On the scaling of probability density functions with apparent power-law exponents less than unity

TL;DR

This paper investigates the finite-size scaling properties of probability density functions with apparent power-law exponents less than one, establishing conditions under which the true scaling exponent equals one or the apparent exponent, and clarifying common misconceptions.

Contribution

It provides a general theoretical framework for understanding the scaling behavior of probability densities with small exponents, correcting previous misunderstandings in the literature.

Findings

For a0< 1, the scaling exponent a0 is at least 1.

If the scaling function approaches a non-zero constant at small arguments, then a0 = a0.

When the scaling function vanishes at small arguments, a0 = 1.

Abstract

We derive general properties of the finite-size scaling of probability density functions and show that when the apparent exponent \tautilde of a probability density is less than 1, the associated finite-size scaling ansatz has a scaling exponent \tau 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 \tau>=1 and \tau>=\tautilde. Moreover, we show that if the scaling function G(x) approaches a non-zero constant for small arguments, \lim_{x-> 0} G(x) > 0, then \tau=\tautilde. However, if the scaling function vanishes for small arguments, \lim_{x-> 0} G(x) = 0, then \tau=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.