5.0037951.

[20]

H. LeeE. Nungesser and P. Tod, The massless Einstein-Boltzmann system with a conformal-gauge singularity in an FLRW background, Classical Quantum Gravity, 37 (2020), 035005.  doi: 10.1088/1361-6382/ab5f41.

[21]

H. Lee, E. Nungesser and P. Tod, On the future of solutions to the massless Einstein-Vlasov system in a Bianchi I cosmology, Gen. Relativity Gravitation, 52 (2020), no. 48. doi: 10.1007/s10714-020-02699-7.

[22]

R. Maartens and F. P. Wolvaardt, Exact non-equilibrium solutions of the Einstein-Boltzmann equations, Classical Quantum Gravity, 11 (1994), 203–225. doi: 10.1088/0264-9381/11/1/021.

[23]

C. Marle, Modele cinétique pour l'établissement des lois de la conduction de la chaleur et de la viscosité en théorie de la relativité, C. R. Acad. Sci. Paris, 260 (1965), 6539–6541.

[24]

C. Marle, Sur l'établissement des equations de l'hydrodynamique des fluides relativistes dissipatifs, I. L'equation de Boltzmann relativiste, Ann. Inst. Henri Poincaré, 10 (1969), 67–127.

[25]

M. Mendoza, I. Karlin, S. Succi and H. J. Herrmann, Relativistic lattice Boltzmann model with improved dissipation, Phys. Rev. D., 87 (2013), 065027. doi: 10.1103/PhysRevD.87.065027.

[26]

E. Molnár, H. Niemi and D. H. Rischke, Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation, Phys. Rev. D., 93 (2016), 114025. doi: 10.1103/PhysRevD.93.114025.

[27]

S. Pennisi and T. Ruggeri, A new BGK model for relativistic kinetic theory of monatomic and polyatomic gases, J. Phys. Conf. Ser., 1035 (2018), 012005. doi: 10.1088/1742-6596/1035/1/012005.

[28]

K. P. Tod, Isotropic cosmological singularities: Other matter models, Class. Quantum Grav., 20 (2003), 521-534.  doi: 10.1088/0264-9381/20/3/309.

[29]

R. M. Wald, General Relativity, University of Chicago Press, Chicago, IL, 1984. doi: 10.7208/chicago/9780226870373.001.0001.

[30]

P. Walender, On the temperature jump in a rarefied gas, Ark, Fys., 7 (1954), 507-553. 

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