Universality (dynamical systems)

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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 that display universality tend to be chaotic and often have a large number of interacting parts. The term universality was popularized by Leo Kadanoff in the late 1970s, but the concept was certainly known since the 1950s.

The notion of universality originated in the study of phase transitions in statistical mechanics. A phase transition occurs when a material changes its properties in a dramatic way: water, as it is heated boils and turns into vapor; or a magnet, when heated, loses its magnetism. Phase transitions are characterized by an order parameter, such as the density or the magnetization, that changes as a function of a parameter of the system, such as the temperature. The special value of the parameter at which the system changes its phase is the system's critical point. For systems that exhibit universality, the closer the parameter is to its critical value, the less sensitively the order parameter depends on the details of the system.

If the parameter β is critical at the value β_c, then the order parameter a will be well approximated by

[a=a_0 \| \beta-\beta_c \|^\alpha.\,]

The exponent α is a critical exponent of the system. The remarkable discovery made in the second half of the twentieth century was that very different systems had the same critical exponents, hence universality.

In 1976 Mitchell Feigenbaum discovered universality in iterated maps.

Universality is also observed in non-equilibrium systems, such as interacting particle systems, reaction-diffusion models, or self-organizing systems.

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