Summary
Reaction-diffusion processes were introduced by Nicolis and Prigogine, and Haken. Existence theorems have been established for most models, but not much is known about ergodic properties. In this paper we study a class of models which have a reversible measure. We show that the stationary distribution is unique and is the limit starting from any initial distribution.
Article PDF
Similar content being viewed by others
References
Boldrighini, C., DeMasi, A,, Pellegrinotti, A.: Non-equilibrium fluctuations in particle systems modelling diffusion-reaction systems (preprint 1989)
Boldrighini, C., DeMasi, A., Pellegrinotti, A., Presutti, E.: Collective phenomena in interacting particle systems. Stoch. Proc. Appl.25, 137–152 (1987)
Chen, M.F., Infinite dimensional reaction diffusion processes. Acta Math. Sin. New Ser.1, 261–273 (1985)
Chen, M.F.: Coupling for jump processes. Acta Math. Sin., New Ser.2, 123–126 (1986a)
Chen, M.F.: Jump processes and particle systems. (In Chinese) Beijing Normal Univ. Press (1986b)
Chen, M.F.: Existence theorems for interacting particle systems with non-compact state space. Sci. Sin., Ser. A,30, 148–156 (1987)
Chen, M.F.: Stationary distributions for infinite particle systems with noncompact state space. Acta Math. Sci.9, 9–19 (1989)
Dewel, G., Borckmans, P., Walgraef, D.: Nonequilibrium phase transitions and chemical instabilities. J. Stat. Phys.24, 119–137 (1981)
Dewel, G., Walgraef, D., Borckmans, P.: Renormalization group approach to chemical instabilities. Z. Phys. B28, 235–237 (1977)
Ding, W.D. and Zheng, X.G.: Ergodic theorems for linear growth processes with diffusion (preprint 1987)
Feng, S. and Zheng, X.G.: Solutions of a class of nonlinear master equations. Carleton University Preprint no. 115 (1988)
Feistel, R.: Nonlinear chemical reactions in diluted solutions In: Ebeling, W., Ulbricht, H. (eds.) Selforganization by nonlinear irreversible processes Proceedings, Mühlungsborn 1985. (Springer Ser. Synergetics, vol. 33) Berlin Heidelberg New York: Springer 1985
Grassberger, P.: On phase transitions in Schlögl's second model. Z. Phys. B58, 229–244 (1982)
Grassberger, P., Torre, A. de la: Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behavior. Ann. Phys.122, 373–396 (1979)
Haken, H.: Synergetics. Berlin Heidelberg New York: Springer 1977
Hanuse, P.: Fluctuations in non-equilibrium phase transitions: critical behavior. In: Vidal, C., Pacault, A. (eds.) Nonlinear phenomena in chemical dynamics. Proceedings, Bordeaux 1981 (Springer Ser. Synergetics, vol. 12) Berlin Heidelberg New York: Springer 1981
Hanuse, P., Blanché, A.: Simulation study of the critical behavior of a chemical model system. In: Garrido, L. (ed.) Systems far from equilibrium. Conference, Barcelona 1980. (Lect. Notes Phys., vol. 132, pp. 337–344) Berlin Heidelberg New York: Springer 1980
Holley, R.: Free energy in a Markovian model of a lattice spin system. Commun. Math. Phys.23, 87–99 (1971)
Holley, R.: An ergodic theorem for interacting systems with attractive interactions. Z. Wahrscheinlichkeitstheor. Verw. Geb.24, 325–334 (1972)
Holley, R., Stroock, D.: A martingale approach to infinite systems of interacting particles. Ann. Probab.4, 195–228 (1976)
Holley, R., Stroock, D.: In one and two dimensions every stationary measure for a stochastic Ising model is a Gibbs state. Commun. Math. Phys.55, 37–45 (1977)
Janssen, H.K.: Stochastisches Reaktionsmodell für einen Nichtgleichgewichts-Phasenübergang. Z. Phys.270, 57–73 (1974)
Janssen, H.K.: On the nonequilibrium phase transition in reaction diffusion systems with an absorbing stationary state. Z. Phys. B42, 151–154 (1981)
Liggett, T.M.: An infinite particle systems with zero range interactions. Ann. Probab.1, 240–253 (1973)
Liggett, T.M.: Interacting Particle Systems. Berlin Heidelberg New York: Springer 1985
Liggett, T.M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 443–468 (1981)
Mountford, T.: The ergodicity of a class of reaction diffusion processes (preprint 1989)
Neuhauser, C.: Untersuchung des Einflusses von Wanderung auf nichtlineare Dynamiken bein Ising-Modell und bein Schlöglmodell. Diplomarbeit thesis, Heidelberg (1988)
Neuhauser, C.: An ergodic theorem for Schlögl models with small migration. Probab. Th. Rel. Fields85, 27–32 (1990)
Nicolis, G., Priogogine, I.: Self-organization in nonequilibrium systems. New York: Wiley 1977
Ohtsuki, T., Keyes, T.: Nonequilibrium critical phenomena in one component reaction diffusion systems. Phys. Rev. A35, 2697–2703 (1987)
Schlögl, F.: Chemical reaction models and non-equilibrium phase transitions. Z. Phys.253, 147–161 (1972)
Shiga, T.: Stepping stone models in population genetics and population dynamics. In: Albeverio, S., et al. (eds.) Stochastic processes in physics and engineering. pp. 345–355. Dordrecht: Reidel 1988
Zheng, X.G., Ding, W.D.: Existence theorems for linear growth processes with diffusion. Acta Math. Sci.7, 25–42 (1987).
Author information
Authors and Affiliations
Additional information
The work was begun while the first author was visiting Cornell and supported by the Chinese government. The initial results (for Schlögl's first model) was generalized while the three authors were visiting the Nankai Institute for Mathematics, Tianjin, People's Republic of China
Partially supported by the National Science Foundation and the Army Research Office through the Mathematical Sciences Institute at Cornell University
Partially supported by NSF grant DMS 86-01800
Rights and permissions
About this article
Cite this article
Ding, WD., Durrett, R. & Liggett, T.M. Ergodicity of reversible reaction diffusion processes. Probab. Th. Rel. Fields 85, 13–26 (1990). https://doi.org/10.1007/BF01377624
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01377624