Journal of Statistical Physics

, Volume 177, Issue 4, pp 608–625 | Cite as

Intermittent Waiting-Time Noises Through Embedding Processes

  • Isaias McHardy
  • Marco Nizama
  • Adrian A. Budini
  • Manuel O. CáceresEmail author


An intermittent two-state noise can be modelled through a renewal process characterized by two different time scales. A (four state) Markovian embedding of this non-Markovian process is presented. The equivalence between the renewal approach and the enlarged master equation is shown. Analytical results for n-time moments of the intermittent dichotomic noise are obtained. The Monte Carlo simulations supports our analytical results. The advantage of using the enlarged master equation for calculating higher order moments is established.


Intermittent noise Waiting-time problem Non-Markov Dichotomous noise 



Funding was provided by CONICET (Grant No. PIP 112-201501-00216, CO).


  1. 1.
    Cox, D.R., Isham, V.: Point Process. Chapman and Hall, New York (1980)zbMATHGoogle Scholar
  2. 2.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971). Ch. XI, Sect. 4zbMATHGoogle Scholar
  3. 3.
    Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes, 2nd edn. Acadepic Press, New York (1975)zbMATHGoogle Scholar
  4. 4.
    Sanford, J., Brewer, W., Smith, F., Baumgardner, J.: The waiting time problem in a model hominin population. Theor. Biol. Med. Model. 12, 18 (2015). CrossRefGoogle Scholar
  5. 5.
    Montroll, E.W., Weiss, G.H.: Random walks on lattices. II. J. Math. Phys. 6, 167 (1965)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Scher, H., Lax, M.: Stochastic transport in a disordered solid. I. Theory Phys. Rev. B 7, 4491 (1973)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Lax, M., Scher, H.: Renewal theory and AC conductivity in random structures. Phys. Rev. Lett. 39, 781 (1977)ADSCrossRefGoogle Scholar
  8. 8.
    Prato, D.P., Pury, P.A.: The waiting time problem. Physica A 157, 1261 (1989)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Godreche, C., Luck, J.M.: Statistics of the occupation time of renewal process. J. Stat. Phys. 104, 489 (2001)CrossRefGoogle Scholar
  10. 10.
    Caceres, M.O.: Anomalous hydrodynamical dispersion and the transport with multiple families of paths in porous media. Phys. Rev. E 69, 036302 (2004)ADSCrossRefGoogle Scholar
  11. 11.
    Caceres, M.O., Insua, G.L.: Passage time of asymmetric anomalous walks with multiple paths. J. Phys. A: Math. Gen. 38, 3711 (2005)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Budini, A.A., Grigolini, P.: Non-Markovian nonstationary completely positive open-quantum-system dynamics. Phys. Rev. A 80, 022103 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    Garcia, O.E.: Stochastic modeling of intermittent scrape-off Layer plasma fluctuations. Phys. Rev. Lett. 108, 265001 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Budini, A.A.: Non-poissonian intermittent fluorescence from complex structured environments. Phys. Rev. A 73, 061802R (2006). 978-3-319-51552-6. Springer (2017)Google Scholar
  15. 15.
    van Kampen, N.G.: Stochastic differential equations. Phys. Rep. 24 C, 171 (1976)ADSCrossRefGoogle Scholar
  16. 16.
    Horsthemke, W., Lefever, R.: Noise-Induced Transitions. Springer, Berlin (1984). and references thereinzbMATHGoogle Scholar
  17. 17.
    Cáceres, M.O.: Non-equilibrium Statistical Physics with Application to Disordered Systems. Springer, Berlin (2017)CrossRefGoogle Scholar
  18. 18.
    Carbone, V., et al.: Intermittency and turbulence in a magnetically confined fusion plasma. Phys. Rev. E 62, R49 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    García, O.E., Horacek, J., Pitts, R.A.: Intermittent fluctuations in the TCV scrape-off layer. Nucl. Fusion 55, 062002 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    Terwiel, R.H.: Projection operator method applied to stochastic linear differential equations. Physica A 74, 248 (1974)MathSciNetGoogle Scholar
  21. 21.
    Bourret, R.C., Frisch, U., Pouquet, A.: Brownian motion of harmonic oscillator with stochastic frequency. Physica A 65, 303 (1973)Google Scholar
  22. 22.
    Cáceres, M.O.: Computing a non-Maxwellian velocity distribution from first principles. Phys. Rev. E 67, 016102 (2003)ADSCrossRefGoogle Scholar
  23. 23.
    Masoliver, J.: Second-order precesses driven by dichotomic noise. Phys. Rev. A 45, 706 (1992)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.CONICET, Centro Atómico BarilocheBarilocheArgentina
  2. 2.Departamento de FísicaUniversidad Nacional del Comahue and CONICETNeuquénArgentina
  3. 3.Universidad Tecnológica Nacional (UTN-FRBA)BarilocheArgentina
  4. 4.Centro Atómico Bariloche, CNEA, and Instituto Balseiro and CONICETBarilocheArgentina

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