Lévy Flights, 1/f Noise and Self Organized Criticality


A new analysis of a previously studied traveling agent model, showed that there is a relation between the degree of homogeneity of the medium where the agents move, agent motion patterns, and the noise generated from their displacements. We proved that for a particular value of homogeneity, the system self organizes in a state where the agents carry out Lévy walks and the displacement signal corresponds to 1/f noise. Using probabilistic arguments, we conjectured that 1/f noise is a fingerprint of a statistical phase transition, from randomness (disorder) to predictability (order), and that it emerges from the contextuality nature of the system which generates it.

Share and Cite:

O. Corona, P. Padilla, O. Escolero, A. Frank and R. Fossion, "Lévy Flights, 1/f Noise and Self Organized Criticality," Journal of Modern Physics, Vol. 4 No. 3, 2013, pp. 337-343. doi: 10.4236/jmp.2013.43046.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. Ott, J. P. Bouchaud, D. Langevin and W. Urbach, “Anomalous Diffusion in ‘Living Polymers’: A Genuine Levy Flight?” Physical Review Letters, Vol. 65, No. 17, 1990, pp. 2201-2204. doi:10.1103/PhysRevLett.65.2201
[2] T. H. Solomon, E. R. Weeks and H. L. Swinney, “Observation of Anomalous Diffusion and Lévy Flights in a Two-Dimensional Rotating Flow,” Physical Review Let ters, Vol. 71, No. 24, 1993, pp. 3975-3978. doi:10.1103/PhysRevLett.71.3975
[3] F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect and C. Cohen-Tannoudji, “Subrecoil Laser Cooling and Lévy Flights,” Physical Review Letters, Vol. 72, No. 2, 1994, pp. 203-206. doi:10.1103/PhysRevLett.72.203
[4] J. Klafter and G. Zumofen, “Lévy Statistics in a Hamiltonian System,” Physical Review E, Vol. 49, No. 6, 1994, pp. 4873-4877. doi:10.1103/PhysRevE.49.4873
[5] O. V. Bychuk and B. O’Shaughnessy, “Anomalous Dif- fusion at Liquid Surfaces,” Physical Review Letters, Vol. 74, No. 10, 1995, pp. 1795-1798. doi:10.1103/PhysRevLett.74.1795
[6] I. M. Sokolov, J. Mai and A. Blumen, “Paradoxal Diffusion in Chemical Space for Nearest-Neighbor Walks over Polymer Chains,” Physical Review Letters, 79, No. 5, 1998, pp. 857-860.
[7] M. Bologna, P. Grigolini and J. Riccardi, “The Levy Dif- fusion as an Effect of Sporadic Randomness,” 1999. http://arxiv.org/abs/cond-mat/9907464
[8] P. Santini, “Lévy Scaling in Random Walks with Fluctuating Variance,” Physical Review E, Vol. 61, No. 1, 2000, pp. 93-99. doi:10.1103/PhysRevE.61.93
[9] W. D. Luedtke and U. Landman, “Slip Diffusion and Lévy Flights of an Adsorbed Gold Nanocluster,” Physical Review Letters, Vol. 82, No. 19, 1999, pp. 3835-3838. doi:10.1103/PhysRevLett.82.3835
[10] G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. da Luz, E. Raposo and H. Stanley, “Optimizing the Success of Random Searches,” Nature, Vol. 401, No. 6756, 1999, pp. 911-914. doi:10.1038/44831
[11] G. Ramos-Fernández, J. L. Mateos, O. Miramontes, G. Cocho, H. Larralde and B. Ayala-Orozco, “Lévy Walk Patterns in the Foraging Movements of Spider Monkeys (Ateles Geoffroyi),” Behavioral Ecology and Sociobiology, Vol. 55, 2004, pp. 223-230. doi:10.1007/s00265-003-0700-6
[12] L. Seuront, A. Duponchel and C. Chapperon, “Statistical Mechanics and Its Applications,” Physica A, Vol. 385, No. 2, 2007, pp. 573-582. doi:10.1016/j.physa.2007.07.029
