Universality classes in phase transitions : how is it precisely defined?

Question

I would like to understand how we can say that two physical systems are in the same universality class.

What I understand from this notion is that two systems are in the same universality class if they have the same critical exponents.

If I take the example of the Ising mode, the order parameter $m$ behave as : $(T_c-T)^{\beta=1/2}$ near the phase transition.

And its derivative with respect to magnetic field behave as $(T-Tc)^{-\gamma=-1}$

But how do I compare it with a liquid gas phase transition for example ? What would be the "equivalent" of the magnetic field to derive the order parameter : the pressure, the number of particle ?

I understand the general idea of universality classes but I don't understand from an accurate point of view how to make the correspondance between the variable in two differents systems (is $H$ the magnetic field equivalent to the pressure or to the number of particle in a liquid gas phase transition for example).

Answer 1

The way it's precisely (though not necessarily rigorously) defined is by considering a field theory description of the statistical system and the renormalization group (a nice intuition of which can be gained from Kadanoff's block scaling).

How the renormalization group applied to a statistical field theory explains the universality classes is nicely summarized in two slides from a Kadanoff's talk:

Also, from Wikipedia:

The renormalization group [...] 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.

As for the question:

But how do I compare it [Ising model] with a liquid gas phase transition for example? What would be the "equivalent" of the magnetic field to derive the order parameter: the pressure, the number of particles?

The equivalent of the Ising external magnetic field for the liquid-gas phase transition is the pressure (the number of particles, I believe, should be an equivalent choice, and the particle density distribution has been used to describe this transition); other equivalences are also automatically defined by the choice of the order parameter, etc. and derivatives involving these quantities $-$ here's a small list.

It's worth checking the related questions Examples of important known universality classes besides Ising and Reasons for violation of universality in statistical mechanics.