On the scaling of probability density functions with apparentpower-law exponents less than unity |
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Authors: | K. Christensen N. Farid G. Pruessner M. Stapleton |
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Affiliation: | (1) Institute for Mathematical Sciences, Imperial College London, 53 Prince's Gate, London, SW7 2PG, UK;(2) Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2AZ, UK;(3) Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK |
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Abstract: | We derive general properties of the finite-size scaling of probability density functions and show that when the apparent exponent of a probability density is less than 1, the associated finite-size scaling ansatz has a scaling exponent τ equal to 1, provided that the fraction of events in the universal scaling part of the probability density function is non-vanishing in the thermodynamic limit. We find the general result that τ≥1 and . Moreover, we show that if the scaling function approaches a non-zero constant for small arguments, , then . However, if the scaling function vanishes for small arguments, , then τ= 1, again assuming a non-vanishing fraction of universal events. Finally, we apply the formalism developed to examples from the literature, including some where misunderstandings of the theory of scaling have led to erroneous conclusions. |
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Keywords: | KeywordHeading" >PACS. 89.75.Da Systems obeying scaling laws 89.75.-k Complex systems 05.65.+b Self-organized systems 89.75.Hc Networks and genealogical trees 05.70.Jk Critical point phenomena |
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