https://doi.org/10.1140/epjb/e2008-00173-2
On the scaling of probability density functions with apparent power-law exponents less than unity
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
Corresponding author: a g.pruessner@imperial.ac.uk
Received:
24
January
2008
Published online:
19
April
2008
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.
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
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2008
