The modern meaning of the term was introduced by Leo Kadanoff in the 1960s, [citation needed] but a simpler version of the concept was already implicit in the van der Waals equation and in the earlier Landau theory of phase transitions, which did not incorporate scaling correctly.
[citation needed] The term is slowly gaining a broader usage in several fields of mathematics, including combinatorics and probability theory, whenever the quantitative features of a structure (such as asymptotic behaviour) can be deduced from a few global parameters appearing in the definition, without requiring knowledge of the details of the system.
Relevant operators are those responsible for perturbations to the free energy, the imaginary time Lagrangian, that will affect the continuum limit, and can be seen at long distances.
The collection of scale-invariant statistical theories define the universality classes, and the finite-dimensional list of coefficients of relevant operators parametrize the near-critical behavior.
The key observation is that near a phase transition or critical point, disturbances occur at all size scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena, as seems to have been put in a formal theoretical framework first by Pokrovsky and Patashinsky in 1965 [4].
However, as the phase transition is approached, the scale-dependent parameters play less and less of an important role, and the scale-invariant parts of the physical description dominate.
[citation needed] The expectation values of operators, such as the rate of flow, the heat capacity, and so on, are obtained by integrating over all possible configurations.
The term has been applied[5] to multi-agent simulations, where the system-level behavior exhibited by the system is independent of the degree of complexity of the individual agents, being driven almost entirely by the nature of the constraints governing their interactions.