11.06.02 · stat-mech-physics / phase-transitions

Mean-field theory of phase transitions: the van der Waals and Curie-Weiss models

shipped3 tiersLean: none

Anchor (Master): Kardar, Statistical Physics of Fields (2007), Ch. 3; Huang, Statistical Mechanics, 2nd ed. (1987), Ch. 14–15

Intuition Beginner

Mean-field theory is the simplest approach to phase transitions. The idea: replace the complicated interactions between many particles with an effective average interaction. Each particle feels the "mean field" produced by all its neighbours, and this field depends on the state of the system as a whole.

For a magnet, each spin feels an effective magnetic field proportional to the average magnetisation of the whole system. If most spins point up, the effective field pushes each spin to point up — a positive feedback loop. Temperature provides the opposing force: thermal fluctuations randomise spins. The competition between ordering (from the mean field) and disordering (from temperature) produces a phase transition at a critical temperature $T_{c}$ .

Below $T_{c}$ , the ordering wins and a spontaneous magnetisation $m$ appears. Above $T_{c}$ , disorder wins and $m = 0$ . The magnetisation grows continuously from zero as you cool below $T_{c}$ , following a power law $m \sim (T_{c} - T)^{1/2}$ .

Mean-field theory gives the same critical exponents for the van der Waals fluid (liquid-gas transition) and the Curie-Weiss magnet (ferromagnetic transition). This universality is not a coincidence — the two systems share the same mathematical structure, with pressure playing the role of magnetic field and density playing the role of magnetisation.

Visual Beginner

The self-consistency equation $m = tanh (T_{c} m / T)$ has a graphical solution. Plot the identity line $y = m$ and the curve $y = tanh (T_{c} m / T)$ . At high temperature ( $T > T_{c}$ ), the tanh curve lies below the identity line and the only intersection is $m = 0$ . At low temperature ( $T < T_{c}$ ), the tanh curve is steeper near the origin and crosses the identity line at three points: $m = 0$ (unstable) and $m = \pm m_{0}$ (stable). The critical temperature is where the two curves are tangent at the origin.

Worked example Beginner

For the Curie-Weiss model, the self-consistency equation is $m = tanh (m T_{c} / T)$ where $T_{c} = J z / k$ ( $J$ is the coupling, $z$ is the number of neighbours). At $T = T_{c} /2$ (half the critical temperature):

$m = tanh (2 m)$ .

Trying $m = 0.96$ : $tanh (1.92) = 0.9576$ . Close but not exact. Trying $m = 0.958$ : $tanh (1.916) = 0.957$ . Converging to $m \approx 0.957$ .

Below $T_{c}$ , the magnetisation rises quickly toward $m = 1$ (fully aligned). At $T = 0$ , $m = 1$ exactly (all spins aligned, no thermal fluctuations).

Check your understanding Beginner

Formal definition Intermediate+

Curie-Weiss model

The Curie-Weiss Hamiltonian for $N$ spins $s_{i} = \pm 1$ is

H = - \frac{J}{2 N} (i = 1 \sum N s_{i})^{2} - h i = 1 \sum N s_{i},

where $J > 0$ is the ferromagnetic coupling and $h$ is the external field. The factor $1/ N$ ensures extensivity (the interaction energy scales as $N$ , not $N^{2}$ ). Every spin interacts with every other spin equally — this is an infinite-range model, not a lattice model with nearest-neighbour interactions.

The partition function is

Z = {s_{i}} \sum exp \frac{β J}{2 N} (i \sum s_{i})^{2} + β h i \sum s_{i} .

Evaluating the sum by the Hubbard-Stratonovich transformation (a Gaussian linearisation trick):

Z = \int_{- \infty}^{\infty} \frac{d x}{2 π / N β J} exp [- \frac{N β J x ^{2}}{2} + N ln (2 cosh (β J x + β h))] .

In the thermodynamic limit $N \to \infty$ , the integral is dominated by its saddle point, obtained by maximising the exponent:

x = tanh (β J x + β h) .

Identifying $m = x$ (the magnetisation per spin), this is the self-consistency equation

m = tanh (\frac{T _{c} m}{T} + \frac{h}{k T}), T_{c} = \frac{J}{k} .

Mean-field free energy

The mean-field free energy per spin is

f (m) = \frac{1}{2} k T_{c} m^{2} - k T ln (2 cosh (T_{c} m / T + h / k T)) .

For $h = 0$ and $T$ near $T_{c}$ , expanding in powers of $m$ :

f (m) \approx - k T ln 2 + \frac{1}{2} k (T - T_{c}) m^{2} + \frac{k T _{c}^{3}}{12 T ^{2}} m^{4} + \dots .

