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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. 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 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. *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. ==Scale-invariant curves and self-similarity== In mathematics, one can consider the scaling properties of a function or curve under rescalings of the variable . That is, one is interested in the shape of for some scale factor , which can be taken to be a length or size rescaling. The requirement for to be invariant under all rescalings is usually taken to be : for some choice of exponent , and for all dilations . This is equivalent to being a homogeneous function. Examples of scale-invariant functions are the monomials , for which =''n'', in that clearly : An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates (''r'', ''θ''), the spiral can be written as : Allowing for rotations of the curve, it is invariant under all rescalings ; that is, is identical to a rotated version of . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「scale invariance」の詳細全文を読む スポンサード リンク
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