Critical phenomena and continuous phase transitions

Unlike a first-order transition (boiling, melting), a continuous (second-order) phase transition has no latent heat and no jump in density — instead an order parameter shrinks smoothly to zero at a critical temperature, while quantities like heat capacity and susceptibility diverge there following power laws whose exponents are shared by whole classes of otherwise unrelated systems.

Practise this concept

Boiling and melting are first-order transitions: the two phases briefly coexist, and turning liquid into gas costs a fixed amount of latent heat even though the temperature doesn't change while it happens. A continuous transition looks nothing like that. A ferromagnet losing its magnetisation as it warms through the Curie point, or helium becoming a superfluid as it cools, both slide smoothly from one phase into the other — there's no latent heat and no two phases coexisting, just an order parameter (magnetisation, in the ferromagnet's case) shrinking continuously to zero.

What makes the critical point interesting isn't the order parameter itself but how sharply the system responds to a small nudge right at that temperature. The heat capacity, the magnetic susceptibility, the correlation length — all of them diverge there, and they do it following power laws: order parameter ~ (Tc − T)^β below Tc, susceptibility ~ |T − Tc|^−γ on either side. β, γ and the other critical exponents are measurable numbers, not free parameters you fit for convenience.

The remarkable part is universality: a fluid at its liquid-gas critical point and a magnet at its Curie point can have the same critical exponents, even though one is about density and the other about spin alignment. What decides the exponents turns out to be only the dimensionality of space and the symmetry of the order parameter — everything about the specific atoms or forces involved washes out near the critical point. That's why critical phenomena get studied as their own subject, independent of which material happens to be transitioning.

Key ideas

Requires: Phase transitions, The statistical definition of entropy

Formulas

m = \left(\frac{T_c - T}{T_c}\right)^{\beta}

Symbol	Name	Unit
$m$	order parameter	1 (dimensionless)
$T$	temperature	K (kelvin)
$Tc$	critical temperature	K (kelvin)
$beta$	critical exponent	1 (dimensionless)

Common misconceptions

Believing every phase transition releases or absorbs latent heat, like boiling or melting — a continuous transition (e.g. a ferromagnet losing its magnetisation at the Curie point) has no latent heat at all; the order parameter simply shrinks to zero smoothly.
Thinking a diverging heat capacity or susceptibility at the critical point means the system's energy becomes infinite — it's the derivative (how sharply a quantity responds to a small change) that diverges, not the quantity itself, which stays finite.
Believing shared critical exponents mean two systems (e.g. a fluid near its boiling-liquid critical point and a magnet near its Curie point) must be made of similar stuff — universality classes are set by dimensionality and symmetry, not by chemistry or microscopic mechanism, which is exactly why wildly different systems can share the same exponents.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

Thermodynamics and statistical physicsLOFY.04.004