Scale invariance

Scale invariance is the property of a system whose relevant structure or laws remain unchanged when its scale changes. It can be exact or approximate over a range of scales.

Scaling transformation: A transformation that multiplies lengths or other quantities by a common scale factor. Scale invariance is tested by applying such transformations and checking what remains unchanged.
Critical phenomenon: Collective behavior near a continuous phase transition, often marked by diverging correlation lengths and power laws. At criticality, fluctuations occur across many scales and produce scale-invariant patterns.
Scale covariance: A property in which a transformation changes an object's form according to a specified scaling rule. Covariance allows quantities to transform, whereas invariance requires the relevant description to stay unchanged.
Symmetry (physics): A transformation that leaves a physical system or its laws unchanged. Scale invariance is one particular kind of symmetry, defined by rescaling.
Dimensional analysis: A method that uses physical dimensions to constrain equations and organize relationships among quantities. It identifies dimensionless combinations that can remain fixed as physical scales change.
Fractal: A geometric set with structure that repeats across scales, often characterized by a non-integer dimension. Fractals provide concrete examples of geometric scale invariance.
Self-similarity: A property in which parts of a structure resemble the whole under rescaling. Self-similarity is a common geometric expression of scale invariance, but the terms are not identical.
Dimensionless quantity: A quantity with no physical units, formed from ratios or products of dimensional quantities. Dimensionless quantities can compare behavior across systems with different characteristic scales.
Power law: A functional relationship in which one quantity varies as a fixed power of another. Power laws often express scale-invariant behavior across a range of magnitudes.
Brownian motion: A stochastic process with continuous paths and independent, normally distributed increments. Its rescaled paths have the same distribution after adjusting time and amplitude.

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