[13] R. Atkinson, C. Rhodes, D. MacDonald and R. Anderson, “Scale-Free Dynamics in the Movement Patterns of Jackals,” Oikos, Vol. 98, No. 1, 2002, pp. 134-140. doi:10.1034/j.1600-0706.2002.980114.x
[14] D. Austin, W. Bowen and J. McMillan, “Intraspecific Variation in Movement Patterns: Modeling Individual Behaviour in a Large Marine Predator,” Oikos, Vol. 105, No. 1, 2004, pp. 15-30. doi:10.1111/j.0030-1299.1999.12730.x
[15] D. W. Sims, E. J. Southall, N. E. Humphries, G. C. Hays, C. J. A. Bradshaw, J. W. Pitchford, A. James, M. Z. Ah- med, A. S. Brierley, M. A. Hindell, D. Morritt, M. K. Musyl, D. Righton, E. L. C. Shepard, V. J. Wearmouth, R. P. Wilson, M. J. Witt and J. D. Metcalfe, “Scaling Laws of Marine Predator Search Behaviour,” Nature, Vol. 451, No. 7182, 2008, pp. 1098-1102. doi:10.1038/nature06518
[16] O. Miramontes, D. Boyer and F. Bartumeus, “The Effects of Spatially Heterogeneous Prey Distributions on Detection Patterns in Foraging Seabirds,” PLoS One, Vol. 7, No. 4, 2012, p. e34317. doi:10.1371/journal.pone.0034317
[17] C. T. Brown, L. S. Liebovitch and R. Glendon, “Lévy Flights in Dobe Ju/’hoansi Foraging Patterns,” Human Ecology, Vol. 35, No. 1, 2007, pp. 129-138. doi:10.1007/s10745-006-9083-4
[18] D. Boyer and O. López-Corona, “Self-Organization Scaling and Collapse in a Coupled Automaton Model of Foragers,” Journal of Physics A, Vol. 42, No. 43, 2009, p. 4014.
[19] A. Downey, “Think Complexity,” O’Reilly Media, 2012, p. 79.
[20] O. Bohigas, M. Giannoni and C. Schmit, “Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws,” Physical Review Letters, Vol. 52, No. 1, 1984, pp. 1-4. doi:10.1103/PhysRevLett.52.1
[21] E. Faleiro, U. Kuhl, R. Molina, L. Mu?oz, A. Relano and J. Retamosa, “Power Spectrum Analysis of Experimental Sinai Quantum Billiards,” Physics Letters A, Vol. 358, No. 4, 2006, pp. 251-255. doi:10.1016/j.physleta.2006.05.029
[22] R. Haq, A. Pandey and O. Bohigas, “Fluctuation Proper- ties of Nuclear Energy Levels: Do Theory and Experiment Agree?” Physical Review Letters, Vol. 48, No. 16, 1982, pp. 1086-1089 doi:10.1103/PhysRevLett.48.1086
[23] J. M. Gómez, E. Faleiro, R. Molina, L. Mu?oz, A. Rela?o, and J. Retamosa, “Chaos and 1/f Noise in Nuclear Spectra,” AIP Conference Proceedings, Vol. 831, 2006, p. 80.
[24] A. Cavagna, A. Cimarelli, et al., “Scale-Free Correlations in Bird Flocks,” 2009. arXiv:0911.4393
[25] J. Buhl, D. Sumpter, et al., “From Disorder to Order in Marching Locusts,” Science, Vol. 312, No. 5778, 2006, pp. 1402-1406. doi:10.1126/science.1125142
[26] A. Goldberger, “Fractal Dynamics in Physiology: Alterations with Disease and Aging,” Proceedings of the National Academy of Sciences, Vol. 99, No. S1, 2002, pp. 2466-2472. doi:10.1073/pnas.012579499
[27] W. H. Press, “Flicker Noises in Astronomy and Elsewhere,” Comments on Astrophysics, Vol. 7, No. 4, 1978, pp. 103-119.
[28] S. L. Miller, W. M. Miller and P. J. McWhorter, “Extremal Dynamics: A Unifying and Activated Processes’ Physical Explanation of Fractals, 1/f Noise,” Journal of Applied Physics, Vol. 73, No. 6, 1993.
[29] R. Fossion, E. Landa, P. Stransky, V. Velazquez, J. C. Lopez Vieyra, I. Garduno, D. Garcia and A. Frank, “Scale Invariance as a Symmetry in Physical and Biological Systems: Listening to Photons, Bubbles and Heartbeats,” AIP Conference Proceedings, Vol. 1323, 2010, pp. 74-90. doi:10.1063/1.3537868
[30] P. Bak and M. Paczuski, “Why Nature Is Complex,” Physics World, Vol. 6, No. 12, 1993, pp. 39-43.
