A scale-free probability distribution must be a power law

When the parameters of some physical systems are precisely tuned, the systems can enter a phase transition in which the behaviour of observables changes dramatically. In particular, the systems can become scale-free in the sense of losing any relationship to scales of measurement, i.e., the systems suddenly switch to behaving the same irrespective of the scales of measurement being used. (For many examples of this, and a discussion of scale invariance arising from phase transitions, visit this website). Among the critical phenomena in the vicinity of these phase transitions we can then get power law behaviours, e.g., for probability distributions of observables in the system. In the present note, I want to record a simple proof that whenever a probability distribution $p(x)$ is scale-free, it must in fact be a power law of the form $p(x) \sim x^{-\alpha}$ .

The scale-free characteristic can be expressed as

$p(bx) = g(b)p(x)$

so that multiplying the argument by a scale factor $b$ simply results in the same probability function multiplied by a scale factor $g(b)$ , where $g$ is some other function. To show that any function having this scale-free characteristic must be a power law, begin by setting $x = 1$ . Then $p(b) = g(b)p(1)$ and therefore

$g(b) = \frac{p(b)}{p(1)}$

so the expression for $p(bx)$ above becomes

$p(bx) = \frac{p(b)}{p(1)} p(x)$

Since this is an identity in the scale factor $b$ , we can differentiate both sides with respect to $b$ to get

$x p^{\prime}(bx) = \frac{p^{\prime}(b)p(x)}{p(1)}$

Setting $b = 1$ in this we get

$x \frac{dp}{dx} = \frac{p^{\prime}(1)}{p(1)}p(x)$

This is a separable first-order differential equation, so

$\int \frac{dp}{p} = \frac{p^{\prime}(1)}{p(1)} \int \frac{dx}{x}$

and therefore

$\ln p = \frac{p^{\prime}(1)}{p(1)} \ln x + c$

Setting $x = 1$ we find $c = \ln p(1)$ , so

$\ln p = \frac{p^{\prime}(1)}{p(1)} \ln x + \ln p(1)$

and thus we arrive at the power law

$p(x) = p(1) x^{-\alpha}$

where

$\alpha = -\frac{p^{\prime}(1)}{p(1)}$

So the power law distribution $p(x) \sim x^{-\alpha}$ is the only function satisfying the scale-free criterion $p(bx) = g(b)p(x)$ . In the vicinity of the critical point of a continuous phase transition at which a physical system becomes scale-free, power law behaviour should be seen among the observables in the system.

A scale-free probability distribution must be a power law

Published by Dr Christian P. H. Salas

Leave a comment Cancel reply