What is the formal definition of a Universality Class?

[AspiringResearcher]

Hi guys,

I have been reading some of the literature recently concerning the Kardar-Parisi-Zhang equation and the words "universality" and "KPZ universality class" keep appearing. I already did a cursory wikipedia search on the subject, but it did not make much sense to me.

Can you please explain to this undergraduate what a universality class is in statistical mechanics (preferably with a formal definition), and how scaling exponents are important in their definition?

 

[Stephen Tashi]

Not that I know anything about the Kardar-Parisi-Zhang equation, but I found this definition in searching the web:

From: http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332006000500030

In the study of interface growth dynamics, one is mostly concerned about the temporal behavior of the interface roughness, which is a measure of the interface width. The most relevant information about the dynamical details of a growth process can be obtained from the temporal behavior of the roughness. In particular, for self-affine interfaces, it is known that the roughness grows with time as a power law, where we define the growth exponent, b. Actually, due to correlations, the roughness does not grow indefinitely with time; the interface eventually reaches a stationary regime where the roughness saturates. Both the saturation roughness and saturation time depend on the system size as a power law, for which we define the roughness exponent, a, and the dynamic exponent, z, respectively.

A set of values for these three roughening exponents, in a given dimension, defines an universality class. Thus, if two or more processes have the same exponents values, one can say that they belong to the same universality class,