# Scale law

Under laws of scale or scaling laws are meant the manifestations of mathematical relationships of the kind

${\ displaystyle f (x) = bc ^ {x}}$,

d. H. exponential relationships, or

${\ displaystyle f (x) = bx ^ {c}}$,

d. H. Power or polynomial relationships, where and represent real constants . Power laws are more common than exponential relationships. ${\ displaystyle b}$${\ displaystyle c}$

Such relationships are so common in nature and society that one can speak of a structure- forming principle. Some of the distributions are purely empirically found, but some of them could be placed on a solid theoretical basis so that one can speak of "laws" in the scientific sense. One of the reasons for this is that

${\ displaystyle x (t) = ce ^ {t}}$

the solution of the simplest linear differential equation

${\ displaystyle {\ dot {x}} = x}$

that describes a self-accelerating process, e.g. B. the growth of a population without resource constraints.

Scale relationships that are based on power laws are scale invariant due to the relationship

${\ displaystyle f (ax) = b (ax) ^ {c} = a ^ {c} bx ^ {c} = a ^ {c} f (x) \ propto f (x)}$

d. that is , that is proportional and the characteristics of do not change. Exponential relationships do not show this scale invariance. ${\ displaystyle f (ax)}$${\ displaystyle f (x)}$${\ displaystyle f}$

## Examples

### statistics

Benford's Law
states that the probability of the occurrence of the digits in the first digit of frequency numbers obtained from natural distributions satisfies the relationship f D = log (1 + 1 / D ). In other words, in a good 30% of all numbers the 1 is in the first position, in 17% the 2 etc.

### biology

Geoffrey West attributes the universality of the laws of scale in biology to the following points:

1. Organisms of all sizes are kept alive by hierarchically branched metabolic supply networks.
2. These networks are space-filling (and often fractal ).
3. The endpoints of these networks are invariant.
4. The Evolution has the energy dissipation minimizes the organisms and / or maximizes the surfaces over which takes place resource sharing.

From these principles at least the allometries seem to be derived with very simple scale laws (the exponents tend to be integral multiples of 1/4).

Examples are the relationships between

### chemistry

Frequency of the chemical elements in the earth's crust ( Goldschmidt diagram)
Modern Goldschmidt diagram

### physics

• Stefan-Boltzmann-Law  : ...
• 1 / f noise : When the 1 / f noise which follows amplitude distribution of the noise signal a scaling law, more precisely a power law : where the amplitude at a frequency referred to, and ; hence the name 1 / f noise (because of ).${\ displaystyle A (f) = f ^ {- \ alpha}}$${\ displaystyle A (f)}$ ${\ displaystyle f}$${\ displaystyle \ alpha \ approx 1}$${\ displaystyle f ^ {- 1} = 1 / f}$
• Statistical Physics : Critical behavior in the case of phase transitions of the second kind. B. , with the magnetization , the critical temperature and the critical exponent , is described under scale invariance .${\ displaystyle M \ sim | T-T _ {\ mathrm {c}} | ^ {\ beta}}$${\ displaystyle M}$${\ displaystyle T _ {\ mathrm {c}}}$${\ displaystyle \ beta}$
• High-energy physics : Here one also actually observes so-called critical exponents , which in the language of high-energy physics are called anomalous dimensions .
• Thomson's velocity scaling : For high energies, the cross section for electron impact ionization in isoelectronic series only depends on .${\ displaystyle \ left ({\ tfrac {E} {m}} \ right) _ {\ text {projectile}}}$

### Internet

The Internet is a vast network with emergent phenomena such as self-similar scaling in the burst - patterns of its traffic and scale-free structure in the connection topology .

#### Weblogs

Other self-linking internet platforms such as weblogs also show a certain connection: new weblogs prefer to link - i. H. with a higher probability - on already popular weblogs and make them even more popular. This linking algorithm is also the rule for creating a scale-free network .

### Economics

Main article: Pareto distribution  : ...