Part III: Phase Transitions | Chapter 1

Phase Transitions

Ehrenfest classification, Landau theory, order parameters, and the physics of first-order and continuous transitions

Historical Context

Phase transitions -- the dramatic transformations between states of matter -- have fascinated scientists since the 18th century. Paul Ehrenfest proposed his classification scheme in 1933, categorizing transitions by the order of the thermodynamic derivative that first shows a discontinuity. While elegant, this classification proved too restrictive for real systems.

Lev Landau revolutionized the field in 1937 with his phenomenological theory based on symmetry and order parameters. His insight that phase transitions involve symmetry breaking -- the ordered phase has lower symmetry than the disordered phase -- became one of the most powerful organizing principles in physics, influencing fields from condensed matter to particle physics and cosmology.

1. Ehrenfest Classification

Derivation 1: Clausius-Clapeyron Equation for First-Order Transitions

A first-order transition has discontinuities in the first derivatives of the Gibbs free energy: entropy \(S = -(\partial G/\partial T)_P\) and volume \(V = (\partial G/\partial P)_T\). At coexistence, the Gibbs free energies of the two phases are equal: \(G_1(T, P) = G_2(T, P)\).

Along the coexistence curve, \(dG_1 = dG_2\):

\[-S_1\,dT + V_1\,dP = -S_2\,dT + V_2\,dP\]

Rearranging:

\[\boxed{\frac{dP}{dT} = \frac{S_2 - S_1}{V_2 - V_1} = \frac{\Delta S}{\Delta V} = \frac{L}{T\Delta V}}\]

where \(L = T\Delta S\) is the latent heat. This is the Clausius-Clapeyron equation. Examples: ice melting (\(\Delta V < 0\), so the slope is negative -- pressure lowers the melting point), water boiling (\(\Delta V > 0\), positive slope).

Classification Summary

First-order transitions: Discontinuities in \(S\), \(V\); latent heat \(L \ne 0\); phase coexistence; metastability (superheating/supercooling); nucleation and growth. Examples: melting, boiling, sublimation.

Second-order (continuous) transitions: \(S\) and \(V\)are continuous; discontinuities or divergences in \(C_P\), \(\kappa_T\),\(\alpha\); no latent heat; no phase coexistence; critical fluctuations. Examples: ferromagnetic transition, superfluid transition, superconducting transition.

2. Order Parameters

Derivation 2: Symmetry Breaking and the Order Parameter Concept

Landau introduced the order parameter \(\phi\) as a thermodynamic quantity that is zero in the disordered (high-symmetry) phase and nonzero in the ordered (low-symmetry) phase:

\[\phi = \begin{cases} 0 & T > T_c \\ \ne 0 & T < T_c \end{cases}\]

The choice of order parameter encodes the symmetry that is broken:

Ferromagnet: \(\phi = M\) (magnetization); breaks rotational symmetry
Liquid-gas: \(\phi = \rho_l - \rho_g\) (density difference); breaks no symmetry -- this transition has a critical point
Superfluid: \(\phi = \langle\hat{\psi}\rangle\) (macroscopic wave function); breaks U(1) gauge symmetry
Superconductor: \(\phi = \Delta\) (gap function); breaks U(1) gauge symmetry
Crystal: \(\phi = \rho_{\mathbf{G}}\) (Fourier component of density); breaks translational symmetry

The dimensionality of the order parameter matters: a scalar (\(n = 1\), Ising), a 2-component vector (\(n = 2\), XY model), or a 3-component vector (\(n = 3\), Heisenberg model) lead to different universality classes.

3. Landau Theory of Continuous Transitions

Figure. Landau free energy F(φ) = aφ² + bφ⁴. Above T_c (blue): single minimum at φ=0. Below T_c (red): two degenerate minima at ±φ₀, signaling spontaneous symmetry breaking.

