Order parameters and the two kinds of transition
A phase transition is a qualitative change in a system's collective state, captured by an order parameter — a quantity that is zero in the disordered phase and nonzero in the ordered one. For a ferromagnet it is the magnetization m; for a liquid-gas transition, the density difference; for a superfluid, the condensate wavefunction.
Transitions come in two kinds. A first-order transition (ice melting) has latent heat and the order parameter jumps discontinuously. A second-order (continuous) transition — the Curie point of a magnet, the critical point of a fluid — has no latent heat; the order parameter rises continuously from zero, but its response functions diverge. Critical phenomena live at these continuous transitions.
The order parameter vanishes as a power law approaching T_c from below. The exponent β is not 1 (a straight line) but a nontrivial number — and it is the same for whole families of systems.
Critical exponents and the diverging correlation length
Near a continuous transition, every observable follows a power law in the reduced temperature t = (T - T_c)/T_c. The exponents \alpha, \beta, \gamma, \nu are the critical exponents — and they, not the messy details, define the physics.
Correlation length ξ, susceptibility χ and heat capacity C all diverge at T_c with their own exponents. The diverging ξ — the size of the largest correlated fluctuation — is the key to everything.
The physical picture: as T \to T_c, patches of order grow until, right at the critical point, correlated fluctuations exist on all length scales simultaneously. This scale-invariance is why a fluid at its critical point turns milky (critical opalescence) — it scatters light of every wavelength.
Universality: why the details don't matter
Here is the astonishing fact. The uniaxial ferromagnet and the liquid-gas critical point — utterly different microscopically — share the same critical exponents. This is universality: near a continuous transition the exponents depend only on a few gross features — the dimensionality of space and the symmetry of the order parameter — and nothing else.
The simplest model in the 3D Ising universality class is the Ising model — spins on a lattice, each \pm 1, favouring alignment with their neighbours. Its exponents (\beta \approx 0.326, \gamma \approx 1.24, \nu \approx 0.630) match real magnets and fluids to remarkable precision, even though a real fluid contains no spins at all.
Landau theory and its honest failure
Landau's mean-field approach expands the free energy in powers of the order parameter, keeping only what the symmetry allows. For a symmetric magnet, only even powers survive.
The Landau free energy. Above T_c the minimum sits at m = 0; below T_c the coefficient of m² turns negative and the minimum moves to nonzero m — a symmetry-breaking phase transition in one equation.
Minimizing predicts β = ½ — the mean-field exponent. Clean, but wrong: for the 3D Ising model the true β is 0.326. Landau theory ignores fluctuations, which dominate exactly where the correlation length diverges.
The renormalization group: where this leads
The resolution — Kadanoff and Wilson's renormalization group (RG) — turns scale-invariance into a calculational tool. Repeatedly coarse-grain the system (average nearby spins into blocks, then rescale) and watch how the effective couplings \{K\} flow.
One RG step shrinks all lengths by a factor b and maps the couplings to new ones. A critical point is a fixed point of this flow (ξ = ∞ is unchanged by rescaling); the exponents come from linearizing the flow near it.
Universality now has a mechanism: different systems flow to the same fixed point, so they inherit the same exponents. The details wash out under coarse-graining exactly as the diverging correlation length suggested. Wilson won the 1982 Nobel Prize for this picture.
Where this leads next: the same RG that tames critical points is the backbone of quantum field theory and the Standard Model, underlies our understanding of superconductivity, and describes phase transitions in the early universe. You have followed one idea — the statistics of the indistinguishable — from counting states all the way to the frontier of theoretical physics.