doi: 10.3934/krm.2021031
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Related Models, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[1]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345

[2]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[3]

Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic & Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029

[4]

Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009

[5]

Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic & Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036

[6]

Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141

[7]

Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341

[8]

Marzia Bisi, Giampiero Spiga. On a kinetic BGK model for slow chemical reactions. Kinetic & Related Models, 2011, 4 (1) : 153-167. doi: 10.3934/krm.2011.4.153

[9]

Young-Pil Choi, Seok-Bae Yun. A BGK kinetic model with local velocity alignment forces. Networks & Heterogeneous Media, 2020, 15 (3) : 389-404. doi: 10.3934/nhm.2020024

[10]

Sebastian Bauer. A non-relativistic model of plasma physics containing a radiation reaction term. Kinetic & Related Models, 2018, 11 (1) : 25-42. doi: 10.3934/krm.2018002

[11]

Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic & Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020

[12]

Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic & Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255

[13]

Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic & Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383

[14]

Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic & Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005

[15]

Hyunjin Ahn, Seung-Yeal Ha, Jeongho Kim. Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021156

[16]

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

[17]

Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047

[18]

Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177

[19]

Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049

[20]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (51)
  • HTML views (65)
  • Cited by (0)

Other articles
by authors

[Back to Top]