Probability Theory and Related Fields

, Volume 98, Issue 4, pp 489–515 | Cite as

Coexistence results for catalysts

  • Rick Durrett
  • Glen Swindle


In this paper we consider a modification of Ziff, Gulari and Barshad's (1986) model of oxidation of carbon monoxide on a catalyst surface in which the reactants are mobile on the catalyst surface. We find regions in the parameter space in which poisoning occurs (the catalyst surface becomes completely occupied by one type of atom) and another in which there is a translation invariant stationary distribution in which the two atoms have positive density. The last result is proved by exploiting a connection between the particle system with fast stirring and a limiting system of reaction diffusion equations.

Mathematics Subject Classification



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  1. Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math.30, 33–76 (1978)Google Scholar
  2. Bramson, M., Neuhauser, C.: A catalytic surface reaction model. J. Comput. Appl. Math.40, 157–161 (1992)Google Scholar
  3. Chuch, K.N., Conley, C.C., Smoller, J.A.: Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J.26, 373–393 (1977)Google Scholar
  4. De Masi, A., Ferrari, P.A., Lebowitz, J.L.: Reaction-diffusion equations for interacting particle systems. J. Stat. Phys.44, 589–644 (1986)Google Scholar
  5. Dickman, R.: Kinetic phase transitions in a surface-reaction model: mean field theory. Phys. Rev. A34, 4246–4250 (1986)Google Scholar
  6. Durrett, R.: Multicolor particle systems with large threshold and range. J. Theoret. Probab.5, 127–152 (1992)Google Scholar
  7. Durrett, R., Neuhauser C.: Particle systems and reaction-diffusion equations. Ann. Probab. (to appear, 1993)Google Scholar
  8. Engel, T., Ertl, G.: Elementary steps in the catalytic oxidation of carbon monoxide in platinum metals. Adv. Catal.28, 1–63 (1979)Google Scholar
  9. Feinberg, M., Terman, D.: Travelling composition waves on isothermal catalyst surfaces. (Preprint No. 1, 1991)Google Scholar
  10. Fife, P.C., Tang, M.M.: Comparison principles for reaction-diffusion systems. J. Differ. Equations40, 168–185 (1981)Google Scholar
  11. Gardner, R.A.: Existence and stability of travelling wave solutions of competition models: a degree theoretic approach. j. Differ. Equations44, 343–364 (1982)Google Scholar
  12. Grannan, E.R., Swindle, G.: Rigorous results on mathematical models of catalyst surfaces. J. Stat. Phys.61, 1085–1103 (1991)Google Scholar
  13. Griffeath, D.: Additive and cancellative interacting particle systems (Lect. Notes Math. vol. 724) Berlin Heidelberg New York: Springer 1979Google Scholar
  14. Harris, T.E.: Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. Math.9, 66–89 (1972)Google Scholar
  15. Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin Heidelberg New York: Springer 1987Google Scholar
  16. McDiarmid, C.: On the method of bounded differences. In: Siemons, J. (ed.) Surveys in Combinatorics. Cambridge: Cambridge University Press 1989Google Scholar
  17. Mountford, T.S., Sudbury, A.: An extension of a result of Swindle and Grannan on the poisoning of catalytic surfaces. (Preprint 1992)Google Scholar
  18. Quastel, J.: Diffusion of color in the simple exclusion process. Commun. Pure Appl. Math.45, 623–679 (1992)Google Scholar
  19. Volpert, V.A., Volpert, A.I.: Application of Leray-Schauder method to the proof of the existence of solutions to parabolic systems. Sov. Math. Dokl.37 (1), 138–141 (1988)Google Scholar
  20. Ziff, R.M., Gulari, E., Barshad, Y.: Kinetic phase transitions in an irreversible surface-reaction model. Phys. Rev. Lett.56, 2553–2556 (1986)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Rick Durrett
    • 1
  • Glen Swindle
    • 2
  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Department of Statistics and Applied ProbabilityUniversity of CaliforniaSanta BarbaraUSA

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