Journal of Statistical Physics

, Volume 53, Issue 5–6, pp 1207–1215 | Cite as

On the problem of evaluating quasistationary distributions for open reaction schemes

  • P. K. Pollett
Articles
Received:
Revised:

Abstract

A number of recent papers have been concerned with the stochastic modeling of autocatalytic reactions. In some instances the birth and death model has been criticized for its apparent inadequacy in being able to describe the long-term behavior of the catalyst, in particular the fluctuations in the concentration of the catalyst about its macroscopically stable state. This criticism has been answered, to some extent, with the introduction of the notion of a quasistationary distribution; a number of authors have established the existence of limiting conditional distributions that can adequately describe these fluctuations. However, much of the work appears only to be appropriate for dealing with closed systems, for attention is usually restricted to finite-state birth and death processes. For open systems it is more appropriate to consider infinite-state processes and, from the point of view of establishing conditions for the existence of quasistationary distributions, extending the results for closed systems is far from straightforward. Here, simple conditions are given for the existence of quasistationary distributions for Markov processes with a denumerable infinity of states. These can be applied to any open autocatalytic system. The results also extend to explosive processes and to processes that terminate with probability less than 1.

Key words

Stochastic processes quasistationary distributions chemical kinetics 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Malek-Mansour and G. Nicolis,J. Stat. Phys. 13:197–217 (1975).Google Scholar
  2. 2.
    R. Gortz and D. F. Walls,Z. Physik B 25:423–427 (1976).Google Scholar
  3. 3.
    D. T. Gillespie,J. Phys. Chem. 81:2340–2361 (1977).Google Scholar
  4. 4.
    J. Keizer,J. Chem. Phys. 67:1473–1476 (1977).Google Scholar
  5. 5.
    J. W. Turner and M. Malek-Mansour,Physica 93A:517–525 (1978).Google Scholar
  6. 6.
    R. W. Parsons and P. K. Pollett,J. Stat. Phys. 46:249–254 (1987).Google Scholar
  7. 7.
    G. Nicolis,J. Stat. Phys. 6:195–222 (1972).Google Scholar
  8. 8.
    I. Oppenheim, K. E. Shuler, and G. H. Weiss,Physica 88A:191–214 (1977).Google Scholar
  9. 9.
    S. Dambrine and M. Moreau,Physica 106A:559–573 (1981).Google Scholar
  10. 10.
    S. Dambrine and M. Moreau,Physica 106A:574–588 (1981).Google Scholar
  11. 11.
    N. G. van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).Google Scholar
  12. 12.
    G. E. H. Reuter,Acta Math. 97:1–46 (1957).Google Scholar
  13. 13.
    J. N. Darroch and E. Seneta,J. Appl. Prob. 4:192–196 (1967).Google Scholar
  14. 14.
    P. K. Pollett, Quasistationary distributions and the Kolmogorov criterion, Research report, Murdoch University (1986).Google Scholar
  15. 15.
    P. K. Pollett,Stochastic Process. Appl. 22:203–221 (1986).Google Scholar
  16. 16.
    F. P. Kelly,Reversibility and Stochastic Networks (Wiley, London, 1979).Google Scholar
  17. 17.
    P. K. Pollett,Adv. Appl. Prob. 20 (1988), to appear.Google Scholar
  18. 18.
    J. F. C. Kingman,Proc. Lond. Math. Soc. (3) 13:337–358 (1963).Google Scholar
  19. 19.
    D. Vere-Jones,Aust. J. Stat. 11:67–78 (1969).Google Scholar
  20. 20.
    D. C. Flaspohler,Ann. Inst. Stat. Math. 26:351–356 (1974).Google Scholar
  21. 21.
    R. L. Tweedie,Q. J. Math. Oxford (2) 25:485–495 (1974).Google Scholar
  22. 22.
    B. Gaveau and L. S. Schulman,J. Phys. A 20:2865–2873 (1987).Google Scholar
  23. 23.
    J. Greensite and M. B. Halperin,Nucl. Phys. B 242:167–188 (1984).Google Scholar
  24. 24.
    R. J. McCraw and L. S. Schulman,J. Stat. Phys. 18:293–301 (1978).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • P. K. Pollett
    • 1
  1. 1.Department of MathematicsUniversity of QueenslandSt. LuciaAustralia

Personalised recommendations