Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part II. Numerical methods

  • 46 Accesses

  • 15 Citations


The scaling exponent and the scaling function for the 1D single-species coagulation model (A+A→A) are shown to be universal, i.e., they are not influenced by the value of the coagulation rate. They are independent of the initial conditions as well. Two different numerical methods are used to compute the scaling properties of the concentration: Monte Carlo simulations and extrapolations of exact finite-lattice data. These methods are tested in a case where analytical results are available. To obtain reliable results from finite-size extrapolations, numerical data for lattices up to ten sites are sufficient.

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


  1. 1.

    P. Argyrakis,Computers Phys. 6:525 (1992).

  2. 2.

    D. ben-Avraham,J. Chem. Phys. 88:941 (1988).

  3. 3.

    P. Argyrakis and R. Kopelman,Phys. Rev. A 41:2114 (1990).

  4. 4.

    K. Krebs, M. P. Pfannmüller, B. Wehefritz, and H. Hinrichsen, Finite-size scaling studies of one-dimensional reaction-diffusion systems, Part I: Analytical results,J. Stat. Phys., this issue.

  5. 5.

    R. Kopelman,J. Stat. Phys. 42:185 (1986).

  6. 6.

    R. Kopelman, S. J. Parus, and J. Prasad,Chem. Phys. 128:209 (1988).

  7. 7.

    R. Kroon, H. Fleurent, and R. Sprik,Phys. Rev. E 47:2462 (1993).

  8. 8.

    R. Kopelman, C. S. Li, and Z.-Y. Shi,J. Lumin. 45:40 (1990).

  9. 9.

    F. C. Alcaraz, M. Droz, M. Henkel, and V. Rittenberg,Ann. Phys. 230:250 (1994).

  10. 10.

    F. C. Alcaraz, M. Henkel, and V. Rittenberg, Unpublished.

  11. 11.

    V. Privman and M. D. Grynberg,J. Phys. A: Math. Gen. 25:6567 (1992).

  12. 12.

    M. Hoyuelos and H. O. Mártin,Phys. Rev. E,48:3309 (1993).

  13. 13.

    D. Zhong and D. ben-Abraham, Diffusion-limited coalescence with finite reaction rates in one dimension, Preprint cond-mat/9407047, SISSA-94-02 (1994).

  14. 14.

    V. Privman, C. R. Doering, and H. L. Frisch,Phys. Rev. E 48:846 (1993).

  15. 15.

    H. Taitelbaum, R. Kopelman, G. H. Weiss, and S. Halvin,Phys. Rev. A 41:3116 (1990).

  16. 16.

    H. Taitelbaum, S. Halvin, and G. H. Weiss,Chem. Phys. 146:351 (1990).

  17. 17.

    L. Braunstein, H. O. Mártin, M. D. Grynberg, and H. E. Roman,J. Phys. A: Math. Gen. 25:L255 (1992).

  18. 18.

    Z.-Y. Shi and R. Kopelman,Chem. Phys. 167:149 (1992).

  19. 19.

    C. R. Doering and D. ben-Avraham,Phys. Rev. A 38:3035 (1988).

  20. 20.

    R. Bulirsch and J. Stoer,Numer. Math. 6:413 (1964); P. Christe and M. Henkel, InIntroduction to Conformal Invariance and Its Application to Critical Phenomena (Springer, Berlin, 1993), Chapter 3.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Krebs, K., Pfannmüller, M.P., Simon, H. et al. Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part II. Numerical methods. J Stat Phys 78, 1471–1491 (1995). https://doi.org/10.1007/BF02180139

Download citation

Key Words

  • Reaction-diffusion systems
  • finite-size scaling
  • Monte Carlo simulations
  • nonequilibrium statistical mechanics
  • coagulation model