Critical properties throughout science are popularly associated with heavy-tailed distributions, but experimental evidence indicates several alternative, and very different, functional forms. Until now there has been no clear understanding of why this is, nor any general criterion as to which form to expect in a given practical situation. Here, a general scaling argument is presented, specific to spatially averaged properties, that indicates the following simple rule: if the mean value increases rapidly with system size then a power-law distribution is appropriate; if it changes slowly then a ‘generalized Gumbel distribution’ is likely, and if it decreases rapidly then an exponentially truncated power-law distribution is appropriate. The three scenarios are connected with the well-established classification of a scaling variable as either irrelevant, marginal or relevant. This result is supported by the current data set and finally renders comprehensible the fact that real critical properties exhibit diverse and apparently unrelated distributions, instead of ubiquitous heavy tails.
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Newman, M. E. J. Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323–351 (2005).
Christensen, K., Danon, L., Scanlon, T. & Bak, P. Universal scaling law for earthquakes. Proc. Natl Acad. Sci. 99, 2509–2513 (2002).
Gumbel, E. J. Statistics of Extremes (Columbia Univ. Press, 1958).
Joubaud, S., Petrosyan, A., Ciliberto, S. & Garnier, N. B. Experimental evidence of non-Gaussian fluctuations near a critical point. Phys. Rev. Lett. 100, 180601 (2008).
Bramwell, S. T., Holdsworth, P. C. W. & Pinton, J.-F. Universality of rare fluctuations in turbulence and critical phenomena. Nature 396, 552–554 (1998).
Pinton, J-F., Holdsworth, P. C. W. & Labbé, R. Power fluctuations in a closed turbulent shear flow. Phys. Rev. E 60, R2452–R2455 (1999).
Portelli, B., Holdsworth, P. C. W. & Pinton, J.-F. Intermittency and non-Gaussian fluctuations of the global energy transfer in fully developed turbulence. Phys. Rev. Lett. 90, 104501 (2003).
Van Milligen, B. P., Sanchez, R., Carreras, B. A., Lynch, V. E. & LaBombard, B. Additional evidence for the universality of the probability distribution of turbulent fluctuations and fluxes in the scrape-off layer region of fusion plasmas. Phys. Plasmas 12, 052507 (2005).
Antal, T., Droz, M., Györgyi, G. & Rácz, Z. 1/f noise and extreme value statistics. Phys. Rev. Lett. 87, 240601 (2001).
Bramwell, S., T., Fennell, T., Holdsworth, P. C. W. & Portelli, B. Universal fluctuations of the Danube water level: A link with turbulence, criticality and company growth. Europhys. Lett. 57, 310–314 (2002).
Dahlstedt, K. & Jensen, H. J. Fluctuation spectrum and size scaling of river flow and level. Phys. A 348, 596–610 (2005).
Pennetta, C., Alfinito, E., Reggiani, L. & Ruffo, S. Non-Gaussianity of resistance fluctuations near electrical breakdown. Semicond. Sci. Technol. 19, S164–S166 (2004).
Chamon, C. & Cugliandolo, L. F. Fluctuations in glassy systems. J. Stat. Mech. P07022 (2007).
Brey, J. J., Garcia de Soria, M. I., Maynar, P. & Ruiz-Montero, M. J. Mesoscopic theory of critical fluctuations in isolated granular gases. Phys. Rev. Lett. 94, 098001 (2005).
Goldburg, W. I., Goldschmidt, Y. Y. & Kellay, H. Fluctuation and dissipation in liquid-crystal electroconvection. Phys. Rev. Lett. 87, 245502 (2001).
Toth-Katona, T. & Gleeson, J. T. Distribution of injected power fluctuations in electroconvection. Phys. Rev. Lett. 91, 264501 (2003).
Bramwell, S. T. et al. Universal fluctuations in correlated systems. Phys. Rev. Lett. 84, 3744–3747 (2000).
Rypdal, K. et al. Scale-free vortex cascade emerging from random forcing in a strongly coupled system. New J. Phys. 10, 093018 (2008).
Chapman, S. C., Rowlands, G. & Watkins, N. W. Extreme statistics: A framework for data analysis. Nonl. Proc. Geophys. 9, 409–418 (2002).
Planet, R., Santucci, S. & Ortín, J. Avalanches and non-Gaussian fluctuations of the global velocity of imbibation fronts. Phys. Rev. Lett. 102, 094502 (2009).
Zheng, B. Generic features of fluctuations in critical systems. Phys. Rev. E 67, 026114 (2003).
