Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation
This paper introduces a new method to show the validity of a continuum description for the deterministic dynamics of many interacting particles. Here the many-particle evolution is analyzed for a hard sphere flow with the addition that after a collision the collided particles are removed from the system. We consider random initial configurations which are drawn from a Poisson point process with spatially homogeneous velocity density f 0(v). Assuming that the moments of order less than three of f 0 are finite and no mass is concentrated on lines, the homogeneous Boltzmann equation without gain term is derived for arbitrary long times in the Boltzmann–Grad scaling. A key element is a characterization of the many-particle flow by a hierarchy of trees which encode the possible collisions. The occurring trees are shown to have favorable properties with a high probability, allowing us to restrict the analysis to a finite number of interacting particles and enabling us to extract a single-body distribution. A counter-example is given for a concentrated initial density f 0 even to short-term validity.
KeywordsBoltzmann equation Boltzmann–Grad limit Validity Kinetic annihilation Deterministic dynamics Random initial data
Mathematics Subject Classification (2000)82C40 76P05 82C22 60K35
Unable to display preview. Download preview PDF.
- Coppex, F., Droz, M., Piasecki, J., Trizac, E., Wittwer, P.: Some exact results for Boltzmann’s annihilation dynamics. Phys. Rev. E 79, 21103 (2003) Google Scholar
- Droz, M., Frachebourg, L., Piasecki, J., Rey, P.-A.: Ballistic annihilation kinetics for a multivelocity one-dimensional ideal gas. Phys. Rev. E 51, 5541–5548 (1995) Google Scholar
- Durrett, R.: Probability: Theory and Examples, 3rd edn. Duxbury, N. Scituate (2004) Google Scholar
- Gallavotti, G.: Rigorous theory of Boltzmann equation in the Lorentz gas. Preprint Nota interna, vol. 358, Univ. di Roma (1970) Google Scholar