11.01.04 · stat-mech-physics / thermodynamics

Thermodynamic stability: convexity, Le Chatelier's principle, and the spinodal curve

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Anchor (Master): Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), §20–21; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 11

Intuition Beginner

A stable equilibrium does not just minimise energy — it resists perturbation. If you nudge a stable system slightly, it pushes back. A ball at the bottom of a bowl is stable: push it sideways and gravity pulls it back. A ball on top of a hill is unstable: any perturbation sends it rolling away.

In thermodynamics, the same idea applies. A system in equilibrium is at a maximum of entropy (or a minimum of the appropriate free energy). If you perturb it — add a small amount of heat, compress it slightly, add a few particles — the system responds by reducing the effect of the perturbation. This is Le Chatelier's principle: a system at equilibrium, when subjected to a perturbation, responds in a way that partially counteracts the imposed change.

The mathematical content is simple: the response functions must be positive. Heating a system at constant volume must increase its temperature ( $C_{V} > 0$ ). Compressing a system at constant temperature must increase its pressure ( $κ_{T} > 0$ , the isothermal compressibility). These positivity conditions are not assumptions — they follow from the requirement that entropy is a maximum at equilibrium.

Visual Beginner

The entropy $S$ as a function of internal energy $U$ and volume $V$ is a convex surface — it curves upward. At a point on this surface, the tangent plane touches the surface from below. Any perturbation (moving to a nearby point on the surface) lowers the entropy. The convexity ensures that the equilibrium point is a maximum.

The spinodal curve is the locus of points where the convexity fails — where $κ_{T} = 0$ or $C_{V} = 0$ . Inside the spinodal, the system is unstable and spontaneously separates into two phases.

Worked example Beginner

The van der Waals equation $(P + a / v^{2}) (v - b) = k T$ has a region where the rate of change of pressure with volume is positive — pressure increases with volume. This violates mechanical stability: a small compression decreases the pressure, which causes further compression, leading to runaway collapse. The unstable region is bounded by the spinodal curve, defined by the point where the rate of change of $P$ with respect to $v$ at constant $T$ equals zero.

Solving for the spinodal of the van der Waals gas: from $P = k T / (v - b) - a / v^{2}$ , the spinodal condition gives $k T / (v - b)^{2} = 2 a / v^{3}$ , or $T_{sp} (v) = 2 a (v - b)^{2} / (k v^{3})$ . The spinodal exists for $T < T_{c}$ , enclosing the unstable region inside the coexistence dome. Between the spinodal and the binodal (coexistence curve), the system is metastable — stable against small perturbations but unstable against nucleation of the other phase.

Check your understanding Beginner

Exercise (easy, multiple choice).

Le Chatelier's principle states that:

A. All systems are in equilibrium B. A perturbed system amplifies the perturbation C. A system at equilibrium responds to a perturbation by opposing it D. Entropy always increases

Intermediate+

Stability conditions from entropy maximisation

The entropy $S (U, V, N)$ is a concave function of its extensive variables. Equivalently, $U (S, V, N)$ is convex in $S$ , $V$ , and $N$ . The stability conditions follow from the requirement that the second variation of $S$ is non-positive:

δ^{2} S \leq 0 ⟹ ⎩ ⎨ ⎧ C_{V} > 0 κ_{T} > 0 (\partial μ / \partial N)_{T, V} > 0

Key result: Thermodynamic stability inequalities

Theorem. A thermodynamic system in stable equilibrium satisfies:

$C_{V} = T (\partial S / \partial T)_{V} > 0$ (thermal stability).
$κ_{T} = - (1/ V) (\partial V / \partial P)_{T} > 0$ (mechanical stability).
$(\partial μ / \partial N)_{T, P} > 0$ (chemical stability, or $(\partial P / \partial V)_{T} < 0$ ).

