Classifying transitions by the derivatives of G
At any transition the Gibbs energy G itself stays continuous — otherwise the system would gain or lose energy for free. Ehrenfest's idea is to classify by which derivative of G first jumps. A first-order transition has a discontinuity in the first derivatives, S=-(\partial G/\partial T)_P and V=(\partial G/\partial P)_T — hence a latent heat and a volume change (boiling, melting).
A first-order transition: the first derivatives of G jump, giving a latent heat L.
The critical point and the order parameter
Walk up the liquid–vapour coexistence line to higher P and T. The density jump between liquid and gas shrinks, and at the critical point it vanishes: liquid and vapour become one indistinguishable fluid. Beyond it you can pass continuously from 'gas' to 'liquid' by looping around — there is no longer any line to cross.
Introduce an order parameter m — a quantity that is nonzero in the ordered phase and zero above the critical temperature T_c. For fluids it is the density difference \rho_{\text{liq}}-\rho_{\text{gas}}; for a ferromagnet it is the magnetization. Approaching T_c from below, it vanishes as a power law.
The order parameter vanishes with a characteristic critical exponent \beta.
Critical exponents and universality
Near T_c every response becomes a power law in the reduced temperature t=(T-T_c)/T_c: the susceptibility (how strongly the order parameter responds to its conjugate field), the correlation length \xi, and the specific heat each diverge with their own critical exponent.
Diverging response functions define the exponents \gamma, \nu, \alpha (with \beta, \delta completing the set).
Here is the astonishing empirical fact. Systems with nothing physically in common — the liquid–gas critical point, a uniaxial ferromagnet, a binary alloy unmixing — share the same numerical exponents. This is universality: near a critical point only the dimensionality of space and the symmetry of the order parameter matter, never the chemistry. Explaining this is the crowning achievement of the modern theory.
Landau's mean field: a worked model
Landau theory makes the boldest possible simplification: near T_c the order parameter m is small, so expand the free energy as a power series in m, keeping only the terms the symmetry allows. If the physics is unchanged under m\to-m (as for a magnet with no external field), only even powers survive.
The Landau free energy — the coefficient a(T) changes sign exactly at T_c.
- Equilibrium minimizes f: set \partial f/\partial m = 0, giving 2a\,m + 4b\,m^3 = 0.
- Factor: m=0 or m^2 = -a/(2b).
- For T>T_c, a>0, so only m=0 is real — the disordered phase.
- For T<T_c, a<0, so m=\pm\sqrt{a_0(T_c-T)/2b} — order switches on continuously.
- Therefore m\propto(T_c-T)^{1/2}: mean-field theory predicts the exponent \beta=1/2.
The van der Waals gas is the mean-field model for the liquid–gas transition. Below T_c its isotherms develop the tell-tale unstable wiggle; the critical point is fixed by the two conditions that the isotherm has a horizontal inflection.
The van der Waals equation of state and the critical-point conditions that locate (T_c,P_c,V_c).
Why mean field fails, and the renormalization group
The clue is the diverging correlation length \xi. At the critical point fluctuations of every size coexist, and the system looks statistically the same at every magnification — it is scale-invariant. Mean field, by averaging fluctuations away, is blind to this self-similarity, and that is precisely why its exponents come out wrong.
The renormalization group (Wilson) turns 'zoom out and rescale' into a calculation: repeatedly coarse-grain the system, absorbing short-wavelength fluctuations into effective couplings. Critical points are fixed points of this flow, and any system flowing to the same fixed point inherits the same exponents. That is why universality holds, and why only dimension and symmetry survive the coarse-graining.