[31] E. R. Scheinerman, “Invitation to Dynamical Systems,” Dover Publications, Mineola, 2012.
[32] L. A. Lipsitz, “Dynamics of Stability the Physiologic Basis of Functional Health and Frailty,” The Journals of Gerontology Series A: Biological Sciences and Medical Sciences, Vol. 57, No. 3, 2002, pp. 115-125. doi:10.1093/gerona/57.3.B115
[33] I. Eliazar and J. Klafter, “A Unified and Universal Explanation for Lévy Laws and 1/f Noises,” Proceedings of the National Academy of Sciences, Vol. 106, No. 30, 2009, pp. 12251-12254.
[34] I. Eliazar and J. Klafter, “Universal Generation of Statistical Self-Similarity: A Randomized Central Limit Theorem,” Physical Review Letters, Vol. 103, No. 4, 2009, Article ID: 40602. doi:10.1103/PhysRevLett.103.040602
[35] D. Boyer, G. Ramos-Fernández, O. Miramontes, J. L. Mateos, G. Cocho, H. Larralde, H. Ramos and F. Rojas, “Scale-Free Foraging by Primates Emerges from Their Interaction with a Complex Environment,” Proceedings of the Royal Society B, Vol. 273, 2006, pp. 1743-1750
[36] D. Boyer, O. Miramontes and H. Larralde, “Lévy-Like Behaviour in Deterministic Models of Intelligent Agents Exploring Heterogeneous Environments,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 43, 2009.
[37] E. Landa, I. Morales, P. Stransky, R. Fossion, V. Velázquez, J. C. Vieyra and A. Frank, “Scale Invariance in Chaotic Time Series: Classical and Quantum Examples, Chaos Theory: Modeling, Simulation and Applications,” In: 3rd Chaotic Modeling and Simulation International Conference (CHAOS), World Scientific, London, 2010.
[38] A. Relano, J. M. Gómez, R. A. Molina, J. Retamosa and E. Faleiro, “Quantum Chaos and 1/f Noise,” Physical Review Letters, Vol. 89, No. 24, 2002, Article ID: 244102. doi:10.1103/PhysRevLett.89.244102
[39] A. L. Goldberger, “Non-Linear Dynamics for Clinicians: Chaos Theory, Fractals, and Complexity at the Bedside,” The Lancet, Vol. 347, No. 9011, 1996, pp. 1312-1314. doi:10.1016/S0140-6736(96)90948-4
[40] A. L. Goldberger, C. K. Peng and L. A. Lipsitz, “What Is Physiologic Complexity and How Does It Change with Aging and Disease?” Neurobiology of Aging, Vol. 23, No. 1, 2002, pp. 23-26. doi:10.1016/S0197-4580(01)00266-4
[41] A. L. Goldberger, et al., “Fractal Dynamics in Physiology: Alterations with Disease and Aging,” Proceedings of the National Academy of Sciences, Vol. 99, No. 1, 2002, pp. 2466-2472. doi:10.1073/pnas.012579499
[42] S. M. Pikkujamsa, T. H. M?kikallio, L. B. Sourander, et al., “Cardiac Interbeat Dynamics from Childhood to Senescence: Comparison of Conventional and New Measures Based on Fractals and Chaos Theory,” Circulation, Vol. 100, No. 4, 1999, pp. 393-399. doi:10.1161/01.CIR.100.4.393
[43] J. Hayano, et al., Heart and Circulatory Physiology: American Journal of Physiology, Vol. 273, 1997, p. 2811.
[44] T. Hennig, P. Maass, J. Hayano and S. Heinrichs, “Exponential Distribution of Long Heart Beat Intervals during Atrial Fibrillation and Their Relevance for White Noise Behaviour in Power Spectrum,” Journal of Biological Physics, Vol. 32, No. 5, 2006, pp. 383-392. doi:10.1007/s10867-006-9022-z
[45] L. Hardy, “Quantum Theory from Five Reasonable Axioms,” 2001. http://arXiv.org/abs/quant-ph/0101012v4
[46] A. Khrennikov, “Probabilistic Foundations of Quantum Mechanics and Quantum Information,” 2003. http://arXiv.org/abs/quant-ph/0309066v1
[47] A. Khrennikov, “Vaxjio Interpretation-2003: Realism of Contexts,” 2004. http://arXiv.org/abs/quant-ph/0401072v1

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.