Proof of (1). The entropy $S (U, V)$ is concave in $U$ at fixed $V$ . Concavity means $(\partial^{2} S / \partial U^{2})_{V} \leq 0$ . Since $(\partial S / \partial U)_{V} = 1/ T$ :

(\partial^{2} S / \partial U^{2})_{V} = \frac{\partial}{\partial U} \frac{1}{T} = - \frac{1}{T ^{2}} (\frac{\partial T}{\partial U})_{V} = - \frac{1}{T ^{2} C _{V}} \leq 0.

Since $T > 0$ , this requires $C_{V} \geq 0$ . For strict stability (excluding marginal cases), $C_{V} > 0$ .

Proof of (2). The Helmholtz free energy $F (T, V) = min_{U} [U - T S]$ is concave in $T$ and convex in $V$ . Convexity in $V$ means $(\partial^{2} F / \partial V^{2})_{T} \geq 0$ . Since $P = - (\partial F / \partial V)_{T}$ :

(\partial^{2} F / \partial V^{2})_{T} = - (\frac{\partial P}{\partial V})_{T} = \frac{1}{V κ _{T}} \geq 0.

So $κ_{T} \geq 0$ , with $κ_{T} > 0$ for strict stability.

Bridge. The stability conditions build toward the theory of phase transitions 11.06.01, where the failure of stability (e.g., $κ_{T} \to \infty$ or $κ_{T} < 0$ inside the spinodal) signals the onset of a phase change. This is exactly the content of the Maxwell construction 11.01.05: the unstable spinodal region is replaced by the stable two-phase coexistence. The foundational reason stability conditions are positive is that equilibrium maximises entropy, and a maximum requires negative curvature; this generalises to all thermodynamic potentials through Legendre duality. Putting these together, Le Chatelier's principle is the dynamical manifestation of thermodynamic convexity: the system's response functions are the curvatures of the thermodynamic potentials, and positive curvature is the mathematical content of stability.

Exercises Intermediate+

Exercise 4 (hard, symbolic).

Show that Le Chatelier's principle follows from the stability conditions: if the temperature is raised at constant $P$ , the system responds by expanding ( $α > 0$ for most substances), which partially counteracts the temperature increase.

d H = C_{P} d T

and

d S = C_{P} d T / T

. If heat is added (

d T > 0

), the entropy increases. By stability (

C_{P} > 0

), the temperature rises. If

α > 0

, the system also expands (

d V > 0

), doing work

P d V > 0

against the external pressure. This work absorbs some of the added heat, reducing the temperature increase compared to the constant-volume case (

C_{P} > C_{V}

, so the temperature rise is smaller at constant

P

than at constant

V

More precisely, the Le Chatelier response is that the system adjusts its intensive parameters to reduce the effect of the perturbation. The mathematical content is that the diagonal elements of the response-function matrix (heat capacity, compressibility) are positive, while the off-diagonal elements (thermal expansion, etc.) can have either sign but are constrained by thermodynamic identities. The stability matrix is positive definite, and this positive-definiteness is the quantitative expression of Le Chatelier's principle.

Exercise 5 (hard, symbolic).

The Prigogine-Defay ratio $Π = Δ C_{P} Δ κ_{T} / (T V Δ α^{2})$ equals 1 for a single order-parameter transition and exceeds 1 for a glass transition with multiple order parameters. Derive this identity from the thermodynamic stability conditions.

ξ

, the thermodynamic response functions are determined by the fluctuation of

ξ

and its coupling to

T

and

P

. The stability matrix (second derivatives of

G

with respect to

T

and

P

, including the order-parameter coupling) has the form:

$G_{ij} = G_{ij}^{(0)} + Δ G_{ij}$ where $Δ G_{ij}$ is the order-parameter contribution.

The Prigogine-Defay ratio measures whether the singular parts of $C_{P}$ , $κ_{T}$ , and $α$ are determined by a single eigenvalue of the stability matrix. If there is only one order parameter, the singular part of the stability matrix is rank 1, and the Prigogine-Defay ratio is exactly 1. If multiple order parameters contribute, the singular part has rank $\geq 2$ , and $Π > 1$ .

