Scale invariance

An example: scale invariance or self-similarity of a Koch curve

Scale invariance or scale independence is a term used in mathematics , particle physics and statistical physics , more precisely statistical mechanics . It describes the property of a state, process, relationship or situation in which, regardless of the scale of the observation variables, the peculiarity or characteristic including its key values remain largely exactly the same. This gives a “self-similar” state, which mostly shows certain universality properties.

mathematics

A function that is dependent on the variable is called scale invariant if the essential properties of the function do not change under a rescaling . As a rule, it is understood to mean that only changes by one factor (which can depend on): ${\ displaystyle \! \ x}$ ${\ displaystyle \! \ f (x)}$ ${\ displaystyle \! \ x \ to ax}$ ${\ displaystyle \! \ f}$ ${\ displaystyle a}$

{\ displaystyle f (ax) = C (a) f (x) \ ,.}

This means, for example, that important properties of the function - such as zeros , extremes , turning points or poles - do not depend on which scale is used. Examples of scale-invariant functions are the monomials . ${\ displaystyle x ^ {p}}$

In generalization for functions of several variables this means: The function is called scale invariant if ${\ displaystyle f (x_ {1}, x_ {2}, \ dots, x_ {n})}$

{\ displaystyle f (ax_ {1}, ax_ {2}, \ dots, ax_ {n}) = C (a) f (x_ {1}, x_ {2}, \ dots, x_ {n}) \, .}

Examples are homogeneous polynomials , the p norms , the Mahalanobis distance, and the correlation coefficient .

Also networks whose linking degree does not follow any scale are called scale-invariant or scale-free networks .

Particle physics

The spatial expansion of quarks in nucleons is described in scattering processes by the structure function. From the invariance of this structure function with respect to the 4-pulse transfer, i.e. the scaling in the momentum space , it is postulated that the quarks as building blocks of the nucleons have no spatial extension, i.e. are point-shaped (see Bjorken scaling ).

Statistical Physics

Systems with phase transitions of the second kind , d. H. Transitions with a continuous course of the order parameter show a scale-invariant behavior of the property at the critical point , which is described by the order parameter.

One example is the transition from non-magnetic ( paramagnetic ) to ferromagnetic behavior of a material that can be described by the Ising model at a critical temperature . (At exactly this temperature, the distribution of uniformly magnetized regions is spin - clusters ) spatially scale invariant, d. that is, there are clusters on all size scales. The order parameter, in this example the magnetization , is still zero at the critical temperature because there are clusters of different directions of magnetization. Illustrative: Regardless of how close you get to the system, i. That is, how much you enlarge it, you will always see the same (magnetic) image.

Scale invariance is a characteristic of conformal field theories that mainly describe two-dimensional systems in statistical mechanics (scale invariance at the critical point) and quantum field theory (e.g. string theory ). The behavior of a system on different scales in these areas (regardless of whether it is scale-invariant or not) can be described by the renormalization group .