Geodesic
Renewal Theory for a System with Internal States
M. Niemann1, 2 ; E. Barkai3 ; H. Kantz2
1 Institut für Physik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
2 Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
3 Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 191-239.

Voir la notice de l'article provenant de la source EDP Sciences

We investigate a stochastic signal described by a renewal process for a system with N states. Each state has an associated joint distribution for the signal’s intensity and its holding time. We calculate multi-point distributions, correlation functions, and the power-spectrum of the signal. Focusing on fat tailed power-law distributed sojourn times in the states of the system, we investigate 1/f noise in this widely applicable model. When the mean waiting time is infinite, the averaged sample spectrum depends both on the age of the process, i.e. the time elapsing from start of the process and the start of observation, and on the total time of observation. Fluctuations of the periodogram estimator of the power-spectrum are investigated for aged systems and are found to be determined by the distribution of the number of renewals in the observation time window. These reduce to the Mittag-Leffler distribution when the start of observation is also the start of the process. When the average waiting time is finite we find a time independent Wienerian spectrum computed from the stationary correlation function of the signal.
DOI : 10.1051/mmnp/201611312

M. Niemann 1, 2 ; E. Barkai 3 ; H. Kantz 2

1 Institut für Physik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
2 Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
3 Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel 
@article{MMNP_2016_11_3_a11,
     author = {M. Niemann and E. Barkai and H. Kantz},
     title = {Renewal {Theory} for a {System} with {Internal} {States}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {191--239},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2016},
     doi = {10.1051/mmnp/201611312},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611312/}
}
TY  - JOUR
AU  - M. Niemann
AU  - E. Barkai
AU  - H. Kantz
TI  - Renewal Theory for a System with Internal States
JO  - Mathematical modelling of natural phenomena
PY  - 2016
SP  - 191
EP  - 239
VL  - 11
IS  - 3
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611312/
DO  - 10.1051/mmnp/201611312
LA  - en
ID  - MMNP_2016_11_3_a11
ER  - 
%0 Journal Article
%A M. Niemann
%A E. Barkai
%A H. Kantz
%T Renewal Theory for a System with Internal States
%J Mathematical modelling of natural phenomena
%D 2016
%P 191-239
%V 11
%N 3
%I mathdoc
%U https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611312/
%R 10.1051/mmnp/201611312
%G en
%F MMNP_2016_11_3_a11
M. Niemann; E. Barkai; H. Kantz. Renewal Theory for a System with Internal States. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 191-239. doi : 10.1051/mmnp/201611312. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611312/

[1] Takuma Akimoto, Eli Barkai Phys. Rev. E 2013 032915

[2] E. Barkai Phys. Rev. Lett. 2003 104101

[3] E. Barkai, E. Aghion, D. Kessler Physical Review X 2014 021036

[4] E. Barkai, Y. C. Cheng J. of Chemical Physics 2003 6167

[5] E. Barkai, Y. Garini, R. Metzler Physics Today 2012 29

[6] E. Barkai, I. Sokolov. Multi-point Distribution Function for the Continuous Time Random Walk. J. of Stat. Mech: Theory and Experiment (2007) P08001.

[7] A. Baule, R. Friedrich EPL 2007 10002

[8] G. Bel, E. Barkai Phys. Rev. Lett. 2005 240602

[9] G.M. Berger, B. Mandelbrot IBM Journal of Research and Development 1963 224

[10] S. Bianco, M. Ignaccolo, M. S. Rider, M. J. Ross, P. Winsor, P. Grigolini Phys. Rev. E 2007 061911

[11] P. Billingsley. Probability and Measure. John Wiley Sons (1995).