Derivation 3: Landau Free Energy Expansion

Near a continuous transition, the order parameter \(\phi\) is small. Landau expanded the free energy in powers of \(\phi\), keeping only terms consistent with the symmetry of the system. For a scalar order parameter with\(\phi \to -\phi\) symmetry (e.g., Ising ferromagnet):

\[F(\phi, T) = F_0(T) + a(T)\phi^2 + b\phi^4 + \cdots\]

where \(b > 0\) for stability and \(a(T)\) changes sign at \(T_c\):

\[a(T) = a_0(T - T_c), \qquad a_0 > 0\]

Minimizing \(F\) with respect to \(\phi\):

\[\frac{\partial F}{\partial\phi} = 2a\phi + 4b\phi^3 = 0\]

Solutions:

\[\boxed{\phi_0 = \begin{cases} 0 & T > T_c \\ \pm\sqrt{\frac{-a}{2b}} = \pm\sqrt{\frac{a_0(T_c - T)}{2b}} & T < T_c \end{cases}}\]

The order parameter vanishes continuously as \(\phi_0 \propto (T_c - T)^{1/2}\), defining the mean-field critical exponent \(\beta = 1/2\).

4. First-Order Transitions in Landau Theory

Derivation 4: Including the Cubic Term

When the \(\phi \to -\phi\) symmetry is absent (e.g., liquid-solid transitions), a cubic term is allowed:

\[F = F_0 + a\phi^2 + c\phi^3 + b\phi^4\]

Even with the symmetry, if \(b < 0\), we must include a sixth-order term (\(d\phi^6\) with \(d > 0\)). The free energy:

\[F = F_0 + a\phi^2 + b\phi^4 + d\phi^6, \qquad b < 0, \, d > 0\]

This produces a first-order transition: as \(a\) decreases, a new minimum appears at finite \(\phi\). The transition occurs when the new minimum has the same free energy as \(\phi = 0\):

\[a^* = \frac{3b^2}{16d}, \qquad \phi^* = \sqrt{\frac{-3b}{8d}}\]

The order parameter jumps discontinuously from 0 to \(\phi^*\) at the transition: this is a first-order transition with latent heat.

5. Response to an External Field

Derivation 5: Susceptibility and Critical Exponents

Adding an external field \(h\) conjugate to the order parameter:

\[F = F_0 + a\phi^2 + b\phi^4 - h\phi\]

The equilibrium condition \(\partial F/\partial\phi = 0\) gives:

\[h = 2a\phi + 4b\phi^3\]

The susceptibility \(\chi = \partial\phi/\partial h\). At \(T > T_c\)(\(\phi \to 0\)): \(\chi = 1/(2a) = 1/(2a_0(T - T_c))\), giving\(\chi \propto |T - T_c|^{-1}\) (Curie-Weiss law, \(\gamma = 1\)).

At \(T = T_c\) (\(a = 0\)): \(h = 4b\phi^3\), so\(\phi \propto h^{1/3}\) (\(\delta = 3\)).

Landau (Mean-Field) Critical Exponents

Order parameter: \(\phi \propto (T_c - T)^{\beta}\), \(\beta = 1/2\)

Susceptibility: \(\chi \propto |T - T_c|^{-\gamma}\), \(\gamma = 1\)

Specific heat: \(C \sim \text{discontinuity}\), \(\alpha = 0\) (jump)

Critical isotherm: \(\phi \propto h^{1/\delta}\), \(\delta = 3\)

Limitations of Landau Theory

Landau theory is a mean-field theory that ignores fluctuations. It gives the correct qualitative picture but wrong critical exponents in dimensions \(d < d_c = 4\)(the upper critical dimension). In 3D, the actual exponents are\(\beta \approx 0.326\), \(\gamma \approx 1.237\) for the Ising universality class. The Ginzburg criterion determines when fluctuations become important: they dominate in a region \(|T - T_c| < T_{\text{Gi}}\) around \(T_c\).

6. Applications

Key Applications

Superconductivity: The Ginzburg-Landau theory (1950) describes the superconducting transition using a complex order parameter (the gap function). It successfully predicts magnetic flux quantization and the Abrikosov vortex lattice.

Electroweak symmetry breaking: The Higgs mechanism in particle physics is mathematically identical to Landau theory with a complex doublet order parameter. The Higgs potential has the “Mexican hat” form.

Liquid crystals: The isotropic-to-nematic transition is first-order (cubic term present due to the tensorial nature of the order parameter). The de Gennes theory uses Landau expansion.

Structural phase transitions: Ferroelectric transitions (e.g., BaTiO\(_3\)) are well described by Landau theory with strain coupling.

7. Computational Exploration

This simulation visualizes the Landau free energy landscape, order parameter behavior, susceptibility divergence, and first-order vs continuous transitions.

