Journal of Statistical Physics

, Volume 27, Issue 4, pp 677–691 | Cite as

Nonunique solutions of kinetic equations

  • M. H. Ernst
  • K. Hellesoe
  • E. H. Hauge


Two very hard particle models are solved and the nonuniqueness of the initial value problem for these (model) kinetic equations is explicitly demonstrated, when distribution functions decaying sufficiently slowly are permitted. The intimate connection between nonuniqueness and violation of conservation laws is made evident. The associated eigenvalue problems are solved. Finally, the general implications of these results for kinetic equations with transition rates that are increasing functions of the state variable, are stated in the form of a number of conjectures. They affect the solution of the Boltzmann equation for realistic intermolecular interactions when the collision rategI(g, χ) is an increasing function of the relative velocityg.

Key words

Linear and nonlinear Boltzmann equation initial value problem in kinetic theory violation of mass or energy conservation high energy tails of distribution functions eigenfunctions with positive and negative eigenvalues 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Aizenman and T. A. Bak,Commun. Math. Phys. 65:203 (1979).Google Scholar
  2. 2.
    H. Cornille and A. Gervois,J. Phys. (Paris) Lett. 41:L581 (1980).Google Scholar
  3. 3.
    J. Piasecki and Y. Pomeau,J. Stat. Phys. (1982).Google Scholar
  4. 4.
    S. Rouse and S. Simons,J. Phys. A 11:423 (1978).Google Scholar
  5. 5.
    M. H. Ernst and E. M. Hendriks,Phys. Lett. 70A:183 (1979).Google Scholar
  6. 6.
    M. H. Ernst,Phys. Rep.,78:1 (1981).Google Scholar
  7. 7.
    C. Cercignani,Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh, 1975).Google Scholar
  8. 8.
    E. H. Hauge and E. Praestgaard,J. Stat. Phys. 24:21 (1981).Google Scholar

Copyright information

© Plenum Publishing Corporation 1982

Authors and Affiliations

  • M. H. Ernst
    • 1
  • K. Hellesoe
    • 2
  • E. H. Hauge
    • 2
  1. 1.Instituut voor Theoretische FysicaRijksuniversiteit UtrechtTA UtrechtThe Netherlands
  2. 2.Institutt for Theoretisk FysikkNorges Tekniske HogskoleTrondheim-NTHNorway

Personalised recommendations