scale+invariance

=Scale Invariance=

toc Read here about scale invariant metrics and/or phenomena, being phenomena that do not change at different scales. The identification of such general, static, non-varying characteristics is critical to the research effort.

=Overview= "Scale invariant" is both an informal idea of non-varying characteristics, and a formal term in physics and mathematics referring to precisely defined phenomena.

=The Naive idea of Scale Invariance= In general parlance, most qualities are considered scale invariant: the general idea of something that is "true" is assumed to be true independent of scale. Any measurable, physical quantity (length, area, volume, time, speed, amount of substance, temperature, luminance, pressure) is naively assumed to be invariant over scales: the quality of a meter, whether it is a billionth of a meter on the atomic scale or a billion meters on the planetary scale, is invariant. In fact, the meter changes significantly across these scales (see Bridgman's account of length) hence the naive idea is problematic. It is the agenda of this research to communicate these variations in simple to understand terms.

=Scale Invariance, Formal Use in Math and Physics= In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor. The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry.

In general, dimensionless quantities are scale invariant. The analogous concept in statistics are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.

Some specific phenomena that are considered scale invariant:
 * In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilatations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.
 * In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.
 * In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
 * In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.
 * Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.

=Details of some Scale Invariant Phenomena=

Universality, or Phase Transitions
The set of different microscopic theories described by the same scale-invariant theory is known as a universality class. Other examples of systems which belong to a universality class are:


 * Avalanches in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales.
 * The frequency of network outages on the Internet, as a function of size and duration.
 * The frequency of citations of journal articles, considered in the network of all citations amongst all papers, as a function of the number of citations in a given paper.
 * The formation and propagation of cracks and tears in materials ranging from steel to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale.
 * The electrical breakdown of dielectrics, which resemble cracks and tears.
 * The percolation of fluids through disordered media, such as petroleum through fractured rock beds, or water through filter paper, such as in chromatography. Power-law scaling connects the rate of flow to the distribution of fractures.
 * The diffusion of molecules in solution, and the phenomenon of diffusion-limited aggregation.
 * The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).

The key observation is that, for all of these different systems, the behaviour resembles a phase transition, and that the language of statistical mechanics and scale-invariant statistical field theory may be applied to describe them.

Lévy Flight
A Lévy flight is a random walk in which the step-lengths are distributed according to a heavy-tailed probability distribution. Specifically, the distribution used is a power law of the form y = x -α where 1 < α < 3 and therefore has an infinite variance.

Lévy flights, named after the French mathematician Paul Pierre Lévy, are Markov processes. After a large number of steps, the distance from the origin of the random walk tends to a stable distribution.

Two-dimensional Lévy flights were described by Benoît Mandelbrot in The Fractal Geometry of Nature (Page 288ff). The exponential scaling of the step lengths gives Lévy flights a scale invariant property, and they are used to model data that exhibits clustering.

=Links= Scale invariance is one of the core concepts of scale studies. For other, related core concepts, see include component="pageList" hideInternal="true" tag="core concept" limit="100"