June  2020, 13(3): 435-478. doi: 10.3934/krm.2020015

High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials

School of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, P. R. China

* Corresponding author: Yu-Long Zhou

Received  May 2019 Revised  November 2019 Published  March 2020

The Boltzmann equation has an angular singularity inherent in the long range interaction between molecules. Angular singularity produces many difficulties in both theoretical and numerical study of Boltzmann equation. As a result, many rely on angular cutoff models to approximate the Boltzmann equation. Cutoff models have no angular singularity and can be solved by existent numerical methods. However, as the singularity goes stronger, the proportion of grazing collision becomes larger, which renders ineffectiveness of the cutoff Boltzmann equation. Based on the theoretical result that the limit of grazing collision is Landau operator, we propose to add a suitably scaled Landau operator to the cutoff equation to form a new approximate equation. This new approximate equation was studied in [27] in the case of hard potentials and its approximation accuracy is proved to be one order higher than that of angular cutoff models, which is a significant improvement in numerical computing. In this work, under moderately soft potentials, we establish the well-posedness theory of the new approximate equation, prove regularity propagation of its solution, check the high order accuracy. The new approximate equation can be solved by existing numerical methods, and this work may provide a theoretical foundation and a new direction to high order numerical methods for solving the Boltzmann equation.

Citation: Zheng-an Yao, Yu-Long Zhou. High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials. Kinetic & Related Models, 2020, 13 (3) : 435-478. doi: 10.3934/krm.2020015
References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.  Google Scholar

[2]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math., 52 (2012), 433-463.  doi: 10.1215/21562261-1625154.  Google Scholar

[3]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70.  doi: 10.1002/cpa.10012.  Google Scholar

[4]

R. J. AlonsoI. M. Gamba and S. H. Tharkabhushanam, Convergence and error estimates for the Lagrangian-based conservative spectral method for Boltzmann equations, SIAM J. Numer. Anal., 56 (2018), 3534-3579.  doi: 10.1137/18M1173332.  Google Scholar

[5]

R. J. Alonso and W. Sun, The radiative transfer equation in the forward-peaked regime, Comm. Math. Phys., 338 (2015), 1233-1286.  doi: 10.1007/s00220-015-2395-8.  Google Scholar

[6]

L. Arkeryd, On the Boltzmann equation, Arch. Ration. Mech. Anal., 45 (1972), 1-16.  doi: 10.1007/BF00253392.  Google Scholar

[7] G. A. Bird, Molecular Gas Dynamics, Clarendon Press, Oxford, 1994.   Google Scholar
[8]

E. A. CarlenM. C. CarvalhoJ. Le RouxM. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 1 (2010), 85-122.  doi: 10.3934/krm.2010.3.85.  Google Scholar

[9]

Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions I: spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8.  Google Scholar

[10]

L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theory Stat. Phys., 21 (1992), 259-276.  doi: 10.1080/00411459208203923.  Google Scholar

[11]

L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Rat. Mech. Anal., 193 (2009), 227-253.  doi: 10.1007/s00205-009-0233-x.  Google Scholar

[12]

L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part i: Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.  Google Scholar

[13]

L. FainsilberP. Kurlberg and B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation, SIAM J. Math. Anal., 37 (2006), 1903-1922.  doi: 10.1137/040618916.  Google Scholar

[14]

F. Filbet and L. Pareschi, A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case, J. Comput. Phys., 179 (2002), 1-26.  doi: 10.1006/jcph.2002.7010.  Google Scholar

[15]

N. Fournier, Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff, J. Stat. Phys., 125 (2006), 927-946.  doi: 10.1007/s10955-006-9208-6.  Google Scholar

[16]

N. Fournier and D. Godinho, Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff, Comm. Math. Phys., 316 (2012), 307-344.  doi: 10.1007/s00220-012-1578-9.  Google Scholar

[17]

N. Fournier and C. Mouhot, On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity, Commun. Math. Phys., 289 (2009), 803-824.  doi: 10.1007/s00220-009-0807-3.  Google Scholar

