Classical statistical mechanics

Phase transitions and critical points

A phase transition changes a thermodynamic state and is marked by nonanalytic free energy in the macroscopic limit; a critical point terminates a first-order coexistence line.

\Delta G=0 \text{ at coexistence}, \qquad \xi\sim |T-T_c|^{-\nu}

Definition: Quantities and meaning

At a first-order transition phases coexist and a first derivative of the free energy can jump; latent heat is an example. At a critical point the correlation length ξ grows very large (diverging in idealized models), long-range fluctuations matter, and mean-field descriptions can fail.

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Quantitative relation

\chi\sim |T-T_c|^{-\gamma}

Example: Worked example

A coexistence line ends at point C. Near C, ξ grows and critical opalescence appears. Why is a small-fluctuation approximation around the mean no longer reliable?

Solution

When ξ is large, fluctuations are correlated over long distances, so changes are not independent and their amplitude is not small; critical effects must be treated beyond linearization.

Example: Example: latent heat and an entropy jump

At fixed pressure, water boils reversibly at temperature T with molar vaporization enthalpy L. Since ΔG=0 between coexisting phases and ΔH=L, ΔS_tr=L/T. Using L≈40.7 kJ·mol⁻¹ at T≈373 K gives ΔS_tr≈109 J·mol⁻¹·K⁻¹. Entropy has a finite jump at a first-order transition, unlike the divergence of response functions near a critical point.

Near a critical point, observables often follow power laws in the reduced temperature t=(T−T_c)/T_c. Critical exponents depend not on microscopic details but on a universality class set by features such as dimension and symmetry, explaining why unlike materials can share the same critical behavior. In a finite sample, ξ cannot exceed the system size, so ideal singularities are rounded into finite peaks. Finite-size scaling uses this rounding to infer the critical behavior of the infinite-system limit. For a one-component fluid, the liquid–vapor critical point ends the coexistence curve, where the two phases lose their density distinction; power laws apply only sufficiently close to that point. At criticality, liquid and vapor become thermodynamically indistinguishable, and a path around the endpoint can connect them without crossing a phase boundary.

Quick check

Which quantity typically grows very large at a critical point in continuum models?

What is the equilibrium condition for two coexisting phases?

References

Charles Kittel and Herbert Kroemer (1980). Thermal Physics
L. D. Landau and E. M. Lifshitz (1980). Statistical Physics