Universality (dynamical Systems)

Universality (dynamical Systems)

In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together. The modern meaning of the term was introduced by Leo Kadanoff in the 1960s, 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.

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.

The renormalization group explains universality. It classifies operators in a statistical field theory into relevant and irrelevant. 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. Irrelevant operators are those that only change the short-distance details. 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.