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Power Laws and Skew Distributions

Published online by Cambridge University Press:  20 August 2015

Reinhard Mahnke ,
Jevgenijs Kaupužs  and
Mārtiņš Brics
Reinhard Mahnke*
Affiliation:
Institute of Physics, Rostock University, D-18051 Rostock, Germany
Jevgenijs Kaupužs*
Affiliation:
Institute of Mathematics and Computer Science, University of Latvia, LV-1459 Riga, Latvia
Mārtiņš Brics*
Affiliation:
Institute of Physics, Rostock University, D-18051 Rostock, Germany
*
Corresponding author.Email:reinhard.mahnke@uni-rostock.de
Email:kaupuzs@latnet.lv
Email:martins.brics2@uni-rostock.de
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Abstract

Power-law distributions and other skew distributions, observed in various models and real systems, are considered. A model, describing evolving systems with increasing number of elements, is considered to study the distribution over element sizes. Stationary power-law distributions are found. Certain non-stationary skew distributions are obtained and analyzed, based on exact solutions and numerical simulations.

Keywords

05.40.-a64.60.F-Power lawcritical exponentevolving systemskew distribution
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 3 , September 2012 , pp. 721 - 731
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Onsager, L., Crystal statistics I: a two-dimensional model with an order-disorder transition, Phys. Rev., 65 (1944), 117–149.CrossRefGoogle Scholar
[2]McCoy, B. and Wu, T. T., The Two-Dimensional Ising Model, Harvard University Press, 1973.Google Scholar
[3]Baxter, R. J., Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1989.Google Scholar
[4]Pelissetto, A. and Vicari, E., Critical phenomena and renormalization-group theory, Phys. Rep., 368 (2002), 549–727.CrossRefGoogle Scholar
[5]Kaupužs, J., Critical exponents predicted by grouping of Feynman diagrams in φ 4 model, Ann. Phys. (Leipzig), 10 (2001), 299–331.Google Scholar
[6]Amit, D. J., Field Theory, the Renormalization Group and Critical Phenomena, World Scientific, Singapore, 1984.Google Scholar
[7]Ma, S. K., Modern Theory of Critical Phenomena, W. A. Benjamin, Inc., New York, 1976.Google Scholar
[8]Zinn-Justin, J., Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1996.Google Scholar
[9]Kleinert, H. and Schulte-Frohlinde, V., Critical Properties of φ 4 Theories, World Scientific, Singapore, 2001.CrossRefGoogle Scholar
[10]Sornette, D., Critical Phenomena in Natural Sciences, Springer, Berlin, 2000.Google Scholar
[11]Singh, R. R. P., Does quantum mechanics play role in critical phenomena?, Physics, 3 (2010), 35.Google Scholar
[12]Lawrie, I. D., Goldstone mode singularities in specific heats and non-ordering susceptibilities of isotropic systems, J. Phys. A Math. Gen., 18 (1985), 1141–1152.Google Scholar
[13]Hasenfratz, P. and Leutwyler, H., Goldstone boson related finite size effects in field theory and critical phenomena with O(N) symmetry, Nucl. Phys. B, 343 (1990), 241–284.Google Scholar
[14]Tuber, U. C. and Schwabl, F., Critical dynamics of the O(n)-symmetric relaxational models below the transition temperature, Phys. Rev. B, 46 (1992), 3337–3361.CrossRefGoogle Scholar
[15]Schäfer, L. and Horner, H., Goldstone mode singularities and equation of state of an isotropic magnet, Z. Phys. B, 29 (1978), 251–263.Google Scholar
[16]Anishetty, R., Basu, R., Dass, N. D. H. and Sharatchandra, H. S., Infrared behaviour of systems with Goldstone bosons, Int. J. Mod. Phys. A, 14 (1999), 3467–3495.Google Scholar
[17]Kaupužs, J., Longitudinal and transverse correlation functions in the φ 4 model below and near the critical point, Prog. Theor. Phys., 124 (2010), 613–643.Google Scholar
[18]Kaupužs, J., Melnik, R. V. N. and Rimšāns, J., Monte Carlo test of Goldstone mode singularity in 3D XY model, Euro. Phys. J. B, 55 (2007), 363–370.Google Scholar
[19]Kaupužs, J., Melnik, R. V. N. and Rimšāns, J., Advanced Monte Carlo study of the Goldstone mode singularity in the 3D XY model, Commun. Comput. Phys., 4 (2008), 124–134.Google Scholar
[20]Kaupužs, J., Melnik, R. V. N. and Rimšāns, J., Monte Carlo estimation of transverse and longi-tudinal correlation functions in the O(4) model, Phys. Lett. A, 374 (2010), 1943–1950.Google Scholar
[21]Schmelzer, J., Röpke, G. and Mahnke, R., Aggregation Phenomena in Complex Systems, Wiley-VCH, Weinheim, 1999.Google Scholar