The equilibrium magnetisation minimises $f$ . For $T > T_{c}$ : the minimum is at $m = 0$ . For $T < T_{c}$ : the quartic term stabilises a minimum at $m \neq = 0$ .

Key result: Mean-field critical exponents

Near $T_{c}$ with $h = 0$ :

Order parameter: $m \sim (T_{c} - T)^{β}$ with $β = 1/2$ for $T < T_{c}$ .
Susceptibility: $χ = (\partial m / \partial h)_{h = 0} \sim ∣ T - T_{c} ∣^{- γ}$ with $γ = 1$ .
Critical isotherm: $h \sim m^{δ}$ at $T = T_{c}$ with $δ = 3$ .
Specific heat: discontinuity at $T_{c}$ with $α = 0$ (jump, not divergence).

These are the mean-field (Landau) critical exponents, shared by the van der Waals gas, the Curie-Weiss model, and all mean-field theories regardless of microscopic details.

Key derivation: Self-consistency and critical exponents Intermediate+

Proposition. The Curie-Weiss self-consistency equation $m = tanh (T_{c} m / T)$ has a non-zero solution for $T < T_{c}$ , and the critical exponents are $β = 1/2$ , $γ = 1$ , $δ = 3$ .

Proof. For $h = 0$ : $m = tanh (T_{c} m / T)$ . Near $T_{c}$ , $m$ is small, and $tanh (x) \approx x - x^{3} /3$ . So:

m \approx \frac{T _{c} m}{T} - \frac{1}{3} (\frac{T _{c} m}{T})^{3} .

For $m \neq = 0$ , divide by $m$ : $1 = T_{c} / T - (T_{c} / T)^{3} m^{2} /3$ . Let $t = (T_{c} - T) / T_{c}$ . To first order in $t$ : $T_{c} / T = 1 + t$ , so $1 = 1 + t - (1 + t)^{3} m^{2} /3 \approx 1 + t - m^{2} /3$ . Hence $t = m^{2} /3$ , giving

m = 3 t = 3 (T_{c} - T) / T_{c} \sim (T_{c} - T)^{1/2}, β = 1/2.

For the susceptibility, add a small field $h$ : $m = tanh (T_{c} m / T + h / k T)$ . Linearising in $h$ : $χ = (\partial m / \partial h)_{h = 0} = (T_{c} χ / T + 1) / (k T) / (1 - T_{c} / T)$ ... more carefully:

$χ^{- 1} = k T (1/ T_{c} - 1/ T) / (1/ T_{c}) = k (T - T_{c})$ for $T > T_{c}$ . So $χ = k / (T - T_{c}) \sim (T - T_{c})^{- 1}$ , giving $γ = 1$ .

For the critical isotherm ( $T = T_{c}$ , $h \neq = 0$ ): $m = tanh (m + h / k T_{c})$ . For small $m$ and $h$ : $m \approx m + h / k T_{c} - m^{3} /3$ , so $h / k T_{c} = m^{3} /3$ , hence $h \sim m^{3}$ and $δ = 3$ . $□$

Bridge. The mean-field critical exponents build toward the renormalization group [11.07.01, 11.07.02], which explains why mean-field exponents are universal and when they are correct. This is exactly the content of the Ginzburg criterion 11.06.04: mean-field theory is self-consistent when fluctuations are small compared to the mean-field order parameter, which happens above the upper critical dimension $d_{c} = 4$ . The foundational reason mean-field theory works at all is that the infinite-range interaction (every spin couples to every other spin) suppresses fluctuations by a factor $1/ N$ , and in the thermodynamic limit fluctuations vanish entirely. This generalises to short-range interactions in high enough dimension ( $d > 4$ ) where the coordination number is effectively infinite, putting these together with the Landau theory 11.06.03 provides the universal framework for all mean-field phase transitions.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Map the van der Waals equation near the critical point to the Curie-Weiss model by identifying: $Δ ρ \leftrightarrow m$ , $Δ P \leftrightarrow h$ , $Δ T \leftrightarrow T - T_{c}$ . Show that both have the same Landau free energy form.

f = a (T - T_{c}) m^{2} + b m^{4} - hm

p = (P - P_{c}) / P_{c}

ϕ = (v - v_{c}) / v_{c}

t = (T - T_{c}) / T_{c}

) expands as

p \approx 4 t - 6 tϕ - (3/2) ϕ^{3}

. The Gibbs free energy per molecule is

g = g_{c} + terms involving p and t

. Integrating

p

with respect to

ϕ

gives a free energy of the form

f (ϕ) = - 4 tϕ + 3 t ϕ^{2} + (3/8) ϕ^{4}

With the mapping $ϕ \leftrightarrow - m$ (or $m = - (v - v_{c}) / v_{c}$ , so $m$ measures the density deviation): $f (m) = 4 t m + 3 t m^{2} + (3/8) m^{4}$ . Near $T_{c}$ , the coefficient of $m^{2}$ changes sign: for $T < T_{c}$ it is negative (ordering), for $T > T_{c}$ it is positive (disordering). The pressure term $4 t m$ plays the role of the external field $h$ . This is exactly the Landau form $f = hm + a (T - T_{c}) m^{2} + b m^{4}$ .

