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Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium
The derivation of the linear Boltzmann equation from a Rayleigh gas particle model
1. | Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom |
2. | Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom |
A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.
References:
[1] |
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander,
Vector-valued Laplace Transforms and Cauchy Problems volume 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, second edition, 2011.
doi: 10.1007/978-3-0348-0087-7. |
[2] |
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. |
[3] |
J. Banasiak and L. Arlotti,
Perturbations of Positive Semigroups with Applications Springer Monographs in Mathematics. Springer-Verlag London, Ltd. , London, 2006. |
[4] |
M. Bisi, J. A. 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. |
[5] |
T. Bodineau, I. 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. |
[6] |
T. Bodineau, I. Gallagher and L. Saint-Raymond,
From hard spheres dynamics to the Stokes-Fourier equations: An L$^2$ analysis of the Boltzmann-Grad limit, Comptes Rendus Mathematique, 353 (2015), 623-627.
doi: 10.1016/j.crma.2015.04.013. |
[7] |
C. Boldrighini, L. A. Bunimovich and Y. G. Sinaǐ,
On the Boltzmann equation for the Lorentz gas, J. Statist. Phys., 32 (1983), 477-501.
doi: 10.1007/BF01008951. |
[8] |
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, The Royal Society, 188 (1946), 10-18.
doi: 10.1098/rspa.1946.0093. |
[9] |
T. Carleman,
Problemes Mathématiques Dans la Théorie Cinétique de Gaz volume 2. Almqvist & Wiksells boktr, 1957. |
[10] |
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. |
[11] |
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. |
[12] |
D. L. Cohn,
Measure Theory Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser/Springer, New York, second edition, 2013.
doi: 10.1007/978-1-4614-6956-8. |
[13] |
L. Desvillettes and V. Ricci,
The Boltzmann-Grad limit of a stochastic Lorentz gas in a force field, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 637-648.
|
[14] |
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. |
[15] |
G. Gallavotti,
Statistical Mechanics. A Short Treatise Theoretical and Mathematical Physics. Springer-Verlag Berlin Heidelberg, 1999.
doi: 10.1007/978-3-662-03952-6. |
[16] |
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. |
[17] |
O. E. Lanford,
Dynamical systems, theory and applications: Battelle seattle 1974 rencontres, Chapter Time Evolution of Large Classical Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 38 (1975), 1-111.
|
[18] |
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. |
[19] |
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. |
[20] |
J. L. Lebowitz and H. Spohn,
Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55.
doi: 10.1007/BF01008247. |
[21] |
H. Lorentz,
The motion of electrons in metallic bodies i, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453.
|
[22] |
A. Pazy, J. Marklof, 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. |
[23] |
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. |
[24] |
K. Matthies and F. Theil,
Validity and non-validity of propagation of chaos, Analysis and Stochastics of growth processes and Interface Models, (2008), 101-119.
doi: 10.1093/acprof:oso/9780199239252.003.0005. |
[25] |
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. |
[26] |
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. |
[27] |
F. A. Molinet,
Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. i, Journal of Mathematical Physics, 18 (1977), 984-996.
doi: 10.1063/1.523380. |
[28] |
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. |
[29] |
M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials Reviews in Mathematical Physics 26 (2014), 1450001, 64 pp.
doi: 10.1142/S0129055X14500019. |
[30] |
H. Spohn,
The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290.
doi: 10.1007/BF01612893. |
[31] |
A. Pazy, H. Spohn, Kinetic equations from Hamiltonian dynamics: The Markovian approximations, In Kinetic theory and gas dynamics, volume 293 of CISM Courses and Lect., pages 183-211. Springer, Vienna, 1988.
doi: 10.1007/978-3-7091-2762-9_6. |
[32] |
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. |
[33] |
K. Uchiyama,
Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297.
|
[34] |
H. van Beijeren, O. E. Lanford, Ⅲ, 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] |
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander,
Vector-valued Laplace Transforms and Cauchy Problems volume 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, second edition, 2011.
doi: 10.1007/978-3-0348-0087-7. |
[2] |
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. |
[3] |
J. Banasiak and L. Arlotti,
Perturbations of Positive Semigroups with Applications Springer Monographs in Mathematics. Springer-Verlag London, Ltd. , London, 2006. |
[4] |
M. Bisi, J. A. 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. |
[5] |
T. Bodineau, I. 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. |
[6] |
T. Bodineau, I. Gallagher and L. Saint-Raymond,
From hard spheres dynamics to the Stokes-Fourier equations: An L$^2$ analysis of the Boltzmann-Grad limit, Comptes Rendus Mathematique, 353 (2015), 623-627.
doi: 10.1016/j.crma.2015.04.013. |
[7] |
C. Boldrighini, L. A. Bunimovich and Y. G. Sinaǐ,
On the Boltzmann equation for the Lorentz gas, J. Statist. Phys., 32 (1983), 477-501.
doi: 10.1007/BF01008951. |
[8] |
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, The Royal Society, 188 (1946), 10-18.
doi: 10.1098/rspa.1946.0093. |
[9] |
T. Carleman,
Problemes Mathématiques Dans la Théorie Cinétique de Gaz volume 2. Almqvist & Wiksells boktr, 1957. |
[10] |
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. |
[11] |
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. |
[12] |
D. L. Cohn,
Measure Theory Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser/Springer, New York, second edition, 2013.
doi: 10.1007/978-1-4614-6956-8. |
[13] |
L. Desvillettes and V. Ricci,
The Boltzmann-Grad limit of a stochastic Lorentz gas in a force field, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 637-648.
|
[14] |
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. |
[15] |
G. Gallavotti,
Statistical Mechanics. A Short Treatise Theoretical and Mathematical Physics. Springer-Verlag Berlin Heidelberg, 1999.
doi: 10.1007/978-3-662-03952-6. |
[16] |
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. |
[17] |
O. E. Lanford,
Dynamical systems, theory and applications: Battelle seattle 1974 rencontres, Chapter Time Evolution of Large Classical Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 38 (1975), 1-111.
|
[18] |
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. |
[19] |
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. |
[20] |
J. L. Lebowitz and H. Spohn,
Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55.
doi: 10.1007/BF01008247. |
[21] |
H. Lorentz,
The motion of electrons in metallic bodies i, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453.
|
[22] |
A. Pazy, J. Marklof, 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. |
[23] |
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. |
[24] |
K. Matthies and F. Theil,
Validity and non-validity of propagation of chaos, Analysis and Stochastics of growth processes and Interface Models, (2008), 101-119.
doi: 10.1093/acprof:oso/9780199239252.003.0005. |
[25] |
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. |
[26] |
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. |
[27] |
F. A. Molinet,
Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. i, Journal of Mathematical Physics, 18 (1977), 984-996.
doi: 10.1063/1.523380. |
[28] |
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. |
[29] |
M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials Reviews in Mathematical Physics 26 (2014), 1450001, 64 pp.
doi: 10.1142/S0129055X14500019. |
[30] |
H. Spohn,
The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290.
doi: 10.1007/BF01612893. |
[31] |
A. Pazy, H. Spohn, Kinetic equations from Hamiltonian dynamics: The Markovian approximations, In Kinetic theory and gas dynamics, volume 293 of CISM Courses and Lect., pages 183-211. Springer, Vienna, 1988.
doi: 10.1007/978-3-7091-2762-9_6. |
[32] |
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. |
[33] |
K. Uchiyama,
Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297.
|
[34] |
H. van Beijeren, O. E. Lanford, Ⅲ, 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. |



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