Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
On Linear Hypocoercive BGK Models Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond

Author : Chiara Saffirio

Published in: From Particle Systems to Partial Differential Equations III

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We review some results concerning the derivation of the Boltzmann equation starting from the many-body classical Hamiltonian dynamics. In particular, the celebrated paper by Lanford III [21] and the more recent papers [13, 23] are discussed.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Modelling of Systems with a Dispersed Phase: “Measuring” Small Sets in the Presence of Elliptic Operators
next chapter Duality Relations for the Periodic ASEP Conditioned on a Low Current
Footnotes
1
The H-Theorem asserts that the kinetic entropy associated to the solution f(t) of the Boltzmann Eq. (1) decreases in time. More precisely, let H(f) be the H-functional defined as the information entropy with a negative sign:
$$\begin{aligned} H(f(t))=\int _{\mathbb {R}^3\times \mathbb {R}^3}dx\,dv\,f(t,x,v)[\log {f(t,x,v)}-1]. \end{aligned}$$
A straightforward computation shows that \(H(f(t))\le H(f(0))\).
 
2
For simplicity, the potential is assumed to be smooth, to ensure the existence and uniqueness of the solution to the Newton equations (6).
 
Literature
1.
Bobylev, A.V., Pulvirenti, M., Saffirio, C.: From particle systems to the Landau equation: a consistency result. Comm. Math. Phys. 319(3), 683–702 (2013)MathSciNetCrossRefMATH
2.
Bodineau, T., Gallagher, I., Saint-Raymond, L.: The Brownian motion as the limit of a deterministic system of hard-spheres. arXiv:​1305.​3397
3.
Bodineau, T., Gallagher, I., Saint-Raymond, L. : From hard sphere dynamics to the Stokes-Fourier equations: an \(L^2\) analysis of the Boltzmann-Grad limit. arXiv:​1511.​03057
4.
Bogoliubov, N.: Problems of dynamical theory in statistical physics. In: de Boer, J., Uhlenbeck, G.E. (eds.) Studies in Statistical Mechanics. Interscience, New York (1962)
5.
Boltzmann, L.: Lectures on Gas Theory. English edition annotated by Brush, S. University of California Press, Berkeley (1964, reprint)
6.
Born, M., Green, H.S.: A general kinetic theory of liquids. I. The molecular distribution functions. Proc. Roy. Soc. Lond. A 188, 10–18 (1946)
7.
8.
Cercignani, C., Gerasimenko, V.I., Petrina, D.I.: Many-Particle Dynamics and Kinetic Equations. Kluwer Academic Publishers, Netherlands (1997)CrossRefMATH
9.
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York (1994)
10.
11.
Desvillettes, L., Pulvirenti, M.: The linear Boltzmann equation for long-range forces: a derivation from particle systems. Model. Methods Appl. Sci. 9, 1123–1145 (1999)MathSciNetCrossRefMATH
12.
Fournier, N.: Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential. Commun. Math. Phys. 299(3), 765–782 (2010)MathSciNetCrossRefMATH
13.
Gallagher, I., Saint Raymond, L., Texier, B.: From Newton to Boltzmann: Hard Spheres and Short-Range Potentials. Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2013)
14.
15.
16.
Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum. Commun. Math. Phys. 105, 189–203 (1986)MathSciNetCrossRefMATH
17.
Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two- and three-dimensional rare gas in vacuum: erratum and improved result. Commun. Math. Phys. 121, 143–146 (1989)MathSciNetCrossRefMATH
18.
King, F.: BBGKY Hierarchy for Positive Potentials, Ph.D. thesis, Department of Mathematics, University of California, Berkeley (1975)
19.
Kirkwood, J.G.: The statistical mechanical theory of transport process I. General theory. J. Chem. Phys 14, 180–202 (1946)CrossRef
20.
Landau, L.D.: Kinetic equation in the case of Coulomb interaction. Phys. Zs. Sow. Union 10, 154 (1936). (in German)
21.
Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol. 38, pp. 1–111. Springer, Berlin (1975)
22.
Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. Roy. Soc. Lond. Ser. A 157, 49–88 (1867)CrossRef
23.
Pulvirenti, M., Saffirio, C., Simonella, S.: On the validity of the Boltzmann equation for short range potentials. Rev. Math. Phys. 26(2), 64 (2014)
24.
Pulvirenti, M., Simonella, S.: The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error. arXiv:​1405.​4676
25.
Ruelle, D.: Statistical Mechanics: Rigorous Results. Imperial College Press, London (1999). Reprint of the 1989 edition. World Scientific Publishing Co. Inc., River Edge, NJ
26.
27.
Spohn, H.: Boltzmann equation and Boltzmann hierarchy. In: Cercignani, C. (ed.) Kinetic Theories and the Boltzmann Equation. Lecture Notes in Mathematics, vol. 1048, pp. 207–220. Springer, Berlin (1984)
28.
29.
Uchiyama, K.: Derivation of the Boltzmann equation from particle dynamics. Hiroshima Math. J. 18, 245–297 (1988)MathSciNetMATH
30.
31.
Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143(3), 273–307 (1998)MathSciNetCrossRefMATH
32.
Yvon, J.: La théorie statistique des fluides et l’équation d’état. Actualités scientifiques et industrielles. Hermann, Paris (1935)
Metadata
Title
Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond
Author
Chiara Saffirio
Copyright Year
2016
Book
From Particle Systems to Partial Differential Equations III

Print ISBN: 978-3-319-32142-4

Electronic ISBN: 978-3-319-32144-8

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-32144-8_15

Premium Partner

    Image Credits