Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T03:35:35.708Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected]
Email:[email protected]
Email:[email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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