Universality in Phase Transitions

Universality refers to the observation that very different physical systems exhibit identical critical behavior near continuous (second-order) phase transitions. The concept reveals deep connections between seemingly unrelated systems, making critical phenomena a cornerstone of modern theoretical physics.

Near the critical point, physical quantities follow power laws:

• Order parameter: \(M \sim |T - T_c|^\beta\)
• Susceptibility: \(\chi \sim |T - T_c|^{-\gamma}\)
• Correlation length: \(\xi \sim |T - T_c|^{-\nu}\)

The exponents \(\beta\), \(\gamma\), \(\nu\), etc., are universal: they depend only on a few key features:

• Dimensionality of space (\(d\))
• Symmetry of the order parameter
• Range of interactions

As an example, a 3D Ising magnet and a liquid-gas transition share the same critical exponents → same universality class. But why would that be? Well, according to the Renormalization group (RG) under coarse-graining, microscopic details become irrelevant. Systems flow toward fixed points that define their universality class.