Phase Transitions: Landau Free Energy, Order Parameters, First-Order vs Continuous

Python

script.py179 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Phase Transitions: Landau Theory Visualization
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Phase Transitions: Landau Theory', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#1a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('#fca5a5')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Landau free energy for continuous transition ---
ax1 = axes[0, 0]
phi = np.linspace(-2, 2, 300)
b = 1.0

colors_t = ['#60a5fa', '#818cf8', '#fbbf24', '#f87171', '#ef4444']
t_labels = ['T >> T_c', 'T > T_c', 'T = T_c', 'T < T_c', 'T << T_c']
a_vals = [2.0, 1.0, 0.0, -1.0, -2.0]

for a_val, color, label in zip(a_vals, colors_t, t_labels):
    F_val = a_val * phi**2 + b * phi**4
    ax1.plot(phi, F_val, color=color, linewidth=2, label=label)

ax1.set_xlabel('Order parameter phi')
ax1.set_ylabel('F - F_0')
ax1.set_title('2nd Order: F = a*phi^2 + b*phi^4')
ax1.set_ylim(-3, 6)
ax1.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Order parameter and susceptibility ---
ax2 = axes[0, 1]
t_range = np.linspace(-2, 2, 500)  # (T - Tc) / Tc
a0 = 1.0
b_val = 1.0

phi_eq = np.where(t_range < 0, np.sqrt(-a0 * t_range / (2 * b_val)), 0.0)
chi_val = np.where(t_range > 0.01, 1.0 / (2 * a0 * t_range),
          np.where(t_range < -0.01, 1.0 / (4 * a0 * np.abs(t_range)), 50.0))
chi_val = np.clip(chi_val, 0, 50)

ax2_twin = ax2.twinx()
ax2.plot(t_range, phi_eq, color='#f87171', linewidth=2.5, label='phi_0(T)')
ax2_twin.plot(t_range, chi_val, color='#60a5fa', linewidth=2, linestyle='--', label='chi(T)')
ax2.axvline(x=0, color='#fbbf24', linestyle=':', alpha=0.5)
ax2.set_xlabel('(T - T_c) / T_c')
ax2.set_ylabel('phi_0', color='#f87171')
ax2_twin.set_ylabel('chi', color='#60a5fa')
ax2_twin.tick_params(colors='#60a5fa')
ax2.tick_params(axis='y', colors='#f87171')
ax2.set_title('Order Parameter and Susceptibility')

# Manual legend
from matplotlib.lines import Line2D
lines = [Line2D([0], [0], color='#f87171', linewidth=2),
         Line2D([0], [0], color='#60a5fa', linewidth=2, linestyle='--')]
ax2.legend(lines, ['phi_0(T)', 'chi(T)'], fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: First-order transition (b < 0, include phi^6) ---
ax3 = axes[1, 0]
phi2 = np.linspace(-2.5, 2.5, 300)
d_val = 0.5  # phi^6 coefficient

b_neg = -1.5  # negative b
a_crit = 3 * b_neg**2 / (16 * d_val)  # transition point

for a_val, color, label in zip([2.0, a_crit + 0.5, a_crit, a_crit - 0.3, -0.5],
                                ['#60a5fa', '#818cf8', '#fbbf24', '#f87171', '#ef4444'],
                                ['a >> a*', 'a > a*', 'a = a* (trans.)', 'a < a*', 'a << a*']):
    F_val = a_val * phi2**2 + b_neg * phi2**4 + d_val * phi2**6
    ax3.plot(phi2, F_val, color=color, linewidth=2, label=label)

ax3.set_xlabel('Order parameter phi')
ax3.set_ylabel('F - F_0')
ax3.set_title('1st Order: F = a*phi^2 + b*phi^4 + d*phi^6')
ax3.set_ylim(-5, 8)
ax3.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Phase diagram (P-T) with first and second order lines ---
ax4 = axes[1, 1]

# Schematic P-T diagram like water
T_triple = 273.16
P_triple = 611.73  # Pa
T_crit = 647.0
P_crit = 2.21e7

# Solid-liquid line (slightly negative slope for water)
T_sl = np.linspace(T_triple, T_triple + 200, 100)
P_sl = P_triple + (T_sl - T_triple) * 1e7 / 100

# Liquid-gas line (Clausius-Clapeyron)
T_lg = np.linspace(T_triple, T_crit, 100)
P_lg = P_triple * np.exp(40.7e3 / 8.314 * (1.0/T_triple - 1.0/T_lg))