[18]

N. Fournier and H. Guerin, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys., 131 (2008), 749-781.  doi: 10.1007/s10955-008-9511-5.  Google Scholar

[19]

N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. Ec. Norm. Super., 50 (2017), 157-199.   Google Scholar

[20]

I. M. Gamba and J. R. Haack, A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit, J. Comput. Phys., 270 (2014), 40-57.  doi: 10.1016/j.jcp.2014.03.035.  Google Scholar

[21]

I. M. Gamba and S. H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), 2012-2036.  doi: 10.1016/j.jcp.2008.09.033.  Google Scholar

[22]

I. M. Gamba and S. H. Tharkabhushanam, Shock and boundary structure formation by spectral-Lagrangian methods for the inhomogeneous Boltzmann transport equation, J. Comput. Phys., 28 (2010), 430-460.  doi: 10.4208/jcm.1003-m0011.  Google Scholar

[23]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9.  Google Scholar

[24]

L.-B. He, Asymptotic analysis of the spatially homogeneous Boltzmann equation: Grazing collisions limit, J. Stat. Phys., 155 (2014), 151-210.  doi: 10.1007/s10955-014-0932-z.  Google Scholar

[25]

L.-B. He, Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction, Comm. Math. Phys., 312 (2012), 447-476.  doi: 10.1007/s00220-012-1481-4.  Google Scholar

[26]

L.-B. He, Sharp bounds for Boltzmann and Landau collision operators, Ann. Sci. Ec. Norm. Super., 51 (2018), 1253-1341.  doi: 10.24033/asens.2375.  Google Scholar

[27]

L.-B. He and Y.-L. Zhou, High order approximation for the Boltzmann equation without angular cutoff, Kinet. Relat. Models, 11 (2018), 547-596.  doi: 10.3934/krm.2018024.  Google Scholar

[28]

Z. HuoY. MorimotoS. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, 1 (2008), 453-489.  doi: 10.3934/krm.2008.1.453.  Google Scholar

[29]

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, Berkeley and Los Angeles, 1956, University of California Press, 171–197.  Google Scholar

[30]

X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021.  Google Scholar

[31]

H. P. Mckean, An exponential formula for solving Boltmann's equation for a Maxwellian gas, J. Combinatorial Theory, 2 (1967), 358-382.  doi: 10.1016/S0021-9800(67)80035-8.  Google Scholar

[32]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.  Google Scholar

[33]

S. MischlerC. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8.  Google Scholar

[34]

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 467-501.  doi: 10.1016/S0294-1449(99)80025-0.  Google Scholar

[35]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal., 173 (2004), 169-212.  doi: 10.1007/s00205-004-0316-7.  Google Scholar

[36]

K. Nanbu, Direct simulation scheme derived from the boltzmann equation I. monocomponent gases, J. Phys. Soc. Japan, 52 (1983), 2042-2049.   Google Scholar

[37]

A. PalczewskiJ. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal., 34 (1997), 1865-1883.  doi: 10.1137/S0036142995289007.  Google Scholar

[38]

A. Palczewski and J. Schneider, Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Stat. Phys., 91 (1998), 307-326.  doi: 10.1023/A:1023000406921.  Google Scholar

[39]

V. A. Panferov and A. G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Math. Methods Appl. Sci., 25 (2002), 571-593.  doi: 10.1002/mma.303.  Google Scholar

[40]

L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Trans. Theo. Stat. Phys., 25 (1996), 369-382.  doi: 10.1080/00411459608220707.  Google Scholar

[41]

L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., 37 (2000), 1217-1245.  doi: 10.1137/S0036142998343300.  Google Scholar

[42]

L. Pareschi and G. Russo, On the stability of spectral methods for the homogeneous Boltzmann equation, Trans. Theo. Stat. Phys., 29 (2000), 431-447.  doi: 10.1080/00411450008205883.  Google Scholar

[43]

L. PareschiG. Russo and G. Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comput. Phys., 165 (2000), 216-236.  doi: 10.1006/jcph.2000.6612.  Google Scholar