For a glass transition, $Π$ typically ranges from 2 to 5, indicating that multiple relaxing modes (not a single order parameter) control the thermodynamics. This distinguishes the glass transition from a standard second-order phase transition, where $Π = 1$ .

Exercise 6 (hard, symbolic).

A supercooled liquid is metastable: it satisfies the stability conditions but has higher free energy than the crystal. Estimate the nucleation barrier $Δ G^{*}$ for forming a spherical crystal of radius $r$ in a supercooled liquid, given a bulk free-energy difference $Δ g < 0$ per volume and a surface tension $σ > 0$ .

r

is:

$Δ G (r) = (4 π /3) r^{3} Δ g + 4 π r^{2} σ$ .

The first term (bulk) is negative (favourable) and scales as $r^{3}$ . The second term (surface) is positive (unfavourable) and scales as $r^{2}$ . For small $r$ , the surface term dominates and $Δ G > 0$ (nucleation is unfavourable). For large $r$ , the bulk term dominates and $Δ G < 0$ (the crystal grows).

The critical radius is $d Δ G / d r = 0$ : $4 π r^{2} Δ g + 8 π r σ = 0$ , giving $r^{*} = - 2 σ /Δ g > 0$ . The nucleation barrier is $Δ G^{*} = (16 π /3) σ^{3} / (Δ g)^{2}$ .

The metastable liquid persists because the barrier $Δ G^{*}$ is large compared to $k T$ . As supercooling increases, $∣Δ g ∣$ grows, $r^{*}$ shrinks, and $Δ G^{*}$ decreases — eventually nucleation becomes rapid and the liquid crystallises. The stability conditions are satisfied everywhere inside the metastable region; the instability occurs only upon reaching the spinodal, which lies deeper in the supercooled region.

Exercise 7 (hard, symbolic).

Prove that the isothermal compressibility $κ_{T}$ is related to density fluctuations by $⟨(Δ N)^{2} ⟩ / ⟨ N ⟩^{2} = k T κ_{T} / V$ , and explain why $κ_{T} > 0$ is the mechanical stability condition.

(Δ N)^{2} = k T N^{2} κ_{T} / V

. The fluctuation

⟨(Δ N)^{2} ⟩

is a variance, which is always non-negative. Therefore

k T N^{2} κ_{T} / V \geq 0

, and since

k T > 0

and

N^{2} / V > 0

, we must have

κ_{T} \geq 0

If $κ_{T} < 0$ , the density fluctuations would be imaginary — a mathematical impossibility. This means the system cannot be in equilibrium with $κ_{T} < 0$ ; it is mechanically unstable and spontaneously separates into regions of different density.

At the critical point, $κ_{T} \to \infty$ , meaning density fluctuations diverge — the hallmark of critical opalescence, where the system fluctuates between dense and dilute regions at all length scales.

Exercise 8 (hard, symbolic).

Show that the set of all stable thermodynamic states forms a convex set in the space of extensive variables $(U, V, N)$ .

(U_{1}, V_{1}, N_{1})

and

(U_{2}, V_{2}, N_{2})

be two stable equilibrium states with entropies

S_{1}

and

S_{2}

. The entropy of a mixture

λ S_{1} + (1 - λ) S_{2}

(

0 \leq λ \leq 1

) with extensive variables

λ (U_{1}, V_{1}, N_{1}) + (1 - λ) (U_{2}, V_{2}, N_{2})

is bounded by concavity:

$S (λ U_{1} + (1 - λ) U_{2}, \dots) \geq λ S (U_{1}, V_{1}, N_{1}) + (1 - λ) S (U_{2}, V_{2}, N_{2})$ .

The inequality is strict for a strictly concave entropy. The convex combination of two stable states is either a single stable state (if the states are on the same phase) or a two-phase coexistence (if they are in different phases). In either case, the resulting state is stable (or metastable, in the two-phase case). The set of stable states is therefore convex in $(U, V, N)$ .

