Skip to main content
SpringerLink
Log in

Gauge Poisson representations for birth/death master equations

  • Published:
The European Physical Journal B - Condensed Matter and Complex Systems Aims and scope Submit manuscript

Abstract.

Poisson representation techniques provide a powerful method for mapping master equations for birth/death processes -- found in many fields of physics, chemistry and biology -- into more tractable stochastic differential equations. However, the usual expansion is not exact in the presence of boundary terms, which commonly occur when the differential equations are nonlinear. In this paper, a gauge Poisson technique is introduced that eliminates boundary terms, to give an exact representation as a weighted rate equation with stochastic terms. These methods provide novel techniques for calculating and understanding the effects of number correlations in systems that have a master equation description. As examples, correlations induced by strong mutations in genetics, and the astrophysical problem of molecule formation on microscopic grain surfaces are analyzed. Exact analytic results are obtained that can be compared with numerical simulations, demonstrating that stochastic gauge techniques can give exact results where standard Poisson expansions are not able to.

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

Access this article

Log in via an institution

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. H. Haken, Synergetics: An Introduction, 2nd edn. (Springer, Berlin, Heidelberg, New York, 1978)

  2. N.G. Van Kampen, Stochastic processes in Physics and Chemistry (North-Holland, Amsterdam, 1992)

  3. C.W. Gardiner, S. Chaturvedi, J. Stat. Phys. 17, 429 (1977); C.W. Gardiner, S. Chaturvedi, J. Stat. Phys. 18, 501 (1978); C.W. Gardiner, Handbook of Stochastic Methods, 2nd edn. (Springer, Berlin, 1985)

    Google Scholar 

  4. S. Chaturvedi, P.D. Drummond, D.F. Walls, J. Phys. A 10, L187 (1977); P.D. Drummond, C.W. Gardiner, J. Phys. A 13, 2353 (1980)

    Google Scholar 

  5. A.M. Smith, C.W. Gardiner, Phys. Rev. A 39, 3511 (1989); R. Schack, A. Schenzle, Phys. Rev. A 44, 682 (1991)

    Article  Google Scholar 

  6. A. Gilchrist, C.W. Gardiner, P.D. Drummond, Phys. Rev. A 55, 3014 (1997)

    Article  Google Scholar 

  7. P. Deuar, P.D. Drummond, Phys. Rev. A 66, 033812 (2002)

    Article  Google Scholar 

  8. M. Eigen, Naturwissenschaften 58, 465 (1971); I. Hanski, Nature 396, 41 (1998); D. Alves, J.F. Fontanari, Phys. Rev. E 57, 7008 (1998); B. Drossel, Adv. Phys. 50, 209 (2001)

    Google Scholar 

  9. S.B. Charnley, Astrophys J. 509, L121 (1998); Astrophys. J. 562, L99 (2001); O. Biham, I. Furman, V. Pirronello, G. Vidali, Astrophys. J. 553, 595 (2001)

    Google Scholar 

  10. D.T. Gillespie, J. Comput. Phys. 22, 403 (1976); D.T. Gillespie, J. Chem. Phys. 81, 2340 (1977)

    Google Scholar 

  11. L.I. Plimak, M.K. Olsen, M.J. Collett, Phys. Rev. A 64, 025801 (2001)

    Article  Google Scholar 

  12. M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991)

  13. O. Deloubriere, L. Frachebourg, H.J. Hilhorst, K. Kitahara, Physica A 308, 135 (2002)

    Google Scholar 

  14. A. Ramani, B. Grammaticos, T. Bountis, Phys. Rep. 180, 159 (1989)

    Article  MathSciNet  Google Scholar 

  15. F. Baras, M. Malek Mansour, Phys. Rev. E 54, 6139 (1996); U.L. Fulco, D.N. Messias, M.L. Lyra, Phys. Rev. E 63, 066118 (2001)

    Article  Google Scholar 

  16. P.D. Drummond, I.K. Mortimer, J. Comp. Phys. 93, 144 (1991)

    MathSciNet  MATH  Google Scholar 

  17. G.R. Collecutt, P.D. Drummond, Comput. Phys. Commun. 142, 219 (2001); http://www.physics.uq.edu.au/xmds/

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Optik, Universität Erlangen-Nürnberg, Staudstrasse 7/B2, 91058, Erlangen, Germany

    P. D. Drummond

Authors
  1. P. D. Drummond
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. D. Drummond.

Additional information

Received: 12 February 2004, Published online: 8 June 2004

PACS:

05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) - 95.30.Ft Molecular and chemical processes and interactions - 87.23.Kg Dynamics of evolution

P.D. Drummond: Permanent address: Centre for Quantum-Atom Optics, University of Queensland, Brisbane, QLD 4072 Australia

Rights and permissions

Reprints and permissions

About this article

Cite this article

Drummond, P.D. Gauge Poisson representations for birth/death master equations. Eur. Phys. J. B 38, 617–634 (2004). https://doi.org/10.1140/epjb/e2004-00157-2

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2004-00157-2

Keywords

Search

Navigation