[44]

S. Rjasanow and W. Wagner, A stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys., 124 (1996), 243-253.  doi: 10.1006/jcph.1996.0057.  Google Scholar

[45]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation, Springer, Berlin, 2005.  Google Scholar

[46]

L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100.  doi: 10.1007/s00220-016-2757-x.  Google Scholar

[47]

Y.-L. Zhou, A refined estimate of the grazing limit from Boltzmann to Landau operator in Coulomb potential, Appl. Math. Lett., 100 (2020), 106039, 8pp. doi: 10.1016/j.aml.2019.106039.  Google Scholar

show all references

References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.  Google Scholar

[2]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math., 52 (2012), 433-463.  doi: 10.1215/21562261-1625154.  Google Scholar

[3]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70.  doi: 10.1002/cpa.10012.  Google Scholar

[4]

R. J. AlonsoI. M. Gamba and S. H. Tharkabhushanam, Convergence and error estimates for the Lagrangian-based conservative spectral method for Boltzmann equations, SIAM J. Numer. Anal., 56 (2018), 3534-3579.  doi: 10.1137/18M1173332.  Google Scholar

[5]

R. J. Alonso and W. Sun, The radiative transfer equation in the forward-peaked regime, Comm. Math. Phys., 338 (2015), 1233-1286.  doi: 10.1007/s00220-015-2395-8.  Google Scholar

[6]

L. Arkeryd, On the Boltzmann equation, Arch. Ration. Mech. Anal., 45 (1972), 1-16.  doi: 10.1007/BF00253392.  Google Scholar

[7] G. A. Bird, Molecular Gas Dynamics, Clarendon Press, Oxford, 1994.   Google Scholar
[8]

E. A. CarlenM. C. CarvalhoJ. Le RouxM. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 1 (2010), 85-122.  doi: 10.3934/krm.2010.3.85.  Google Scholar

[9]

Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions I: spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8.  Google Scholar

[10]

L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theory Stat. Phys., 21 (1992), 259-276.  doi: 10.1080/00411459208203923.  Google Scholar

[11]

L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Rat. Mech. Anal., 193 (2009), 227-253.  doi: 10.1007/s00205-009-0233-x.  Google Scholar

[12]

L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part i: Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.  Google Scholar

[13]

L. FainsilberP. Kurlberg and B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation, SIAM J. Math. Anal., 37 (2006), 1903-1922.  doi: 10.1137/040618916.  Google Scholar

[14]

F. Filbet and L. Pareschi, A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case, J. Comput. Phys., 179 (2002), 1-26.  doi: 10.1006/jcph.2002.7010.  Google Scholar

[15]

N. Fournier, Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff, J. Stat. Phys., 125 (2006), 927-946.  doi: 10.1007/s10955-006-9208-6.  Google Scholar

[16]

N. Fournier and D. Godinho, Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff, Comm. Math. Phys., 316 (2012), 307-344.  doi: 10.1007/s00220-012-1578-9.  Google Scholar

[17]

N. Fournier and C. Mouhot, On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity, Commun. Math. Phys., 289 (2009), 803-824.  doi: 10.1007/s00220-009-0807-3.  Google Scholar

[18]

N. Fournier and H. Guerin, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys., 131 (2008), 749-781.  doi: 10.1007/s10955-008-9511-5.  Google Scholar

[19]

N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. Ec. Norm. Super., 50 (2017), 157-199.   Google Scholar

[20]

I. M. Gamba and J. R. Haack, A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit, J. Comput. Phys., 270 (2014), 40-57.  doi: 10.1016/j.jcp.2014.03.035.  Google Scholar

[21]

I. M. Gamba and S. H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), 2012-2036.  doi: 10.1016/j.jcp.2008.09.033.  Google Scholar

[22]

I. M. Gamba and S. H. Tharkabhushanam, Shock and boundary structure formation by spectral-Lagrangian methods for the inhomogeneous Boltzmann transport equation, J. Comput. Phys., 28 (2010), 430-460.  doi: 10.4208/jcm.1003-m0011.  Google Scholar