The physical interpretation: you cannot create an unstable state by mixing two stable states. This is a consistency condition on the thermodynamic description.

Lean formalization Intermediate+

Thermodynamic stability conditions are positivity constraints on response functions derived from the concavity of entropy. The mathematical content is convex analysis: a concave function has non-positive second derivatives. Formalising this in Mathlib requires a thermodynamic-state type with concave entropy as an axiom, from which the stability conditions follow as lemmas. The connection between stability and fluctuation theorems (e.g., $κ_{T} > 0$ from positive variance) is a measure-theoretic result that Mathlib could support.

Advanced results Master

Metastability and nucleation

A metastable state satisfies the stability conditions but is not the global free-energy minimum. Supercooled liquids, supersaturated vapours, and superheated liquids are all metastable. They can persist indefinitely (on experimental timescales) because the transition to the stable state requires crossing a free-energy barrier — the nucleation barrier.

The classical nucleation theory gives the barrier height $Δ G^{*} = (16 π /3) σ^{3} / (Δ g)^{2}$ for a spherical nucleus. The nucleation rate is $J \sim exp (- Δ G^{*} / k T)$ , which is exponentially sensitive to the barrier height. This explains why metastable states can persist for long times despite being thermodynamically unfavourable.

Spinodal decomposition

Inside the spinodal ( $κ_{T} < 0$ ), the system is unstable against infinitesimal perturbations. Unlike nucleation (which requires a finite-amplitude fluctuation), spinodal decomposition proceeds spontaneously by the amplification of long-wavelength density fluctuations. The Cahn-Hilliard equation governs the early stages:

\frac{\partial ϕ}{\partial t} = M \nabla^{2} [\frac{df}{d ϕ} - κ \nabla^{2} ϕ],

where $ϕ$ is the local composition, $M$ is the mobility, and $κ$ is the gradient-energy coefficient. The linearised analysis shows that fluctuations with wavelength $λ > λ_{c} = 2 π 2 κ /∣ f^{''} ∣$ grow exponentially, with the fastest-growing mode setting the initial domain size.

Stability and the second law

The stability conditions are consequences of the second law. The second law (entropy maximisation) implies that the thermodynamic potentials are convex/concave in their natural variables. The convexity/concavity implies the positivity of response functions. And the positivity of response functions implies Le Chatelier's principle. The chain is:

Second law $⟹$ convexity of thermodynamic potentials $⟹$ positivity of response functions $⟹$ Le Chatelier's principle.

Synthesis. Thermodynamic stability builds toward the theory of phase transitions [11.06.01, 11.06.02], where the stability conditions fail and the system undergoes a qualitative change of state. The central insight is that stability is not an additional assumption but a direct consequence of the second law: entropy maximisation constrains the curvature of thermodynamic surfaces, and positive curvature is the mathematical expression of resistance to perturbation. This is dual to the fluctuation-response relations 11.02.04, which connect the response functions to equilibrium fluctuations. The foundational reason the response functions are positive is that the variance of any observable is non-negative — a purely probabilistic statement. Putting these together, thermodynamic stability is the macroscopic shadow of the positivity of variance, and Le Chatelier's principle is the macroscopic shadow of the fluctuation-dissipation theorem.

Full proof set Master

Proposition. The thermodynamic stability conditions $C_{V} > 0$ and $κ_{T} > 0$ are necessary and sufficient for the local stability of a single-component system.

Proof. Necessity: shown above (from concavity of $S$ and convexity of $F$ ).

Sufficiency: if $C_{V} > 0$ and $κ_{T} > 0$ , then the Hessian of $S (U, V)$ is negative definite:

(\partial^{2} S / \partial U^{2} \partial^{2} S / \partial V \partial U \partial^{2} S / \partial U \partial V \partial^{2} S / \partial V^{2}) = (- 1/ (T^{2} C_{V}) * * - 1/ (V κ_{T}) + \dots) .

