Scale invariance

Scale invariance or scale independence is a term from mathematics, nuclear physics and statistical physics and statistical mechanics.

The term describes the property of a state, process, relationship, or a situation where the nature or characteristics, including its basic parameters remain largely in the / in spite of changing the viewing size (scaling ) is exactly the same, so that a " self-similar " condition exists, the most certain universality properties.

Mathematics

A dependent of the variable function is called scale-invariant if the essential properties of the function do not change under a rescaling. In general, it refers to that only by a factor ( which may depend on ) changes:

This means for example that important properties of the function - as zeros, extrema, inflection points or poles - not depend on which scale you use. Examples skaleninvarianter functions are the monomials.

The generalization for functions of several variables is obvious: the function is called scale-invariant if examples are homogeneous polynomials, the p- norms, the Mahalanobis distance and the correlation coefficient.

Also, networks whose degree does not follow a link scale are called scale-invariant or scale- free networks.

Nuclear physics

The spatial extent of quarks in nucleons is described by the so-called structure function. From the invariance of this structure function against the 4- momentum transfer is postulated that the quarks as building blocks of nucleons have no spatial extent, so are point-like.

Statistical Physics

Systems with phase transitions of the second kind, ie: transitions with a continuous history of the order parameter, show a scale invariant behavior at the critical point of the property that is described by the order parameter. An example is the transition from non-magnetic ( paramagnetic ) behavior for ferromagnetic behavior of a writable by the Ising model material at a critical temperature. When exactly this temperature, the distribution of uniformly magnetized regions (spin clusters ) is spatially scale invariant, that is, there are clusters on all scales. The order parameter, in this example, the magnetization is zero at the critical temperature or, as there are clusters of different magnetization directions. Clearly: Regardless of how closely one approaches the system, that is, how much you enlarge it, you will always see the same (magnetic) picture.