Exercise 7 (hard, symbolic).

Compute the mean-field critical exponents for the Blume-Capel model (spin-1 Ising model with spins $s_{i} = - 1, 0, + 1$ and Hamiltonian $H = - J \sum_{⟨ ij ⟩} s_{i} s_{j} + D \sum_{i} s_{i}^{2} - h \sum_{i} s_{i}$ ) in the mean-field approximation.

m = (2 sinh (β J z m + β h)) / (2 cosh (β J z m + β h) + e^{β D})

where

z

is the coordination number.

h_{eff} = J z m + h

Z_{1} = e^{β h_{eff}} + 1 + e^{- β h_{eff}} = 2 cosh (β h_{eff}) + e^{β D}

The magnetisation is $⟨ s ⟩ = \partial ln Z_{1} / \partial (β h_{eff}) = 2 sinh (β h_{eff}) / (2 cosh (β h_{eff}) + e^{β D})$ .

Setting $m = ⟨ s ⟩$ with $h = 0$ : $m = 2 sinh (β J z m) / (2 cosh (β J z m) + e^{β D})$ .

Near the critical point ( $m$ small): $m \approx 2 β J z m / (2 + e^{β D})$ , so $T_{c} = 2 J z / (k (2 + e^{β_{c} D}))$ . This is a self-consistent equation for $T_{c}$ because $β_{c}$ appears on both sides.

The critical exponents are the same mean-field values: $β = 1/2$ , $γ = 1$ , $δ = 3$ . The Blume-Capel model has the same universality class as the Ising model in mean-field theory. The crystal-field $D$ shifts $T_{c}$ but does not change the critical exponents — a manifestation of universality.

Exercise 8 (hard, symbolic).

Show that the Ginzburg criterion for the validity of mean-field theory in $d$ dimensions is $∣ T - T_{c} ∣^{(4 - d) /2} ≫ 1$ (i.e., mean-field theory is valid for $d > 4$ or far from $T_{c}$ ). Use the fluctuation-dissipation theorem and the mean-field correlation length.

⟨(δ m)^{2} ⟩ \sim k T χ / ξ^{d}

(from the fluctuation-dissipation theorem) with the square of the mean-field order parameter

m^{2} \sim ∣ T - T_{c} ∣

m^{2} \sim ∣ T - T_{c} ∣

. The mean-square fluctuation of the order parameter in a correlation volume

ξ^{d}

⟨(δ m)^{2} ⟩ \sim k T χ / ξ^{d}

where

χ \sim ∣ T - T_{c} ∣^{- 1}

and

ξ \sim ∣ T - T_{c} ∣^{- 1/2}

So $⟨(δ m)^{2} ⟩ \sim ∣ T - T_{c} ∣^{- 1} \cdot ∣ T - T_{c} ∣^{d /2} = ∣ T - T_{c} ∣^{(d - 2) /2}$ .

Mean-field theory is self-consistent when $⟨(δ m)^{2} ⟩ ≪ m^{2}$ , i.e., $∣ T - T_{c} ∣^{(d - 2) /2} ≪ ∣ T - T_{c} ∣$ . This gives $∣ T - T_{c} ∣^{(4 - d) /2} ≫ 1$ .

For $d > 4$ : $(4 - d) /2 < 0$ , so $∣ T - T_{c} ∣^{(4 - d) /2} \to \infty$ as $T \to T_{c}$ . Mean-field theory is always valid near $T_{c}$ in $d > 4$ .

For $d < 4$ : $(4 - d) /2 > 0$ , so $∣ T - T_{c} ∣^{(4 - d) /2} \to 0$ as $T \to T_{c}$ . Mean-field theory always fails sufficiently close to $T_{c}$ .

For $d = 4$ : marginal case — logarithmic corrections to mean-field behaviour.

The upper critical dimension for the Ising universality class is $d_{c} = 4$ .

