Skip to main content
Log in

On phase transitions in Schlögl's second model

  • Published:
Zeitschrift für Physik B Condensed Matter

Abstract

We study Schlögl's second model, characterized by chemical reactions

$$\begin{array}{*{20}c} {2X\underset{{k_2 }}{\overset{{k_1 }}{\longleftrightarrow}}3X,} & {X\underset{{k_4 }}{\overset{{k_3 }}{\longleftrightarrow}}0,} \\ \end{array} $$

ind-dimensional space. The reactions are assumed to be local; local fluctuations are fully taken into account, and particle transport occurs via diffusion.

In contrast to previous investigations, we find no phase transition whenk 4≠0 andd<4. Fork 4=0,k 3≠0, and 1≦d<4, we find a second-order phase transition which is in the same universality class as the transition in Schlögl's first model. Only ford≧4 we do find the first-order transition found also by previous authors.

These claims are supported by extensive Monte Carlo calculations for various realizations of this process on discrete space-time lattices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Haken, H.: Synergetics, 2nd edn. Berlin, Heidelberg, New York: Springer-Verlag 1979

    Google Scholar 

  2. Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems. New York: Wiley 1977

    Google Scholar 

  3. Schlögl, F.: Z. Phys.253, 147 (1972)

    Google Scholar 

  4. McNeil, K.J., Walls, D.F.: J. Stat. Phys.10, 439 (1974)

    Google Scholar 

  5. Janssen, H.K.: Z. Phys.270, 67 (1974)

    Google Scholar 

  6. Matheson, I., Walls, D.F., Gardiner, C.N.: J. Stat. Phys.12, 21 (1975)

    Google Scholar 

  7. Jähnig, F., Richter, P.H.: J. Chem. Phys.64, 4645 (1976)

    Google Scholar 

  8. Nicolis, G., Turner, J.W.: Physica89 A, 326 (1977)

    Google Scholar 

  9. Gardiner, C.W., McNeil, K.J., Walls, D.F., Matheson, I.S.: J. Stat. Phys.14, 307 (1976)

    Google Scholar 

  10. Grassberger, P., Scheunert, M.: Fortschr. Phys.28, 547 (1980)

    Google Scholar 

  11. Moshe, M.: Phys. Rep. C37, 255 (1978)

    Google Scholar 

  12. Grassberger, P., Sundermeyer, K.: Phys. Lett. B77, 220 (1978)

    Google Scholar 

  13. Broadbent, S.R., Hammersley, J.M.: Camb. Philos. Soc.53, 629 (1957)

    Google Scholar 

  14. Cardy, J.L., Sugar, R.L.: J. Phys. A13, L423 (1980)

    Google Scholar 

  15. Harris, T.E.: Ann. Prob.2, 969 (1974)

    Google Scholar 

  16. Griffeath, D.: Additive and cancellative interacting particle systems. Springer Lecture Notes in Mathematics. Vol.724. Berlin, Heidelberg, New York: Springer-Verlag 1979

    Google Scholar 

  17. Amati, D., Ciafaloni, M., Le Bellac, M., Marchesini, G.: Nucl. Phys. B112, 107 (1976);

    Google Scholar 

  18. Brower, R.C., Furman, M.A., Subbarao, K.: Phys. Rev. D15, 1756 (1977)

    Google Scholar 

  19. Grassberger, P., de la Torre, A.: Ann. Phys. (N.Y.)122, 373 (1979)

    Google Scholar 

  20. Abarbanel, H.D.I., Bronzan, J.B.: Phys. Rev. D9, 2397 (1974)

    Google Scholar 

  21. Brower, R.C., Furman, M.A., Moshe, M.: Phys. Lett.76B, 213 (1978)

    Google Scholar 

  22. Kinzel, W., Yeomans, J.M.: J. Phys. A14, L163 (1981)

    Google Scholar 

  23. Domany, E., Kinzel, W.: Weizmann Institute preprint (1981)

  24. Janssen, H.K.: Z. Phys. B — Condensed Matter42, 151 (1981)

    Google Scholar 

  25. Dewel, G., Walgraef, D., Borckmans, P.: Z. Phys. B — Condensed Matter28, 235 (1977); J. Stat. Phys.24, 119 (1981); Adv. Chem. Phys.49, 311 (1982)

    Google Scholar 

  26. Brachet, M.E., Tirapegui, E.: Phys. Lett.81 A, 211 (1981);

    Google Scholar 

  27. Langouche, F., Roekaerts, D., Tirapegui, E.: Leuven preprints KUL-TF-80/27 (1980) and-81/3 (1981)

  28. Malek-Mansour, M., Broeck, C. van den, Nicolis, G., Turner, J.W.: Univ. Libre des Bruxelles preprint (1980)

  29. Nicolis, G., Malek-Mansour, M.: J. Stat. Phys.22, 495 (1980)

    Google Scholar 

  30. Hanusse, P., Blanché, A.: In. Systems far from equilibrium. Sitges proceedings. Garrido, L. (ed.). Lecture Notes in Physics. Vol. 132, p. 335. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  31. Hanusse, P.: In: Nonlinear phenomena in chemical dynamics. Bordeaux proceedings. Vidal, C., Pacault, A. (eds.). Berlin, Heidelberg, New York: Springer 1981

    Google Scholar 

  32. Doi, M.: J. Phys. A9, 1465, 1479 (1976)

    Google Scholar 

  33. Nicolis, G., Turner, J.W.: In: Bifurcation theory and applications in scientific disciplines. Ann. NY Acad. Sci.316, 251 (1979)

  34. Malek-Mansour, M., Nicolis, G.: J. Stat. Phys.13, 197 (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Address from April 1-September 30, 1982: Chemical Physics Department, Weizmann Institute of Science, Rehovot, Israel

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grassberger, P. On phase transitions in Schlögl's second model. Z. Physik B - Condensed Matter 47, 365–374 (1982). https://doi.org/10.1007/BF01313803

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01313803

Keywords

Navigation