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.
Citation: |
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.
![]() |
|
L. Arlotti
and B. Lods
, Integral representation of the linear {B}oltzmann operator for granular gas dynamics with applications, Journal of Statistical Physics, 129 (2007)
, 517-536.
doi: 10.1007/s10955-007-9402-1.![]() ![]() ![]() |
|
J. Banasiak and L. Arlotti,
Perturbations of Positive Semigroups with Applications Springer Monographs in Mathematics. Springer-Verlag London, Ltd. , London, 2006.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
T. Bodineau
, I. 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.![]() ![]() ![]() |
|
C. Boldrighini
, L. 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.![]() ![]() ![]() |
|
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.
![]() |
|
T. Carleman,
Problemes Mathématiques Dans la Théorie Cinétique de Gaz volume 2. Almqvist & Wiksells boktr, 1957.
![]() |
|
C. Cercignani,
The Boltzmann Equation and Its Applications volume 67 of Applied Mathematical Sciences Springer-Verlag, New York, 1988.
![]() |
|
C. Cercignani, R. Illner and M. Pulvirenti,
The Mathematical Theory of Dilute Gases volume 106 of Applied Mathematical Sciences Springer-Verlag, 1994.
![]() |
|
D. L. Cohn,
Measure Theory Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser/Springer, New York, second edition, 2013.
![]() |
|
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.
![]() ![]() |
|
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.
![]() |
|
G. Gallavotti,
Statistical Mechanics. A Short Treatise Theoretical and Mathematical Physics. Springer-Verlag Berlin Heidelberg, 1999.
![]() |
|
F. Golse
, On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse Math., 17 (2008)
, 735-749.
doi: 10.5802/afst.1200.![]() ![]() ![]() |
|
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.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. L. Lebowitz
and H. Spohn
, Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982)
, 39-55.
doi: 10.1007/BF01008247.![]() ![]() ![]() |
|
H. Lorentz
, The motion of electrons in metallic bodies ⅰ, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905)
, 438-453.
![]() |
|
J. Marklof, Kinetic transport in crystals, In XVIth International Congress on Mathematical
Physics, pages 162-179. World Sci. Publ. , Hackensack, NJ, 2010.
![]() |
|
J. Marklof
and A. Strömbergsson
, The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. of Math., 174 (2011)
, 225-298.
doi: 10.4007/annals.2011.174.1.7.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
F. A. Molinet
, Existence, uniqueness and properties of the solutions of the {B}oltzmann kinetic equation for a weakly ionized gas. ⅰ, Journal of Mathematical Physics, 18 (1977)
, 984-996.
doi: 10.1063/1.523380.![]() ![]() ![]() |
|
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations volume 44 of Applied Mathematical Sciences Springer-Verlag, 1983.
![]() |
|
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.
![]() |
|
H. Spohn
, The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978)
, 277-290.
doi: 10.1007/BF01612893.![]() ![]() ![]() |
|
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.
![]() |
|
H. Spohn,
Large Scale Dynamics of Interacting Particles Texts and Monographs in Physics. Springer Berlin Heidelberg, 1991.
![]() |
|
K. Uchiyama
, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988)
, 245-297.
![]() ![]() |
|
H. van Beijeren
, O. E. Lanford
, 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.![]() ![]() ![]() |
The collision parameter
Two example trees
In the case