[12] J. Bouchaud Journal de Physique I France 1992 1705 1713

[13] J. P. Bouchaud, A. Georges Phys. Rep. 1990 127

[14] S. Burov, R. Metzler, E. Barkai Proceedings of the National Academy of Sciences 2010 13228

[15] A. Dechant, E. Lutz Phys. Rev. Lett. 2015 080603

[16] J. L. Doob Transactions of the American Mathematical Society 1948 422 438

[17] E. B. Dynkin Selected Translations Math. Stat. Prob 1961 417

[18] S. Ferraro, M. Manzini, A. Masoero, E. Scalas Physica (Amsterdam) 2009 3991

[19] H.C. Fogedby Phys. Rev. E 1994

[20] C. Godrèche, J. M. Luck J. Stat. Phys. 2001 489 524

[21] C. Godrèche, S. N. Majumdar, G. Schehr. Statistics of the longest interval in renewal processes. J. Stat. Mech. (2015) P03014.

[22] J.-H. Jeon, E. Barkai, R. Metzler J. Chemical Physics 121916

[23] Y. Jung, E. Barkai, R. J. Silbey Chemical Physics 2002 181 194

[24] Tosio Kato. Perturbation Theory for Linear Operators. Volume 132 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 1980.

[25] J. Klafter, M. F. Shlesinger, G. Zumofen Physics Today 1996

[26] N. Leibovich, E. Barkai Phys. Rev. Lett. 2015

[27] S. B. Lowen, M. C. Teich Phys. Rev. E 1993 992 1001

[28] F. Mainardi, R. Gorenflo, A. Vivoli Fractional Calculus and Applied Analysis 2005 7 38

[29] B. Mandelbrot. Some noises with 1 /f spectrum, a bridge between direct current and white noise. IEEE Trans. Inform. Theory, IT-13 (1967) 289-298.

[30] C. Manzo, J. A, P. Massignan, G. J. Lapeyre, M. Lewenstein, M. F. Garcia Parajo Phys. Rev. X 2015 011021

[31] G. Margolin, E. Barkai J. Stat. Phys. 2006

[32] M. M. Meerschaert, H. P. Scheffler J. Appl. Prob. 2004 623 638

[33] M. M. Meerschaert, P. Straka Mathematical Modelling of Natural Phenomena 2013 1 16

[34] M. M. Meerschaert, P. Straka Annals of Probability 2014 1699 1723

[35] R. Metzler, J. H. Jeon, A. G. Cherstvy, E. Barkai Phys. Chem. Chem. Phys. 2014

[36] R. Metzler, J. Klafter Phys. Rep. 2000 1

[37] A. Moinet, M. Starnini, R. Pastor-Satorras Phys. Rev. Lett. 2015

[38] C. Monthus, J-P. Bouchaud J. Phys. A: Math. Gen. 1996 3847

[39] M. Niemann, H. Kantz Phys. Rev. E. 2008

[40] M. Niemann, H. Kantz, E. Barkai Phys. Rev. Lett. 2013

[41] A. V. Oppenheim, R. W. Schafer. Discrete-Time Signal Processing. Pearson Education Limited (2013).

[42] M. Politi, T. Kaizoji, E. Scalas Europhys. Lett. 2011 20004

[43] M. A. Rodriguez Phys. Rev. E 2015 012112

[44] S. Sadegh, E. Barkai, D. Krapf New. J. of Physics 2014 113054

[45] F. Sanda, S. Mukamel J. Chem. Physics 154107 2007

[46] J. H. P. Schulz, E. Barkai, R. Metzler Phys. Rev. Lett. 2013

[47] F. D. Stefani, J. P. Hoogenboom, E. Barkai Physics Today 2009

[48] K. A. Takeuchi, T. Akimoto. Characteristic Sign Renewals of Kardar-Parisi-Zhang Fluctuations. [cond-mat.stat-mech] (2015) arXiv:1509.03082.

[49] S. Vajna, B. Tóth, J. Kertész New J. Phys. 2013

[50] A.V. Weigel, B. Simon, M.M. Tamkun, D. Krapf Proceedings of the National Academy of Sciences 2011

[51] I.Y. Wong, M.L. Gardel, D.R. Reichmann, E.R. Weeks, M.T. Valentine, A.R. Bausch, D.A. Weitz Phys. Rev. Lett. 2004 1781011

Cité par Sources :