Lean formalization Intermediate+

Mean-field theory requires: (1) a self-consistency equation as a fixed-point condition; (2) a proof that the fixed point bifurcates at a critical temperature; (3) extraction of critical exponents from power-series expansions near the bifurcation. The bifurcation analysis is within Mathlib's scope (implicit function theorem, Taylor expansion), but the physical context (magnetisation as an order parameter, critical exponents as observables) requires a domain-specific framework that does not exist.

Advanced results Master

Universality of mean-field exponents

All mean-field theories — Curie-Weiss, van der Waals, Landau, Bragg-Williams — give the same critical exponents $β = 1/2$ , $γ = 1$ , $δ = 3$ , $α = 0$ , $ν = 1/2$ , $η = 0$ . This universality holds regardless of the microscopic details because mean-field theory replaces all interactions by an average field, erasing information about the lattice structure, the interaction range, and the dimensionality. The only remaining information is the symmetry of the order parameter (scalar for Ising, vector for Heisenberg) and the analytic structure of the free energy.

Mean-field theory for the Heisenberg model

For a model with $O (n)$ symmetry (e.g., the Heisenberg model with $n = 3$ ), the mean-field free energy is $f = a (T - T_{c}) ∣ m ∣^{2} + b ∣ m ∣^{4}$ . The critical exponents are the same as for the Ising model ( $β = 1/2$ , etc.) because the quartic term still stabilises the ordered phase. The $O (n)$ symmetry affects the structure of fluctuations (Goldstone modes for $n \geq 2$ in $d \leq 2$ , by the Mermin-Wagner theorem) but not the mean-field critical exponents.

Relation to exact results

The 1D Ising model has no phase transition at $T > 0$ (exact result), yet mean-field theory predicts $T_{c} = J z / k > 0$ for any coordination $z$ . The failure occurs because 1D fluctuations are too strong. The 2D Ising model has $T_{c} > 0$ but with exponents $β = 1/8$ , $γ = 7/4$ , $δ = 15$ — far from the mean-field values. The 4D Ising model (and above) has mean-field exponents, confirming the Ginzburg criterion.

Synthesis. Mean-field theory builds toward the renormalization group [11.07.01, 11.07.02], which explains both why mean-field exponents are universal and why they fail below $d = 4$ . The central insight is that mean-field theory neglects spatial fluctuations, treating the whole system as uniform. This is dual to the Landau approach 11.06.03, which retains spatial variation but still treats fluctuations perturbatively. The foundational reason mean-field theory gives the correct exponents above $d = 4$ is that the coordination number grows with dimension, making the effective interaction range infinite in high enough dimension; putting these together, mean-field theory is the large- $d$ limit of the exact theory, and the Ginzburg criterion 11.06.04 quantifies the correction at finite $d$ . The generalisation from mean-field to RG is the passage from uniform to scale-dependent order parameters, and this is exactly the content of the block-spin transformation 11.07.02.

Full proof set Master

Proposition. The Landau free energy $f (m) = a (T - T_{c}) m^{2} + b m^{4} - hm$ (with $a, b > 0$ ) reproduces all four mean-field critical exponents.

Proof. Equilibrium: $\partial f / \partial m = 2 a (T - T_{c}) m + 4 b m^{3} - h = 0$ .

Exponent $β$ ( $h = 0$ , $T < T_{c}$ ): $2 a (T - T_{c}) m + 4 b m^{3} = 0$ . Non-zero solution: $m^{2} = a (T_{c} - T) / (2 b)$ , so $m \sim (T_{c} - T)^{1/2}$ , $β = 1/2$ .

Exponent $γ$ ( $h = 0$ , $T > T_{c}$ ): Linearise in $h$ : $m \approx h / (2 a (T - T_{c}))$ . So $χ = m / h = 1/ (2 a (T - T_{c})) \sim (T - T_{c})^{- 1}$ , $γ = 1$ .

Exponent $δ$ ( $T = T_{c}$ ): $4 b m^{3} = h$ , so $h \sim m^{3}$ , $δ = 3$ .

Exponent $α$ ( $h = 0$ ): For $T > T_{c}$ , $m = 0$ and $C = - T \partial^{2} f / \partial T^{2} = 0$ (the free energy is independent of $T$ at $m = 0$ ). For $T < T_{c}$ : $f = a (T - T_{c}) m^{2} + b m^{4} = a (T - T_{c}) \cdot a (T_{c} - T) / (2 b) + b \cdot a^{2} (T_{c} - T)^{2} / (4 b^{2}) = - a^{2} (T_{c} - T)^{2} / (4 b)$ . So $C = - T \partial^{2} f / \partial T^{2} = T a^{2} / (2 b)$ — a constant. The specific heat has a finite jump $Δ C = T_{c} a^{2} / (2 b)$ at $T_{c}$ , giving $α = 0$ (discontinuity). $□$

