Scale invariance

<p>In <a href="/facts/Physics/a7aH2y08">physics</a>, <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, 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, and thus represent a universality.
</p><p>The technical term for this <a href="/facts/Transformation_(mathematics)/Yr2J3o8H">transformation</a> is a dilatation (also known as dilation). Dilatations can form part of a larger <a href="/facts/Conformal_symmetry/DlegNm7N">conformal symmetry</a>.
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<ul><li>In mathematics, scale invariance usually refers to an invariance of individual <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> or <a href="/facts/Curve/xBQ0pTBW">curves</a>. A closely related concept is <a href="/facts/Self-similarity/9Nvo7edY">self-similarity</a>, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the <a href="/facts/Probability_distribution/EpsKKVRu">probability distributions</a> of <a href="/facts/Random_process/Ng1n1dnB">random processes</a> to display this kind of scale invariance or self-similarity.</li>
<li>In <a href="/facts/Classical_field_theory/7HuKY1qu">classical field theory</a>, 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.</li>
<li>In <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, scale invariance has an interpretation in terms of <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a>. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.</li>
<li>In <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a>, scale invariance is a feature of <a href="/facts/Phase_transition/eLP2BY8R">phase transitions</a>. The key observation is that near a phase transition or <a href="/facts/Critical_point_(thermodynamics)/pkM7RLJK">critical point</a>, 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 <a href="/facts/Statistical_field_theory/EeYlVGoB">statistical field theories</a>, and are formally very similar to scale-invariant quantum field theories.</li>
<li><a href="/facts/Universality_(dynamical_systems)/SkgeJqF3">Universality</a> 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.</li>
<li>In general, <a href="/facts/Dimensionless_quantities/TINrpQIq">dimensionless quantities</a> are scale-invariant. The analogous concept in <a href="/facts/Statistics/DNPsKGYU">statistics</a> are <a href="/facts/Standardized_moment/oF0dAe8w">standardized moments</a>, which are scale-invariant statistics of a variable, while the unstandardized moments are not.</li></ul>

Scale invariance open-in-new

Scale invariance