Abstract
We consider the reactive Boltzmann equations for a mixture of different species of molecules, including a fixed background. We propose a scaling in which the collisions involving this background are predominant, while the inelastic (reactive) binary collisions are very rare. We show that, at the formal level, the solutions of the Boltzmann equations converge toward the solutions of a reaction-diffusion system. The coefficients of this system can be expressed in terms of the cross sections of the Boltzmann kernels. We discuss various possible physical settings (gases having internal energy, presence of a boundary, etc.), and present one rigorous mathematical proof in a simplified situation (for which the existence of strong solutions to the Boltzmann equation is known).
Similar content being viewed by others
References
M. Abramowitz and I. A. Stegun (Eds.). Handbook of Mathematical Functions (Dover, New York, 1965).
M. Bisi, M. Groppi, and G. Spiga, Grad's distribution functions in the kinetic equations for a chemical reaction. Contin. Mech. Thermodyn. 14:207–222 (2002).
C. Cercignani. The Boltzmann Equation and its Applications (Springer Verlag, New York, 1988).
S. Chapman, and T. G. Cowling. The Mathematical Theory of Non-uniform Gases (Cambridge University Press, Cambridge, 1970).
V. Giovangigli. Multicomponent Flow Modeling (Birkhäuser, Boston, 1999).
R. Dautray and J. L. Lions. Analyse mathématique et calcul numéerique pour les sciences et les techniques, Vol. 9, Evolution: numérique, transport (Paris, MASSON, 1988).
P. Degond and B. Lucquin–Desreux. Transport coefficients of plasmas and disparate mass binary gases.Transport Theory Statist. Phys. 25:595–633 (1996).
P. Degond and B. Lucquin–Desreux. The asymptotics of collision operators for two species of particles of disparate masses. Math. Models Methods Appl. Sci. 6:405–436 (1996).
A. De Masi, P. A. Ferrari and J. L. Lebowitz. Reaction–Diffusion equations for interacting particle systems. J. Stat. Phys. 44:589–644 (1986).
A. De Masi and E. Presutti. Mathematical methods for hydrodynamic limits (Springer–Verlag, Berlin, 1991).
L. Desvillettes and F. Golse. A remark concerning the Chapman–Enskog asymptotics. Ser. Adv. Math. Appl. Sci. 22:191–203 (1992).
L. Desvillettes, R. Monaco and F. Salvarani. A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. European Journal of Mechanics B/ Fluids 24:219–236 (2005).
T. Goudon and P. Lafitte. A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Preprint.
M. Groppi and G. Spiga. Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas. J. Math. Chem. 26:197–219 (1999).
A. Rossani and G. Spiga. A note on the kinetic theory of chemically reacting gases. Physica A 272:563–573 (1999).
F. Rothe. Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics (Springer, Berlin, 1984).
R. Spigler and D. H. Zanette. Reaction–diffusion models from the Fokker–Planck formulation of chemical processes. IMA J. Appl. Math. 49:217–229 (1992).
R. Spigler and D. H. Zanette. Asymptotic analysis and reaction–diffusion approximation for BGK kinetic models of chemical processes in multispecies gas mixtures. J. Appl. Math. Phys. (ZAMP) 44:812–827 (1993).
R. Spigler and D. H. Zanette. A BGK model for chemical processes: The reaction–diffusion approximation. Math. Mod. Meth. Appl. Sci. 4:35–47 (1994).
A. S. Sznitman. Propagation of Chaos for a System of Annihilating Brownian Spheres. Comm. Pure Appl. Math. 40:663–690 (1987).
D. H. Zanette. Linear and nonlinear diffusion and reaction-diffusion equations from discrete-velocity kinetic models. J. Phys. A: Math. Gen. 26:5339–5349 (1993).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bisi, M., Desvillettes, L. From Reactive Boltzmann Equations to Reaction–Diffusion Systems. J Stat Phys 124, 881–912 (2006). https://doi.org/10.1007/s10955-005-8075-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-005-8075-x