Sources of exponents

Examples are given of scaling relationships governed by fractional exponents. Additional examples are presented with heavy tail probability distributions governed by a fractional exponent. This includes the analysis of the Boltzmann–Gibbs entropy, subject to an appropriate constraint, to generate long-tailed probability distributions. The stretched exponential distribution is employed to derive a time scale for the glass transition that contains an essential singularity at a critical temperature. These examples demonstrate that there are many routes to producing fractional exponents in physical systems.