Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-20T15:33:19.999Z Has data issue: false hasContentIssue false
  • English
  • Français

FOURIER ANALYSIS AND THE KINETIC THEORY OF GASES

Part of: Harmonic analysis in one variable Limit theorems Infinite-dimensional Hamiltonian systems

Published online by Cambridge University Press:  16 December 2011

József Beck
József Beck*
Affiliation:
Mathematics Department, Rutgers University New Brunswick, New Jersey, U.S.A. (email: [email protected])
Get access

Abstract

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior force is acting on them, (4) there is no collision between the particles, (5) the collisions against the walls of the container are according to the law of elastic collision, we deduce from Newtonian mechanics two basic probabilistic limit laws about particle-counting. The first one is a global central limit theorem, and the second one is a local Poisson law. The key fact is that we can prove the “realistic limit”: first the time interval is fixed and the number of particles tends to infinity (where the density particle/volume remains a fixed constant), and then, in the second step, the time tends to infinity.

MSC classification

Secondary: 37K99: None of the above, but in this section 42A99: None of the above, but in this section 60F05: Central limit and other weak theorems
Type
Research Article
Information
Mathematika , Volume 58 , Issue 1 , January 2012 , pp. 93 - 208
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Beck, J., Deterministic approach to the kinetic theory of gases. J. Stat. Phys. 138 (2010), 160–269.CrossRefGoogle Scholar
[2]Chernov, N. and Makarian, R., Chaotic Billiards (Mathematical Surveys and Monographs 127), American Mathematical Society (Providence, RI, 2006).CrossRefGoogle Scholar
[3]Egerváry, E. and Turán, P., On a certain point of the kinetic theory of gases. Studia Math. 12 (1951), 170–180.CrossRefGoogle Scholar
[4]Feller, W., An Introduction to Probability Theory and its Applications, 2nd edn., Vol. 2, Wiley (New York, 1971).Google Scholar
[5]Feynman, R. P., Leighton, R. B. and Sands, M. L., The Feynman Lectures on Physics, Vol. 1, Addison-Wesley (Reading, MA, 1963).Google Scholar
[6]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 5th edn, Clarendon Press (Oxford, 1979).Google Scholar
[7]Hlawka, E., Mathematische Modelle der kinetischen Gastheorie (Edmund Hlawka Selecta) (eds Gruber, P. M. and Schmidt, W. M.), Springer (Berlin, 2000), 361–375, original paper published 1974.Google Scholar
[8]Kac, M., Statistical Independence in Probability, Analysis and Number Theory (The Carus Mathematical Monographs 12), The Mathematical Association of America (Washington, 1959).CrossRefGoogle Scholar
[9]Kac, M. and Steinhaus, H., Sur les fonctions indépendantes (IV). Studia Math. 7 (1938), 1–15.CrossRefGoogle Scholar
[10]Knuth, D. E., The Art of Computer Programming, 2nd edn., Vol. 1, Addison-Wesley (Reading, MA, 1973).Google Scholar
[11]König, D. and Szücs, A., Rend. Circ. Mat. Palermo 36 (1913), 79–90.CrossRefGoogle Scholar
[12]Salem, R. and Zygmund, A., Trigonometric series whose terms have random signs. Acta Math. 91 (1954), 245–301.CrossRefGoogle Scholar
[13]Sinai, Ya. G., Introduction to Ergodic Theory (Mathematical Notes), Princeton University Press (Princeton, NJ, 1976).Google Scholar
[14]Sinai, Ya. G. and Volkovsky, K. L., Ergodic properties of an ideal gas with infinitely many degrees of freedom. Funct. Anal. Appl. 5(4) (1971), 19–21.Google Scholar
[15]Steinhaus, H., Sur les fonctions indépendantes (VII). Studia Math. 10 (1948), 1–20.CrossRefGoogle Scholar
[16]Steinhaus, H., Sur les fonctions indépendantes (X) (Equipartition de molécules dans un récipient cubique). Studia Math. 13 (1953), 1–17.CrossRefGoogle Scholar
[17]Thompson, C. J., Mathematical Statistical Mechanics, Macmillan (London, 1972).Google Scholar
[18]Uhlenbeck, G. E. and Ford, G. W., Lectures in Statistical Mechanics, American Mathematical Society (Providence, RI, 1963).Google Scholar