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July 2018, 38(7): 3299-3355. doi: 10.3934/dcds.2018143

Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Received  June 2017 Revised  February 2018 Published  April 2018

A linear Boltzmann equation with non-autonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution systems for Kolmogorov equations of associated probability measures on collision histories.

Citation: Karsten Matthies, George Stone. Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3299-3355. doi: 10.3934/dcds.2018143
References:
[1]

L. Arlotti and B. Lods, Integral representation of the linear Boltzmann operator for granular gas dynamics with applications, Journal of Statistical Physics, 129 (2007), 517-536. doi: 10.1007/s10955-007-9402-1.

[2]

L. ArlottiB. Lods and M. Mokhtar-Kharroubi, Non-autonomous honesty theory in abstract state spaces with applications to linear kinetic equations, Commun. Pure Appl. Anal., 13 (2014), 729-771. doi: 10.3934/cpaa.2014.13.729.

[3]

N. Ayi, From Newton's law to the linear Boltzmann equation without cut-off, Comm. Math. Phys., 350 (2017), 1219-1274. doi: 10.1007/s00220-016-2821-6.

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2006.

[5]

P. Billingsley, Probability and Measure, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.

[6]

M. BisiJ. Cañizo and B. Lods, Entropy dissipation estimates for the linear Boltzmann operator, Journal of Functional Analysis, 269 (2015), 1028-1069. doi: 10.1016/j.jfa.2015.05.002.

[7]

T. BodineauI. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones Mathematicae, 203 (2016), 493-553. doi: 10.1007/s00222-015-0593-9.

[8]

T. BodineauI. Gallagher and L. Saint-Raymond, From hard spheres dynamics to the Stokes-Fourier equations: an L2 analysis of the Boltzmann-Grad limit, Comptes Rendus Mathematique, 353 (2015), 623-627. doi: 10.1016/j.crma.2015.04.013.

[9]

C. BoldrighiniL. A. Bunimovich and Y. G. Sinaĭ, On the Boltzmann equation for the Lorentz gas, J. Statist. Phys., 32 (1983), 477-501. doi: 10.1007/BF01008951.

[10]

M. Born and H. S. Green, A general kinetic theory of liquids. ⅰ. the molecular distribution functions, In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 188, pages 10-18. The Royal Society, 1946. doi: 10.1098/rspa.1946.0093.

[11]

T. Carleman, Problemes Mathématiques dans la Théorie Cinétique de Gaz, volume 2. Almqvist & Wiksells boktr, 1957.

[12]

C. Cercignani, The Boltzmann Equation and Its Applications, volume 67 of Applied Mathematical Sciences, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[13]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, volume 106 of Applied Mathematical Sciences, Springer-Verlag, 1994. doi: 10.1007/978-1-4419-8524-8.

[14]

L. Desvillettes and M. Pulvirenti, The linear Boltzmann equation for long-range forces: A derivation from particle systems, Math. Models Methods Appl. Sci., 9 (1999), 1123-1145. doi: 10.1142/S0218202599000506.

[15]

S. Friedlander and D. Serre, editors, Handbook of Mathematical Fluid Dynamics, Vol. Ⅰ, North-Holland, Amsterdam, 2002.

[16]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics. European mathematical society, 2013.

[17]

G. Gallavotti, Statistical Mechanics. A Short Treatise, Theoretical and Mathematical Physics. Springer-Verlag Berlin Heidelberg, 1999. doi: 10.1007/978-3-662-03952-6.

[18]

F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse Math. (6), 17 (2008), 735-749. doi: 10.5802/afst.1200.

[19]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two-and three-dimensional rare gas in vacuum. Erratum and improved result, Comm. Math. Phys., 121 (1989), 143-146.

[20]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Comm. Math. Phys., 105 (1986), 189-203.

[21]

O. E. Lanford, Dynamical Systems, Theory and Applications: Battelle Seattle 1974 Rencontres, chapter Time evolution of large classical systems, pages 1-111. Springer Berlin Heidelberg, Berlin, Heidelberg, 1975.

[22]

J. L. Lebowitz and H. Spohn, Transport properties of the Lorentz gas: Fourier's law, J. Statist. Phys., 19 (1978), 633-654. doi: 10.1007/BF01011774.

[23]

J. L. Lebowitz and H. Spohn, Microscopic basis for Fick's law for self-diffusion, J. Statist. Phys., 28 (1982), 539-556. doi: 10.1007/BF01008323.

