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A power law in statistics and in the mathematics of physical sciences is a relationship between quantities such that one is proportional to a (constant) power of the other. Typically such relations hold only over a particular range of values. Form of such laws (for quantities x and y):
y=axk
where a and k are constants. Subtypes include "square law" and "inverse-square law". Some physics examples:
The term power law spectrum is commonly used for an EMR spectrum such that the flux at each frequency is related to that frequency by a power law, the exponent being referred to as its spectral index. Such a spectrum can only fit this description over a particular regime, lest it reach infinities. Synchrotron radiation is described as such a power law spectrum. The full black-body spectrum, which peaks in the middle, is not, but its longer-wavelength portion is usefully treated as a power law spectrum (Rayleigh-Jeans law).
Note that the term power law spectrum is also used with the more general definition of the term spectrum; for example, the distribution of cosmic-ray kinetic energies is termed a spectrum, which is a power law spectrum.
A common way of demonstrating (or discovering) a power law is a log-log plot, i.e., where each axis represents the log of a quantity of interest: a power law forms a straight line on such a plot.