Abstract
A special model of a rarefied hard-sphere gas is considered. The hard-sphere particles undergo absolutely elastic collisions. It is assumed that particles can collide only if their nonzero velocities are orthogonal to each other. The model makes it possible to proceed from the Boltzmann equation to the Smoluchowski coagulation equation, where coagulation means that the kinetic energies of the colliding particles are added. A Monte Carlo scheme for simulation of the phenomenon is described, and the convergence of the simulation algorithm is proved. The convergence of numerical results to exact solutions of the Smoluchowski equation and to finite-difference solutions is tested.
Similar content being viewed by others
References
C. Cercignani, Theory and Application of the Boltzmann Equation (Elsevier, New York, 1975; Mir, Moscow, 1978).
M. V. Smoluchowski, “Versach einer mathematischen Theorie der Coagulationkinetik colloider Lösungen,” Z. Phys. Chem. 92, 129–168 (1917).
V. A. Galkin, The Smoluchowski Equation (Fizmatlit, Moscow, 2001) [in Russian].
V. M. Voloshchuk and Yu. S. Sedunov, Coagulation Processes in Disperse Systems (Gidrometeoizdat, Leningrad, 1975) [in Russian].
M. N. Kogan, Dynamics of Rarefied Gases (Nauka, Moscow, 1967) [in Russian].
V. A. Galkin, “Convergence of Finite-Difference Schemes and Direct Simulation Methods to Solutions of the Smoluchowski Equation in the Kinetic Theory of Coagulation,” Dokl. Akad. Nauk 397, 7–11 (2004) [Dokl. Math. 70, 491–494 (2004)].
Yu. A. Rozanov, Probability Theory, Stochastic Processes, and Mathematical Statistics (Nauka, Moscow, 1985) [in Russian].
I. R. Bagdasarova and V. A. Galkin, “Modeling of Coagulation in the Spatially Inhomogeneous Case,” Mat. Model. 11(6), 82–112 (1999).
V. A. Galkin and D. Yu. Osetskii, Proceedings of II International Conference “Chebyshev’s Mathematical Ideas and Their Applications to Modern Problems in the Natural Sciences” (Obninsk, 2004), pp. 58–61.
Author information
Authors and Affiliations
Additional information
Original Russian Text © V.A. Galkin, D.Yu. Osetskii, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 3, pp. 536–549.
Rights and permissions
About this article
Cite this article
Galkin, V.A., Osetskii, D.Y. Case of a Boltzmann gas leading to the Smoluchowski coagulation equation. Comput. Math. and Math. Phys. 46, 514–526 (2006). https://doi.org/10.1134/S0965542506030171
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0965542506030171