Bertin, E. Global fluctuations and Gumbel statistics. Phys. Rev. Lett. 95, 170601 (2005).
Clusel, M., Fortin, J.-Y. & Holdsworth, P. C. W. Criterion for universality-class-independent critical fluctuations: Example of the two-dimensional Ising model. Phys. Rev. E 70, 046112 (2004).
van Wijland, F. Phonon displacement distribution at T=0. Physica A 332, 360–366 (2004).
Bertin, E. & Clusel, M. Generalised extreme value statistics and sum of correlated variables. J. Phys. A 39, 7607–7619 (2006).
Bertin, E. & Clusel, M. Global fluctuations in physical systems: A subtle interplay between sum and extreme value statistics. Int. J. Mod. Phys B 22, 3311–3368 (2008).
Bruce, A. D. Critical finite-size scaling of the free energy. J. Phys. A 28, 3345–3349 (1995).
Labit, B. et al. Universal statistical properties of drift-interchange turbulence in TORPEX plasmas. Phys. Rev. Lett. 98, 255002 (2007).
Farago, J. Injected power fluctuations in Langevin equation. J. Stat. Phys. 107, 781–803 (2002).
Goldenfeld, N. Lectures on Phase Transitions and the Renormalization Group (Addison–Wesley, 1992).
Malakis, A. & Fytas, N. G. Universal features and tail analysis of the order-parameter distribution of the two-dimensional Ising model: An entropic sampling Monte Carlo study. Phys. Rev. E 73, 056114 (2006).
Tsypin, M. M. & Blöte, H. W. J. Probability distribution of the order parameter for the three-dimensional Ising-model universality class: A high-precision Monte Carlo study. Phys. Rev. E 62, 73–76 (2000).
Bramwell, S. T. et al. Magnetic fluctuations in the classical XY model: The origin of an exponential tail in a complex system. Phys. Rev. E 63, 041106 (2001).
Berezinskii, V. L. Destruction of long range order on one-dimensional and two-dimensional systems having a continuous symmetry, I—classical systems. J. Exp. Theor. Phys. 32, 493–500 (1971).
Banks, S. T. & Bramwell, S. T. Temperature-dependent fluctuations in the two-dimensional XY model. J. Phys. A 38, 5603–5615 (2005).
Ricardo Paredes, V. & Botet, R. Scanning the critical fluctuations: Application to the phenomenology of the two-dimensional XY model. Phys. Rev. E 74, 060102 (2006).
Foltin, G., Oerding, K., Rácz, Z., Workman, R. L. & Zia, R. K. P. Width distribution for random-walk interfaces. Phys. Rev. E 50, R639–R642 (1994).
Portelli, B., Holdsworth, P. C. W., Sellito, M. & Bramwell, S. T. Universal magnetic fluctuations with a field-induced length scale. Phys. Rev. E 64, 036111 (2001).
Oono, Y. Large deviation and statistical physics. Prog. Theor. Phys. Suppl. 99, 165–205 (1989).
Boucher, C., Ellis, R.S. & Turkington, B. Spatializing random measures: Doubly indexed processes and the large deviation principle. Ann. Probab. 27, 297–324 (1999).
Salazar, R., Toralb, R. & Plastinoc, A. R. Numerical determination of the distribution of energies for the XY-model. Physica A 305, 144–147 (2002).
Barré, J., Bouchet, F., Dauxois, T. & Ruffo, S. Large deviation techniques applied to systems with long-range interactions. J. Stat. Phys. 119, 677–713 (2005).
Fisher, R. A. & Tippett, L. H. C. Limiting forms of the frequency distribution of the largest or smallest member of sample. Proc. Camb. Phil. Soc. 24, 180–190 (1928).
Györgyi, G., Moloney, N. R., Ozogány, K. & Rácz, Z. Finite-size scaling in extreme statistics. Phys. Rev. Lett. 100, 210601 (2008).
Cassandro, M. & Jona-Lasinio, G. Critical point behaviour and probability theory. Adv. Phys. 27, 913–941 (1978).
Zucker, I. J. & Robertson, M. M. Exact values of some two-dimensional lattice sums. J. Phys. A 8, 874–881 (1975).
McPhedran, R. C., Botten, L. C., Nicorovici, N. A. & Zucker, I. J. Systematic investigation of two-dimensional static array sums. J. Math. Phys. 48, 033501 (2007).
It is a pleasure to thank Maxime Clusel and Peter Holdsworth for very valuable comments and criticisms.
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Bramwell, S. The distribution of spatially averaged critical properties. Nature Phys 5, 444–447 (2009). https://doi.org/10.1038/nphys1268
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