[23]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9.  Google Scholar

[24]

L.-B. He, Asymptotic analysis of the spatially homogeneous Boltzmann equation: Grazing collisions limit, J. Stat. Phys., 155 (2014), 151-210.  doi: 10.1007/s10955-014-0932-z.  Google Scholar

[25]

L.-B. He, Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction, Comm. Math. Phys., 312 (2012), 447-476.  doi: 10.1007/s00220-012-1481-4.  Google Scholar

[26]

L.-B. He, Sharp bounds for Boltzmann and Landau collision operators, Ann. Sci. Ec. Norm. Super., 51 (2018), 1253-1341.  doi: 10.24033/asens.2375.  Google Scholar

[27]

L.-B. He and Y.-L. Zhou, High order approximation for the Boltzmann equation without angular cutoff, Kinet. Relat. Models, 11 (2018), 547-596.  doi: 10.3934/krm.2018024.  Google Scholar

[28]

Z. HuoY. MorimotoS. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, 1 (2008), 453-489.  doi: 10.3934/krm.2008.1.453.  Google Scholar

[29]

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, Berkeley and Los Angeles, 1956, University of California Press, 171–197.  Google Scholar

[30]

X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021.  Google Scholar

[31]

H. P. Mckean, An exponential formula for solving Boltmann's equation for a Maxwellian gas, J. Combinatorial Theory, 2 (1967), 358-382.  doi: 10.1016/S0021-9800(67)80035-8.  Google Scholar

[32]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.  Google Scholar

[33]

S. MischlerC. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8.  Google Scholar

[34]

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 467-501.  doi: 10.1016/S0294-1449(99)80025-0.  Google Scholar

[35]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal., 173 (2004), 169-212.  doi: 10.1007/s00205-004-0316-7.  Google Scholar

[36]

K. Nanbu, Direct simulation scheme derived from the boltzmann equation I. monocomponent gases, J. Phys. Soc. Japan, 52 (1983), 2042-2049.   Google Scholar

[37]

A. PalczewskiJ. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal., 34 (1997), 1865-1883.  doi: 10.1137/S0036142995289007.  Google Scholar

[38]

A. Palczewski and J. Schneider, Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Stat. Phys., 91 (1998), 307-326.  doi: 10.1023/A:1023000406921.  Google Scholar

[39]

V. A. Panferov and A. G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Math. Methods Appl. Sci., 25 (2002), 571-593.  doi: 10.1002/mma.303.  Google Scholar

[40]

L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Trans. Theo. Stat. Phys., 25 (1996), 369-382.  doi: 10.1080/00411459608220707.  Google Scholar

[41]

L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., 37 (2000), 1217-1245.  doi: 10.1137/S0036142998343300.  Google Scholar

[42]

L. Pareschi and G. Russo, On the stability of spectral methods for the homogeneous Boltzmann equation, Trans. Theo. Stat. Phys., 29 (2000), 431-447.  doi: 10.1080/00411450008205883.  Google Scholar

[43]

L. PareschiG. Russo and G. Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comput. Phys., 165 (2000), 216-236.  doi: 10.1006/jcph.2000.6612.  Google Scholar

[44]

S. Rjasanow and W. Wagner, A stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys., 124 (1996), 243-253.  doi: 10.1006/jcph.1996.0057.  Google Scholar

[45]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation, Springer, Berlin, 2005.  Google Scholar

[46]

L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100.  doi: 10.1007/s00220-016-2757-x.  Google Scholar

[47]

Y.-L. Zhou, A refined estimate of the grazing limit from Boltzmann to Landau operator in Coulomb potential, Appl. Math. Lett., 100 (2020), 106039, 8pp. doi: 10.1016/j.aml.2019.106039.  Google Scholar

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Yong-Kum Cho, Hera Yun. On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials. Kinetic & Related Models, 2012, 5 (4) : 769-786. doi: 10.3934/krm.2012.5.769

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Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169

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