zbMATH — the first resource for mathematics

Coexistence results for catalysts. (English) Zbl 0794.60106
We consider a modification of R. M. Ziff, E. Gulari and Y. Barshad’s [Phys. Rev. Lett. 56, 2553-2556 (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 stiring and a limiting system of reaction- diffusion equations.
Reviewer: R.Durrett

60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI
[1] Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math.30, 33-76 (1978) · Zbl 0407.92014
[2] Bramson, M., Neuhauser, C.: A catalytic surface reaction model. J. Comput. Appl. Math.40, 157-161 (1992) · Zbl 0753.60065
[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) · Zbl 0368.35040
[4] De Masi, A., Ferrari, P.A., Lebowitz, J.L.: Reaction-diffusion equations for interacting particle systems. J. Stat. Phys.44, 589-644 (1986) · Zbl 0629.60107
[5] Dickman, R.: Kinetic phase transitions in a surface-reaction model: mean field theory. Phys. Rev. A34, 4246-4250 (1986)
[6] Durrett, R.: Multicolor particle systems with large threshold and range. J. Theoret. Probab.5, 127-152 (1992) · Zbl 0751.60095
[7] Durrett, R., Neuhauser C.: Particle systems and reaction-diffusion equations. Ann. Probab. (to appear, 1993) · Zbl 0799.60093
[8] Engel, T., Ertl, G.: Elementary steps in the catalytic oxidation of carbon monoxide in platinum metals. Adv. Catal.28, 1-63 (1979)
[9] Feinberg, M., Terman, D.: Travelling composition waves on isothermal catalyst surfaces. (Preprint No. 1, 1991) · Zbl 0744.35059
[10] Fife, P.C., Tang, M.M.: Comparison principles for reaction-diffusion systems. J. Differ. Equations40, 168-185 (1981) · Zbl 0458.35009
[11] Gardner, R.A.: Existence and stability of travelling wave solutions of competition models: a degree theoretic approach. j. Differ. Equations44, 343-364 (1982) · Zbl 0477.35013
[12] Grannan, E.R., Swindle, G.: Rigorous results on mathematical models of catalyst surfaces. J. Stat. Phys.61, 1085-1103 (1991) · Zbl 0741.60108
[13] Griffeath, D.: Additive and cancellative interacting particle systems (Lect. Notes Math. vol. 724) Berlin Heidelberg New York: Springer 1979 · Zbl 0412.60095
[14] Harris, T.E.: Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. Math.9, 66-89 (1972) · Zbl 0267.60107
[15] Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin Heidelberg New York: Springer 1987 · Zbl 0635.60021
[16] McDiarmid, C.: On the method of bounded differences. In: Siemons, J. (ed.) Surveys in Combinatorics. Cambridge: Cambridge University Press 1989 · Zbl 0712.05012
[17] Mountford, T.S., Sudbury, A.: An extension of a result of Swindle and Grannan on the poisoning of catalytic surfaces. (Preprint 1992) · Zbl 0925.82150
[18] Quastel, J.: Diffusion of color in the simple exclusion process. Commun. Pure Appl. Math.45, 623-679 (1992) · Zbl 0769.60097
[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)
[20] Ziff, R.M., Gulari, E., Barshad, Y.: Kinetic phase transitions in an irreversible surface-reaction model. Phys. Rev. Lett.56, 2553-2556 (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.