Stochastic storage models and noise-induced phase transitions

  • S. ShpyrkoEmail author
  • V. V. Ryazanov
Statistical and Nonlinear Physics


The most frequently used in physical application diffusive (based on the Fokker-Planck equation) model leans upon the assumption of small jumps of a macroscopic variable for each given realization of the stochastic process. This imposes restrictions on the description of the phase transition problem where the system is to overcome some finite potential barrier, or systems with finite size where the fluctuations are comparable with the size of a system. We suggest a complementary stochastic description of physical systems based on the mathematical stochastic storage model with basic notions of random input and output into a system. It reproduces statistical distributions typical for noise-induced phase transitions (e.g. Verhulst model) for the simplest (up to linear) forms of the escape function. We consider a generalization of the stochastic model based on the series development of the kinetic potential. On the contrast to Gaussian processes in which the development in series over a small parameter characterizing the jump value is assumed [R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics, Springer Series in Synergetics (Springer Verlag, 1994), Vol. 59], we propose a series expansion directly suitable for storage models and introduce the kinetic potential generalizing them.


05.70.Ln Nonequilibrium and irreversible thermodynamics 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)  


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. W. Horsthemke, R. Lefever, Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology (Springer Verlag, Berlin, 1984) Google Scholar
  2. N.U. Prabhu, Stochastic storage processes (Springer Verlag, Berlin, 1980) Google Scholar
  3. P.J. Brockwell, S.I. Resnick, R.L. Tweedie, Adv. Appl. Prob. 14, 392 (1982) CrossRefMathSciNetzbMATHGoogle Scholar
  4. H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1984) Google Scholar
  5. V.V. Ryazanov, in Aerosols: Sciense, Indystry, Health and Environment, 1 band (Pergamon Press, Kyoto, 1990), p. 142; V.V. Ryazanov, S.G. Shpyrko, Journal of Aerosol Science 28, 647 (1997); V.V. Ryazanov, S.G. Shpyrko, Journal of Aerosol Science 28, 624 (1997) CrossRefGoogle Scholar
  6. V.V. Ryazanov, S.G. Shpyrko, in Proc. Int. Conf. on Probabilistic Safety Assessment Methodology and Applications. PSA'95, Seoul, 1995 (Atomic Energy Research Institute, Seoul, Korea, 1995), Vol. 1, p. 121 Google Scholar
  7. V.V. Ryazanov, Ukr. Phys. J. 38, 615 (1993) MathSciNetGoogle Scholar
  8. V.V. Ryazanov, S. Shpyrko, Cond. Matt. Phys. 9, 71 (2006) Google Scholar
  9. A.V. Chechkin et al., Europhys. Lett. 72, 348 (2005); A.V. Chechkin et al., J. Phys. A: Math. Gen. 36, L537 (2003) CrossRefADSMathSciNetGoogle Scholar
  10. W. Feller, An Introduction to Probability Theory and its Applications (John Wiley & Sons, Inc., New York, 1971), Vol. 2 Google Scholar
  11. N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland Personal Library, 1992) Google Scholar
  12. C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, 3rd edn. (Springer Verlag, Berlin, 2004), Vol. 13 Google Scholar
  13. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000) CrossRefADSMathSciNetzbMATHGoogle Scholar
  14. A.V. Chechkin, V.Yu. Gonchar, J. Klafter, R. Metzler, Phys. Rev. E 72, 010101 (R) (2005) CrossRefADSMathSciNetGoogle Scholar
  15. A.V. Chechkin et al., Phys. Rev. E 67, 010102(R) (2003) CrossRefGoogle Scholar
  16. A.V. Chechkin, R. Gorenflo, I.M. Sokolov, Phys. Rev. E 66, 046129 (2002) CrossRefADSGoogle Scholar
  17. I.M. Sokolov, R. Metzler, Phys. Rev. E 67, 010101(R) (2003) CrossRefADSGoogle Scholar
  18. V.V. Uchaikin, Uspekhi fizicheskih nauk 173, 847 (2003) [Physics - Uspekhi 46, 821 (2003)] CrossRefGoogle Scholar
  19. V.M. Zolotarev, One-Dimensional Stable Distributions, AMS, Providence, RI, 1986 Google Scholar
  20. S. Jespersen, R. Metzler, H.S. Fogedby, Phys. Rev. E 59, 2736 (1999) CrossRefADSGoogle Scholar
  21. M.-O. Hongler, R. Filliger, P. Blanchard, Physica A 370, 301 (2006) CrossRefADSGoogle Scholar
  22. R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics, Springer Series in Synergetics (Springer Verlag, Berlin, 1994), Vol. 59 Google Scholar
  23. P.F. Zweifel, Reactor Physics (McGraw-Hill, New York, 1973) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute for Nuclear ResearchKievUkraine
  2. 2.Dept. of Optics, Palacký UniversityOlomoucCzech Republic

Personalised recommendations