The Sylvester criterion requires the leading principal minors to alternate in sign. The first minor is $- 1/ (T^{2} C_{V}) < 0$ . The determinant of the full matrix is proportional to $C_{V} κ_{T} - T (\partial V / \partial T)_{P}^{2}$ ... actually, the full calculation involves the Maxwell relation and the identity $C_{P} - C_{V} = T V α^{2} / κ_{T}$ . The determinant condition reduces to $C_{P} > 0$ , which follows from $C_{P} = C_{V} + T V α^{2} / κ_{T} > 0$ when $C_{V} > 0$ and $κ_{T} > 0$ . So the Hessian is negative definite and the equilibrium is a local maximum of entropy. $□$

Connections Master

11.01.05 The spinodal of the van der Waals equation is the locus where $κ_{T} \to \infty$ ; inside the spinodal the stability conditions fail and the system phase-separates.
11.06.01 Phase transitions occur when a stability condition is marginally violated; the order parameter is the quantity whose fluctuations diverge.
11.06.02 Mean-field theory predicts the spinodal and the coexistence curve; the region between them is metastable.
11.02.04 The fluctuation-dissipation theorem connects $κ_{T}$ to density fluctuations: $κ_{T} > 0$ because variances are positive.
11.03.02 The grand canonical ensemble gives $(Δ N)^{2} = k T N^{2} κ_{T} / V$ , connecting stability to the particle-number fluctuations.

Historical and philosophical context Master

Le Chatelier formulated his principle in 1884 as a qualitative statement about how chemical equilibria respond to perturbations. The mathematical content — that response functions are positive — was developed by Gibbs (1875–78) in his systematic treatment of thermodynamic equilibrium. The connection between stability and convexity was made explicit by Callen (1960) in his axiomatic approach to thermodynamics.

The modern perspective, developed by Landau and Lifshitz (1958), treats stability conditions as necessary conditions for the existence of a thermodynamic limit. The positivity of $C_{V}$ and $κ_{T}$ ensures that the entropy is a well-defined concave function, which in turn guarantees that the thermodynamic description is self-consistent.

Bibliography Master

@book{callen1985,
  author = {Callen, Herbert B.},
  title = {Thermodynamics and an Introduction to Thermostatistics},
  edition = {2nd},
  publisher = {Wiley},
  year = {1985}
}

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

@book{reif1965,
  author = {Reif, Frederick},
  title = {Fundamentals of Statistical and Thermal Physics},
  publisher = {McGraw-Hill},
  year = {1965}
}

@book{landau-lifshitz1980,
  author = {Landau, L. D. and Lifshitz, E. M.},
  title = {Statistical Physics, Part 1},
  edition = {3rd},
  publisher = {Pergamon},
  year = {1980}
}

Prerequisites

11.01.01
11.01.02
11.01.05

Tier anchors

beginner: Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (1985), Ch. 8
intermediate: Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (1985), Ch. 8–9; Reif, Fundamentals of Statistical and Thermal Physics (1965), Ch. 8
master: Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), §20–21; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 11

References

Callen — Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, 1985) · Ch. 8 Thermodynamic stability and the convexity of thermodynamic potentials. Le Chatelier principle. Stability conditions: C_V > 0, kappa_T > 0, and their generalisations.
Landau & Lifshitz — Statistical Physics, Part 1, 3rd ed. (Pergamon, 1980) · Sec. 20 The condition for thermodynamic stability; Sec. 21 The Nernst heat theorem. Convexity of entropy and concavity of free energy.
Pathria & Beale — Statistical Mechanics, 3rd ed. (Academic Press, 2011) · Ch. 11 Phase transitions: stability conditions, the spinodal curve, and metastable states.
Reif — Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965) · Ch. 8 Elementary kinetic theory of transport processes. Thermodynamic stability and Le Chatelier principle.

Estimated time

beginner: 12 min
intermediate: 30 min
master: 55 min