Abstract
An approach to intermittent systems based on renewal processes is reviewed. The Waiting Times (WTs) between events are the main variables of interest in intermittent systems. A crucial role is played by the class of critical events, characterized by Non-Poisson statistics and non-exponential WT distribution. A particular important case is given by WT distributions with power tail. Critical events play a crucial role in the behavior of a property known as Renewal Aging. Focusing on the role of critical events, the relation between superstatistics and non-homogeneous Poisson processes is discussed, and the role of Renewal Aging is illustrated by comparing a Non-Poisson model with a Poisson one, both of them modulated by a periodic forcing. It is shown that the analysis of Renewal Aging is sensitive to the presence of critical events and that this property can be exploited to detect Non-Poisson statistics in a time series. As a consequence, it is claimed that, apart from the characterization of superstatistical features such as the distribution of the intensive parameter or the separation of the time scales, the Renewal Aging property can give some effort to better determine the role of Non-Poisson critical events in intermittent systems.
[1] C. Beck, Phys. Rev. Lett. 87, 180601 (2001) http://dx.doi.org/10.1103/PhysRevLett.87.18060110.1103/PhysRevLett.87.180601Search in Google Scholar
[2] C. Beck, Continuum Mech. Therm. 16, 293 (2004) http://dx.doi.org/10.1007/s00161-003-0145-110.1007/s00161-003-0145-1Search in Google Scholar
[3] E. G. D. Cohen, Physica D 193(1–4), 35 (2004) http://dx.doi.org/10.1016/j.physd.2004.01.00710.1016/j.physd.2004.01.007Search in Google Scholar
[4] C. Beck, Physica D 193(1–4), 195 (2004) http://dx.doi.org/10.1016/j.physd.2004.01.02010.1016/j.physd.2004.01.020Search in Google Scholar
[5] C. Beck, Europhys. Lett. 64(2), 151 (2003) http://dx.doi.org/10.1209/epl/i2003-00498-410.1209/epl/i2003-00498-4Search in Google Scholar
[6] S. Abe, C. Beck, E. G. D. Cohen, Phys. Rev. E 76, 031102 (2007) http://dx.doi.org/10.1103/PhysRevE.76.03110210.1103/PhysRevE.76.031102Search in Google Scholar PubMed
[7] T. Zhou, F. Moss, P. Jung, Phys. Rev. A 42, 3161 (1990) http://dx.doi.org/10.1103/PhysRevA.42.316110.1103/PhysRevA.42.3161Search in Google Scholar
[8] A. Longtin, A. Bulsara, F. Moss, Phys. Rev. Lett. 67, 656 (1991) http://dx.doi.org/10.1103/PhysRevLett.67.65610.1103/PhysRevLett.67.656Search in Google Scholar PubMed
[9] E. Barkai, Y. Jung and R.J. Silbey, Annu. Rev. Phys. Chem. 55, 457 (2004) http://dx.doi.org/10.1146/annurev.physchem.55.111803.14324610.1146/annurev.physchem.55.111803.143246Search in Google Scholar PubMed
[10] G. Baldini, F. Cannone, G. Chirico, Science 309, 1096 (2005) http://dx.doi.org/10.1126/science.111500110.1126/science.1115001Search in Google Scholar PubMed
[11] S. Bianco, P. Grigolini, P. Paradisi, Chem. Phys. Lett. 438, 336 (2007) http://dx.doi.org/10.1016/j.cplett.2007.03.01310.1016/j.cplett.2007.03.013Search in Google Scholar
[12] M. Kuno, D. P. Fromm, H.F. Hamann, A. Gallagher, D. J. Nesbitt, J. Chem. Phys. 112, 3117 (2000) http://dx.doi.org/10.1063/1.48089610.1063/1.480896Search in Google Scholar
[13] P. Paradisi, P. Allegrini, F. Barbi, S. Bianco, P. Grigolini, AIP Conf. Proc. 800, 92 (2005) http://dx.doi.org/10.1063/1.213859910.1063/1.2138599Search in Google Scholar
[14] S. Bianco, P. Grigolini, P. Paradisi, J. Chem. Phys. 123, 174704 (2005) http://dx.doi.org/10.1063/1.210290310.1063/1.2102903Search in Google Scholar PubMed
[15] D. R. Cox, Renewal Theory (Chapman and Hall, New York, 1967) Search in Google Scholar
[16] P. Allegrini, F. Barbi, P. Grigolini, P. Paradisi, Phys. Rev. E 73, 046136 (2006) http://dx.doi.org/10.1103/PhysRevE.73.04613610.1103/PhysRevE.73.046136Search in Google Scholar PubMed
[17] P. Allegrini, F. Barbi, P. Grigolini, P. Paradisi, Chaos Soliton. Fract. 34, 11 (2007) http://dx.doi.org/10.1016/j.chaos.2007.01.04510.1016/j.chaos.2007.01.045Search in Google Scholar
[18] L. Devroye, Non-uniform random variate generation (Springer-Verlag, New York, 1986) 10.1007/978-1-4613-8643-8Search in Google Scholar
[19] P. Paradisi, P. Grigolini, S. Bianco, O. C. Akin, Int. J. Bifurcat. Chaos 18, 2681 (2008) http://dx.doi.org/10.1142/S021812740802189010.1142/S0218127408021890Search in Google Scholar
[20] P. Paradisi, R. Cesari, D. Contini, A. Donateo, L. Palatella, Eur. Phys. J.-Special Topics (in press) Search in Google Scholar
[21] C. Godréche, J. M. Luck, J. Stat. Phys. 104, 489 (2001) http://dx.doi.org/10.1023/A:101036400325010.1023/A:1010364003250Search in Google Scholar
[22] P. Allegrini, G. Aquino, P. Grigolini, L. Palatella, A. Rosa, Phys. Rev. E 68, 056123 (2003) http://dx.doi.org/10.1103/PhysRevE.68.05612310.1103/PhysRevE.68.056123Search in Google Scholar PubMed
[23] G. Aquino, M. Bologna, P. Grigolini, B. J. West, Phys. Rev. E 70, 036105 (2004) http://dx.doi.org/10.1103/PhysRevE.70.03610510.1103/PhysRevE.70.036105Search in Google Scholar PubMed
[24] O. C. Akin, P. Paradisi, P. Grigolini, Physica A, 371, 157 (2006) http://dx.doi.org/10.1016/j.physa.2006.04.05410.1016/j.physa.2006.04.054Search in Google Scholar
[25] L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998) http://dx.doi.org/10.1103/RevModPhys.70.22310.1103/RevModPhys.70.223Search in Google Scholar
[26] B. Mc Namara, K. Wiesenfeld, Phys. Rev. A 39, 4854 (1989) http://dx.doi.org/10.1103/PhysRevA.39.485410.1103/PhysRevA.39.4854Search in Google Scholar PubMed
[27] O. C. Akin, P. Paradisi, P. Grigolini, J. Stat. Mech.-Theory E., 01013 (2009) 10.1088/1742-5468/2009/01/P01013Search in Google Scholar
[28] C. Beck, E. G. D. Cohen, H. L. Swinney, Phys. Rev. E 72, 056133 (2005) http://dx.doi.org/10.1103/PhysRevE.72.05613310.1103/PhysRevE.72.056133Search in Google Scholar PubMed
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