# Solid-gas line
T_sg = np.linspace(200, T_triple, 50)
P_sg = P_triple * np.exp(51.0e3 / 8.314 * (1.0/T_triple - 1.0/T_sg))

ax4.semilogy(T_sl, P_sl, color='#f87171', linewidth=2.5, label='Solid-Liquid (1st order)')
ax4.semilogy(T_lg, P_lg, color='#60a5fa', linewidth=2.5, label='Liquid-Gas (1st order)')
ax4.semilogy(T_sg, P_sg, color='#34d399', linewidth=2.5, label='Solid-Gas (1st order)')
ax4.plot(T_crit, P_crit, 'o', color='#fbbf24', markersize=10, zorder=5, label='Critical point')
ax4.plot(T_triple, P_triple, 's', color='#a78bfa', markersize=10, zorder=5, label='Triple point')

ax4.text(320, 2e6, 'LIQUID', color='#60a5fa', fontsize=11, fontweight='bold')
ax4.text(500, 3e4, 'GAS', color='#34d399', fontsize=11, fontweight='bold')
ax4.text(230, 5e6, 'SOLID', color='#f87171', fontsize=11, fontweight='bold')

ax4.set_xlabel('Temperature (K)')
ax4.set_ylabel('Pressure (Pa)')
ax4.set_title('Schematic Phase Diagram (Water-like)')
ax4.set_xlim(180, 700)
ax4.set_ylim(100, 1e9)
ax4.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white', loc='lower right')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

# Numerical results
print("\n" + "=" * 60)
print("PHASE TRANSITIONS: NUMERICAL RESULTS")
print("=" * 60)

print("\n--- Landau Mean-Field Critical Exponents ---")
exponents = [
    ("alpha (specific heat)", "0 (discontinuity)"),
    ("beta (order parameter)", "1/2"),
    ("gamma (susceptibility)", "1"),
    ("delta (critical isotherm)", "3"),
    ("nu (correlation length)", "1/2"),
    ("eta (correlation function)", "0"),
]
for name, val in exponents:
    print("  " + name + " = " + val)

print("\n--- Continuous Transition: Order Parameter ---")
for t in [0.0, -0.1, -0.3, -0.5, -1.0, -2.0]:
    if t >= 0:
        phi_val = 0.0
    else:
        phi_val = np.sqrt(-t / 2.0)
    print("  (T-Tc)/Tc = " + str(t) + ": phi = " + str(round(phi_val, 4)))

print("\n--- First-Order Transition Parameters ---")
b_neg = -1.5
d_val = 0.5
a_star = 3 * b_neg**2 / (16 * d_val)
phi_star = np.sqrt(-3 * b_neg / (8 * d_val))
print("  b = " + str(b_neg) + ", d = " + str(d_val))
print("  a* (transition) = " + str(round(a_star, 4)))
print("  phi* (jump) = " + str(round(phi_star, 4)))
latent_heat = -a_star * phi_star**2 - b_neg * phi_star**4 - d_val * phi_star**6
print("  Latent heat (proportional) = " + str(round(abs(latent_heat), 4)))

print("\n--- Clausius-Clapeyron for Water ---")
L_vap = 2260e3  # J/kg
T_boil = 373.15
rho_l = 958  # kg/m^3 at 100C
rho_g = 0.598  # kg/m^3 at 100C
dPdT = L_vap * rho_l * rho_g / (T_boil * (rho_l - rho_g))
print("  L_vap = 2260 kJ/kg")
print("  dP/dT at 100C = " + str(int(dPdT)) + " Pa/K")
print("  (= " + str(round(dPdT / 133.3, 1)) + " mmHg/K)")
print("\n" + "=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

8. Summary and Key Results

Core Formulas

Clausius-Clapeyron: \(dP/dT = L/(T\Delta V)\)
Landau: \(F = F_0 + a\phi^2 + b\phi^4\)
Order parameter: \(\phi \propto (T_c - T)^{1/2}\)
Susceptibility: \(\chi \propto |T - T_c|^{-1}\)

Physical Insights

Phase transitions involve symmetry breaking
First-order: latent heat, coexistence, metastability
Continuous: diverging fluctuations, no latent heat
Landau theory fails near \(T_c\) in \(d < 4\)

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