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Abstract
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. The distributions of the sizes of cities or earthquakes, for example, follow a power law, but in physical systems different distributions of critical properties are usually seen. A scaling argument provides a practical rule to relate the type of distribution to an experimental quantity.Cite this article
Bramwell, S. The distribution of spatially averaged critical properties. Nature Phys 5, 444–447 (2009). https://doi.org/10.1038/nphys1268 