[24]

J. L. Lebowitz and H. Spohn, Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55. doi: 10.1007/BF01008247.

[25]

H. Lorentz, The motion of electrons in metallic bodies ⅰ, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453.

[26]

Kinetic transport in crystals, In XVIth International Congress on Mathematical Physics, pages 162-179. World Sci. Publ., Hackensack, NJ, 2010. doi: 10.1142/9789814304634_0009.

[27]

J. Marklof and A. Strömbergsson, The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. of Math. (2), 174 (2011), 225-298. doi: 10.4007/annals.2011.174.1.7.

[28]

K. MatthiesG. Stone and F. Theil, The derivation of the linear Boltzmann equation from a Rayleigh gas particle model, Kinetic and Related Models, 11 (2018), 137-177. doi: 10.3934/krm.2018008.

[29]

K. Matthies and F. Theil, Validity and failure of the Boltzmann approximation of kinetic annihilation, Journal of Nonlinear Science, 20 (2010), 1-46. doi: 10.1007/s00332-009-9049-y.

[30]

K. Matthies and F. Theil, A semigroup approach to the justification of kinetic theory, SIAM Journal on Mathematical Analysis, 44 (2012), 4345-4379. doi: 10.1137/120865598.

[31]

R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, In Evolution equations, semigroups and functional analysis (Milano, 2000), volume 50 of Progr. Nonlinear Differential Equations Appl., pages 279-293. Birkhäuser, Basel, 2002.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

M. Pulvirenti, Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum, Comm. Math. Phys., 113 (1987), 79-85.

[34]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Reviews in Mathematical Physics 2014, 1450001, 64 pp. doi: 10.1142/S0129055X14500019.

[35]

H. Spohn, The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290. doi: 10.1007/BF01612893.

[36]

H. Spohn, Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics. Springer Berlin Heidelberg, 1991. doi: 10.1007/978-3-642-84371-6.

[37]

K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297.

[38]

S. Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem, Japan Journal of Industrial and Applied Mathematics, 18 (2001), 383-392. doi: 10.1007/BF03168581.

[39]

H. van BeijerenO. E. LanfordⅢ, J. L. Lebowitz and H. Spohn, Equilibrium time correlation functions in the low-density limit, J. Statist. Phys., 22 (1980), 237-257. doi: 10.1007/BF01008050.

show all references

References:
[1]

L. Arlotti and B. Lods, Integral representation of the linear Boltzmann operator for granular gas dynamics with applications, Journal of Statistical Physics, 129 (2007), 517-536. doi: 10.1007/s10955-007-9402-1.

[2]

L. ArlottiB. Lods and M. Mokhtar-Kharroubi, Non-autonomous honesty theory in abstract state spaces with applications to linear kinetic equations, Commun. Pure Appl. Anal., 13 (2014), 729-771. doi: 10.3934/cpaa.2014.13.729.

[3]

N. Ayi, From Newton's law to the linear Boltzmann equation without cut-off, Comm. Math. Phys., 350 (2017), 1219-1274. doi: 10.1007/s00220-016-2821-6.

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2006.

[5]

P. Billingsley, Probability and Measure, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.

[6]

M. BisiJ. Cañizo and B. Lods, Entropy dissipation estimates for the linear Boltzmann operator, Journal of Functional Analysis, 269 (2015), 1028-1069. doi: 10.1016/j.jfa.2015.05.002.

[7]

T. BodineauI. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones Mathematicae, 203 (2016), 493-553. doi: 10.1007/s00222-015-0593-9.

[8]

T. BodineauI. Gallagher and L. Saint-Raymond, From hard spheres dynamics to the Stokes-Fourier equations: an L2 analysis of the Boltzmann-Grad limit, Comptes Rendus Mathematique, 353 (2015), 623-627. doi: 10.1016/j.crma.2015.04.013.

[9]

C. BoldrighiniL. A. Bunimovich and Y. G. Sinaĭ, On the Boltzmann equation for the Lorentz gas, J. Statist. Phys., 32 (1983), 477-501. doi: 10.1007/BF01008951.

[10]

M. Born and H. S. Green, A general kinetic theory of liquids. ⅰ. the molecular distribution functions, In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 188, pages 10-18. The Royal Society, 1946. doi: 10.1098/rspa.1946.0093.

