Physic Labs

Classical statistical mechanics

Phase transitions and critical points

Probe thermodynamic behavior near the critical point: diverging correlation length ξ∼∣T−Tc∣−ν\xi \sim |T - T_c|^{-\nu} and susceptibility χ∼∣T−Tc∣−γ\chi \sim |T - T_c|^{-\gamma}. Contrast a first-order transition with the critical regime.

Advanced

Equipment

  • Normalized thermodynamic plots of free energy and order parameter
  • Temperature slider sweeping across the critical point
  • Parameter slider adjusting the external field/scale
  • Two quantity readouts for the two figure areas

Procedure

  1. Sweep temperature through the critical point

    Drag the temperature slider from below to above TcT_c: the order parameter or free-energy curve changes continuously but its slope sharpens. At a first-order transition, ΔG=0\Delta G = 0 marks two-phase coexistence; at TcT_c the coexistence line ends.

  2. Gauge the correlation-length divergence

    Read the quantity representing ξ\xi or χ\chi as you approach TcT_c: it balloons as a power law ∣T−Tc∣−ν|T - T_c|^{-\nu}. Try from both sides of TcT_c and note the symmetric behavior — a consequence of critical-exponent universality.

  3. Vary the external parameter and predict

    Raise the parameter slider (acting as external field/pressure): watch the critical feature sharpen or wash out — an external field breaks symmetry and smears the transition. Predict first which signature on the plot tells whether the system has passed the critical point.

Simulation

Experiment history

In 1869 Thomas Andrews measured CO₂ isotherms and discovered the "critical point" — above it gas and liquid are indistinguishable. In 1873 Johannes van der Waals gave an equation of state explaining the coexistence curve and critical point, earning the 1910 Nobel Prize. Pierre Curie showed in 1895 that magnetism too has a critical point (the Curie point). Lev Landau in 1937 built the order parameter into the free energy, unifying phase-transition description; in the 1960s–70s Kenneth Wilson's renormalization group explained the universal exponents ν\nu, γ\gamma — Nobel 1982.

Related physicists

Rudolf ClausiusWilliam Thomson (Lord Kelvin)James Prescott JoulePierre CurieLev Davidovich LandauLudwig Eduard BoltzmannJames Clerk Maxwell

Related library topics

Phase transitions and critical points