Abstract
Two species of Brownian particles on the unit circle are considered; both have diffusion coefficient σ>0 but different velocities (drift), 1 for one species and −1 for the other. During the evolution the particles randomly change their velocity: if two particles have the same velocity and are at distance ⩽ε (ε being a positive parameter), they both may simultaneously flip their velocity according to a Poisson process of a given intensity. The analogue of the Boltzmann-Grad limit is studied when ε goes to zero and the total number of particles increases like ε−1. In such a limit propagation of chaos and convergence to a limiting kinetic equation are proven globally in time, under suitable assumptions on the initial state. If, furthermore, σ depends on ε and suitably vanishes when ε goes to zero, then the limiting kinetic equation (for the density of the two species of particles) is the Carleman equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
O. Lanford, Time evolution of large classical systems, inLectures Notes in Physics, Vol. 38, in E. J. Moser, ed. (Springer, 1975), pp. 1–111.
R. Illner and M. Pulvirenti,Commun. Math. Phys. 105:189–203 (1986); Global validity of the Boltzmann equation for a two and three-dimensional rare gas in vacuum: Erratum and improved results,Commun. Math. Phys. (1988).
R. Caflisch,Commun. Pure Appl. Math. 33:645–666 (1980); M. Lachowitz,Math. Meth. Appl. Sci. 9(3):342–366.
E. Presutti, inProceedings of the 1st World Congress of the Bernoulli Society (VNU Science Press, 1987), Vol. 1, pp. 395–413.
T. G. Kurtz,Trans. Am. Math. Soc. 186:259–272 (1973).
H. P. McKean,Israel J. Math. 21:54–92 (1975).
K. Uchiyama, On the Boltzmann-Grad limit for the Broadwell model of the Boltzmann equation,J. Stat. Phys. 52:331 (1988).
R. Illner,Math. Meth. Appl. Sci. 1:187–193 (1979).
Author information
Authors and Affiliations
Additional information
Dedicated to the memory of Paola Calderoni.
Rights and permissions
About this article
Cite this article
Caprino, S., De Masi, A., Presutti, E. et al. A stochastic particle system modeling the Carleman equation. J Stat Phys 55, 625–638 (1989). https://doi.org/10.1007/BF01041601
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01041601