[11]

T. Carleman, Problemes Mathématiques dans la Théorie Cinétique de Gaz, volume 2. Almqvist & Wiksells boktr, 1957.

[12]

C. Cercignani, The Boltzmann Equation and Its Applications, volume 67 of Applied Mathematical Sciences, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[13]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, volume 106 of Applied Mathematical Sciences, Springer-Verlag, 1994. doi: 10.1007/978-1-4419-8524-8.

[14]

L. Desvillettes and M. Pulvirenti, The linear Boltzmann equation for long-range forces: A derivation from particle systems, Math. Models Methods Appl. Sci., 9 (1999), 1123-1145. doi: 10.1142/S0218202599000506.

[15]

S. Friedlander and D. Serre, editors, Handbook of Mathematical Fluid Dynamics, Vol. Ⅰ, North-Holland, Amsterdam, 2002.

[16]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics. European mathematical society, 2013.

[17]

G. Gallavotti, Statistical Mechanics. A Short Treatise, Theoretical and Mathematical Physics. Springer-Verlag Berlin Heidelberg, 1999. doi: 10.1007/978-3-662-03952-6.

[18]

F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse Math. (6), 17 (2008), 735-749. doi: 10.5802/afst.1200.

[19]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two-and three-dimensional rare gas in vacuum. Erratum and improved result, Comm. Math. Phys., 121 (1989), 143-146.

[20]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Comm. Math. Phys., 105 (1986), 189-203.

[21]

O. E. Lanford, Dynamical Systems, Theory and Applications: Battelle Seattle 1974 Rencontres, chapter Time evolution of large classical systems, pages 1-111. Springer Berlin Heidelberg, Berlin, Heidelberg, 1975.

[22]

J. L. Lebowitz and H. Spohn, Transport properties of the Lorentz gas: Fourier's law, J. Statist. Phys., 19 (1978), 633-654. doi: 10.1007/BF01011774.

[23]

J. L. Lebowitz and H. Spohn, Microscopic basis for Fick's law for self-diffusion, J. Statist. Phys., 28 (1982), 539-556. doi: 10.1007/BF01008323.

[24]

J. L. Lebowitz and H. Spohn, Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55. doi: 10.1007/BF01008247.

[25]

H. Lorentz, The motion of electrons in metallic bodies ⅰ, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453.

[26]

Kinetic transport in crystals, In XVIth International Congress on Mathematical Physics, pages 162-179. World Sci. Publ., Hackensack, NJ, 2010. doi: 10.1142/9789814304634_0009.

[27]

J. Marklof and A. Strömbergsson, The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. of Math. (2), 174 (2011), 225-298. doi: 10.4007/annals.2011.174.1.7.

[28]

K. MatthiesG. Stone and F. Theil, The derivation of the linear Boltzmann equation from a Rayleigh gas particle model, Kinetic and Related Models, 11 (2018), 137-177. doi: 10.3934/krm.2018008.

[29]

K. Matthies and F. Theil, Validity and failure of the Boltzmann approximation of kinetic annihilation, Journal of Nonlinear Science, 20 (2010), 1-46. doi: 10.1007/s00332-009-9049-y.

[30]

K. Matthies and F. Theil, A semigroup approach to the justification of kinetic theory, SIAM Journal on Mathematical Analysis, 44 (2012), 4345-4379. doi: 10.1137/120865598.

[31]

R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, In Evolution equations, semigroups and functional analysis (Milano, 2000), volume 50 of Progr. Nonlinear Differential Equations Appl., pages 279-293. Birkhäuser, Basel, 2002.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

M. Pulvirenti, Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum, Comm. Math. Phys., 113 (1987), 79-85.

[34]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Reviews in Mathematical Physics 2014, 1450001, 64 pp. doi: 10.1142/S0129055X14500019.

[35]

H. Spohn, The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290. doi: 10.1007/BF01612893.

[36]

H. Spohn, Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics. Springer Berlin Heidelberg, 1991. doi: 10.1007/978-3-642-84371-6.

[37]

K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297.

[38]

S. Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem, Japan Journal of Industrial and Applied Mathematics, 18 (2001), 383-392. doi: 10.1007/BF03168581.

[39]

H. van BeijerenO. E. LanfordⅢ, J. L. Lebowitz and H. Spohn, Equilibrium time correlation functions in the low-density limit, J. Statist. Phys., 22 (1980), 237-257. doi: 10.1007/BF01008050.

Figure 1.  The parameters of a collision between the tagged particle and particle $j$.
Figure 2.  A tree with two collisions, the time of the final collision is $\tau$.
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