Connections Master

11.06.01 The Ising model is the microscopic theory; mean-field theory is its approximate solution. The exact 2D Ising solution gives different exponents, demonstrating the limitations of mean-field theory.
11.06.03 Landau theory generalises mean-field theory by expanding the free energy in powers of the order parameter, providing a universal framework for all mean-field phase transitions.
11.06.04 The Ginzburg criterion determines when mean-field theory fails by comparing fluctuation magnitudes to the mean-field order parameter.
11.07.01 The renormalization group explains why mean-field exponents are correct above $d = 4$ and provides the corrected exponents for $d < 4$ .
11.01.05 The van der Waals gas is the fluid analogue of the Curie-Weiss magnet; both are mean-field theories with identical critical exponents.

Historical and philosophical context Master

Pierre Weiss proposed the molecular-field hypothesis in 1907 to explain ferromagnetism. His insight was that each magnetic atom experiences an internal field proportional to the average magnetisation — a self-consistent condition that produces a phase transition. The Weiss theory correctly predicted the existence of a Curie temperature and the general shape of the magnetisation curve, though the critical exponents were later found to be inaccurate in two and three dimensions.

The Curie-Weiss model (an infinite-range Ising model) was introduced as a mathematically tractable version of the Ising model that can be solved exactly. Its partition function is computable by the Hubbard-Stratonovich transformation, and the thermodynamic limit can be taken rigorously. This makes it a benchmark for testing approximation methods and a starting point for understanding when mean-field theory is exact.

The connection between the van der Waals fluid and the Curie-Weiss magnet — the density-magnetisation mapping — was recognised by Landau (1937), who unified all mean-field theories into a single framework based on the order parameter and the symmetry of the free energy.

Bibliography Master

@article{weiss1907,
  author = {Weiss, Pierre},
  title = {L'hypoth\`ese du champ mol\'eculaire et la propri\'et\'e ferromagn\'etique},
  journal = {J. Phys. Th\'eor. Appl.},
  volume = {6},
  pages = {661--690},
  year = {1907}
}

@book{goldenfeld1992,
  author = {Goldenfeld, Nigel},
  title = {Lectures on Phase Transitions and the Renormalization Group},
  publisher = {Addison-Wesley},
  year = {1992}
}

@book{kardar2007,
  author = {Kardar, Mehran},
  title = {Statistical Physics of Fields},
  publisher = {Cambridge University Press},
  year = {2007}
}

@book{pathria-beale2011,
  author = {Pathria, R. K. and Beale, Paul D.},
  title = {Statistical Mechanics},
  edition = {3rd},
  publisher = {Academic Press},
  year = {2011}
}

@book{huang1987,
  author = {Huang, Kerson},
  title = {Statistical Mechanics},
  edition = {2nd},
  publisher = {Wiley},
  year = {1987}
}

Prerequisites

11.04.01
11.06.01
11.01.05

Used in

11.06.03
11.06.04

Tier anchors

beginner: Schroeder, An Introduction to Thermal Physics (2000), Ch. 8 (Phase transitions); Susskind, Statistical Mechanics: The Theoretical Minimum (2014), Lecture 11
intermediate: Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992), Ch. 3–4; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 11
master: Kardar, Statistical Physics of Fields (2007), Ch. 3; Huang, Statistical Mechanics, 2nd ed. (1987), Ch. 14–15

References

Weiss 1907 — Lhypothese du champ moleculaire et la propriete ferromagnetique · J. Phys. Theor. Appl. 6, 661–690. The molecular-field hypothesis: each spin feels an average field proportional to the net magnetisation, giving the self-consistency equation m = tanh(Jzm/kT) for a lattice of coordination z.
Goldenfeld — Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992) · Ch. 3 Mean field theory of phase transitions; Ch. 4 Mean field theory of the Ising model and the van der Waals fluid. Self-consistency equations, critical exponents, and the Maxwell construction.
Kardar — Statistical Physics of Fields (Cambridge, 2007) · Ch. 3 The Landau-Ginzburg Hamiltonian. Mean-field critical exponents, susceptibility, and the breakdown of mean-field theory.
Pathria & Beale — Statistical Mechanics, 3rd ed. (Academic Press, 2011) · Ch. 11 Phase transitions: mean field, exact results, and the renormalization group.
Huang — Statistical Mechanics, 2nd ed. (Wiley, 1987) · Ch. 14 Phase transitions: mean-field theory, critical exponents, and the Landau approach. Ch. 15 The renormalization group.

Estimated time

beginner: 15 min
intermediate: 40 min
master: 75 min