Scale invariance - Revision history

Karl Jones at 15:39, 7 April 2016

2016-04-07T15:39:04Z

Karl Jones at 10:18, 7 February 2016

2016-02-07T10:18:57Z

Karl Jones: /* See also */

2015-09-06T14:17:21Z

‎See also

Karl Jones: Created page with "In physics, mathematics, statistics, and economics, '''scale invariance''' is a feature of objects or laws that do not change if scales of length, energy, or o..."

2015-09-06T14:16:50Z

Created page with "In physics, mathematics, statistics, and economics, '''scale invariance''' is a feature of objects or laws that do not change if scales of length, energy, or o..."

New page

In [[physics]], [[mathematics]], [[statistics]], and [[economics]], '''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.

== Description ==

The technical term for this [[transformation (mathematics)|transformation]] is a '''dilatation''' (also known as '''dilation'''), and the dilatations can also form part of a larger [[conformal symmetry]].

*In mathematics, scale invariance usually refers to an invariance of individual [[function (mathematics)|functions]] or [[curve]]s. 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 distribution]]s of [[random process]]es 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 transition]]s. The key observation is that near a phase transition or [[critical point (thermodynamics)|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 theory|statistical field theories]], and are formally very similar to scale-invariant quantum field theories.

*[[universality (dynamical systems)|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.

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

(TO DO: organize, cross-ref)

== See also ==

* [[Scale (ratio)]]

== External links ==

* [https://en.wikipedia.org/wiki/Scale_invariance Scale invariance] @ Wikipedia

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	*In general, [[dimensionless quantities]] are scale invariant. The analogous concept in [[statistics]] are [[standardized moment]]s, which are scale invariant statistics of a variable, while the unstandardized moments are not.		*In general, [[dimensionless quantities]] are scale invariant. The analogous concept in [[statistics]] are [[standardized moment]]s, which are scale invariant statistics of a variable, while the unstandardized moments are not.
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 == See also ==
 * [[Scale (ratio)]]
 == External links ==
 * [https://en.wikipedia.org/wiki/Scale_invariance Scale invariance] @ Wikipedia