[22]Nagel, K. and Schreckenberg, M., A cellular automaton model for freeway traffic, J. Phys. I France, 2 (1992), 2221–2229.CrossRefGoogle Scholar
[23]Emmerich, H. and Rank, E., An improved cellular automaton model for traffic flow simulation, Phys. A, 234 (1997), 676–686.Google Scholar
[24]Helbing, D. and Schreckenberg, M., Cellular automata simulating experimental properties of traffic flow, Phys. Rev. E, 59 (1999), R2505–R2508.Google Scholar
[25]Fukui, M. and Ishibashi, Y., Traffic flow in 1D cellular automaton model including cars moving with high speed, J. Phys. Soc. Japan, 65 (1996), 1868–1870.Google Scholar
[26]Wolf, D. E., Cellular automata for traffic simulations, Phys. A, 263 (1999), 438–451.Google Scholar
[27]Simon, P. and Gutowitz, H. A., Cellular automaton model for bidirectional traffic, Phys. Rev. E, 57 (1998), 2441–2444.CrossRefGoogle Scholar
[28]Tokihiro, T., Takahashi, D., Matsukidaira, J. and Satsuma, J., From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76 (1996), 3247–3250.Google Scholar
[29]Schadschneider, A., The Nagel-Schreckenberg model revisited, Euro. Phys. J. B, 10 (1999), 573–582.Google Scholar
[30]Chowdhury, D., Santen, L. and Schadschneider, A., Statistical physics of vehicular traffic and some related systems, Phys. Rep., 329 (2000), 199–329.CrossRefGoogle Scholar
[31]Mahnke, R. and Pieret, N., Stochastic master-equation approach to aggregation in freeway traffic, Phys. Rev. E, 56 (1997), 2666–2671.CrossRefGoogle Scholar
[32]Mahnke, R., Kaupužs, J. and Lubashevsky, I., Probabilistic description of traffic flow, Phys. Rep., 408 (2005), 1–130.CrossRefGoogle Scholar
[33]Mahnke, R., Kaupužs, J. and Lubashevsky, I., Physics of Stochastic Processes: How Randomness Acts in Time, Wiley-VCH, Weinheim, 2009.Google Scholar
[34]Newman, M. E. J. and Power laws, Pareto distributions and Zipf’s law, Contemp. Phys., 46 (2005), 323–351.Google Scholar
[35]Saichev, A., Malevergne, Y. and Sornette, D., Theory of Zipf’s Law and Beyond, Springer, Berlin, 2009.Google Scholar
[36]Clauset, A., Shalizi, C. R. and Newman, M. E. J., Power-law distributions in empirical data, SIAM Rev., 51 (2009), 661–703.Google Scholar
[37]Frank, S., The common patterns of nature, J. Evol. Bio., 22 (2009), 1563–1585.CrossRefGoogle ScholarPubMed
[38]Kuninaka, H. and Matsushita, M., Why does Zipf’s law break down in rank-size distribution of cities?, J. Phys. Soc. Japan, 77 (2008), 114801.Google Scholar
[39]Moriyama, O., Itoh, H., Matsushita, S. and Matsushita, M., Long-tailed duration distributions for disability in aged people, J. Phys. Soc. Japan, 72 (2003), 2409–2412.Google Scholar
[40]Wada, T., A nonlinear drift which leads to κ-generalized distributions, Euro. Phys. J. B, 73 (2010), 287–291.Google Scholar
[41]Frisch, U. and Sornette, D., Extreme deviations and applications, J. Phys. I France, 7 (1997), 1155–1171.CrossRefGoogle Scholar
[42]Malevergne, Y. and Sornette, D., Extreme Financial Risks: From Dependence to Risk Management, Springer, Berlin, 2006.Google Scholar
[43]Malevergne, Y., Pisarenko, V. and Sornette, D., Empirical distributions of log-returns: between the stretched exponential and the power law?, Quant. Financ., 5 (2005), 379–401.Google Scholar
[44]Choi, M.Y., Choi, H., Fortin, J.-Y. and Choi, J., How skew distributions emerge in evolving systems, Europhys. Lett., 85 (2009), 30006.CrossRefGoogle Scholar
[45]Kaupužs, J., Mahnke, R. and Weber, H., Comment on “How skew distributions emerge in evolving systems” by M. Y., Choiet al., Europhys. Lett., 91 (2010), 30004.Google Scholar
[46]Levy, M. and Solomon, S., Power laws are logarithmic Boltzmann laws, Int. J. Mod. Phys. C, 7 (1996), 595–601.Google Scholar
[47]Malcai, O., Biham, O. and Solomon, S., Power-law distributions and Lévy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements, Phys. Rev. E, 60 (1999), 1299–1303.Google Scholar
[48]Blank, A. and Solomon, S., Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components), Phys. A, 287 (2000), 279–288.Google Scholar
[49]Sornette, D. and Davy, P., Fault growth model and the universal fault length distribution, Geophys. Res. Lett., 18 (1991), 1079–1081.Google Scholar
[50]Takayasu, H., Takayasu, M., Provata, A. and Huber, G., Statistical properties of aggregation with injection, J. Stat. Phys., 65 (1991), 725–745.Google Scholar
[51]Majumdar, S.N. and Sire, C., Exact dynamics of a class of aggregation models, Phys. Rev. Lett., 71 (1993), 3729–3732.Google Scholar