Critical Isotherm and Order-Parameter Equation of State

A continuous phase transition is organised around its critical point, where the distinction between the two phases vanishes and thermodynamic response functions diverge. Three quantities control the local thermodynamics: the order parameter $m$ (magnetisation per spin, or a liquid–gas density difference), the conjugate ordering field $h$ (magnetic field, or a chemical-potential offset), and the reduced temperature $t=(T-T_c)/T_c$ . The equation of state is the functional relationship $h=h(m,t)$ in the vicinity of $(m,t,h)=(0,0,0)$ . The central claim of the scaling hypothesis is that this equation of state is not an arbitrary function of two variables but a generalized homogeneous function: all singular behaviour is captured by a single scaling function of one combined variable. Setting $t=0$ isolates the critical isotherm, where $h\propto|m|^{\delta}\mathrm{sgn}(m)$ . Setting $h\to0^+$ below $T_c$ gives the spontaneous order $m\propto(-t)^{\beta}$ . Differentiating at fixed $t$ gives the susceptibility $\chi\propto|t|^{-\gamma}$ . The same scaling function generates all of these limits, which is why the exponents obey relations such as Widom's $\gamma=\beta(\delta-1)$ . Mean-field (Landau) theory furnishes the simplest concrete realization: minimizing a free energy $\phi=\tfrac12 a t m^2+\tfrac14 b m^4-hm$ yields $h=a t m+b m^3$ , immediately giving $\delta=3$ , $\beta=\tfrac12$ , $\gamma=1$ . These are correct above the upper critical dimension $d_c=4$ but quantitatively wrong in $d=2,3$ , where fluctuations shift the exponents into the appropriate universality class. The renormalization group explains both the existence of the scaling form and the precise non-classical values. Beyond exponents, the full scaling function $f$ (the Widom–Griffiths function) is itself universal up to two non-universal amplitude scales. Plotting rescaled $h/m^{\delta}$ against $t/m^{1/\beta}$ collapses experimental and numerical data for chemically diverse systems onto one curve — the most direct empirical signature of universality.

Key takeaways

Near a continuous transition the order parameter $m$ , its conjugate field $h$ , and the reduced temperature $t$ are tied by a single scaling equation of state $h=m^{\delta}f(t/m^{1/\beta})$ .
On the critical isotherm ( $t=0$ ) field and order parameter obey the pure power law $h\sim|m|^{\delta}\mathrm{sgn}(m)$ , which defines the exponent $\delta$ .
The exponents $\beta$ (coexistence), $\delta$ (critical isotherm), $\gamma$ (susceptibility) are not independent: the Widom relation $\gamma=\beta(\delta-1)$ ties them together.
Mean-field/Landau theory gives $\delta=3,\ \beta=\tfrac12,\ \gamma=1$ ; the 3D Ising class gives $\delta\approx4.79,\ \beta\approx0.326,\ \gamma\approx1.237$ .
The scaled equation of state collapses all data near criticality onto one universal Widom–Griffiths curve — a sharp prediction of universality.
Key idea: Near criticality the singular equation of state is a generalized homogeneous function $h(m,t)=m^{\delta}f(t/m^{1/\beta})$ ; one scaling function of one variable encodes the critical isotherm, the coexistence curve and the susceptibility, forcing $\gamma=\beta(\delta-1)$ and $\alpha+2\beta+\gamma=2$ .

The core insight

Near criticality the singular part of the equation of state is a generalized homogeneous function $h(m,t)=m^{\delta}f(t/m^{1/\beta})$ . A single scaling function $f$ of one variable encodes the critical isotherm, the coexistence curve, and the susceptibility, forcing the exponent relations $\gamma=\beta(\delta-1)$ and $\alpha+2\beta+\gamma=2$ .

Where this lives today

The scaling equation of state is the working tool for analysing critical data. Experimentalists on fluids near their liquid–gas critical point, ferromagnets, and binary alloys fit their isotherms to the Widom–Griffiths form and test data collapse across systems; a good collapse onto one curve is the sharpest confirmation that the scaling function itself, not just the exponents, is universal. The Arrott plot — $h/m$ versus $m^2$ — remains the standard way to locate $T_c$ in a magnet and to see the crossover from mean-field ( $\delta=3$ , straight lines) to non-classical behaviour. In lattice simulations, finite-size scaling of the magnetization response at $T_c$ delivers $\delta$ , and checking Widom $\gamma=\beta(\delta-1)$ against independently measured $\beta,\gamma$ is a routine internal consistency test.

The physical picture

The critical point is a place where the system has forgotten every preferred scale: the response to a tiny field is anomalously huge because order-parameter fluctuations span all length scales. The power law $h\sim|m|^{\delta}$ is the thermodynamic shadow of this scale invariance — a pure power, with no characteristic field strength. Off-critical there always is a scale, so the $h$ – $m$ curve has finite slope at the origin; exactly at $T_c$ that slope vanishes and the curve flattens into the pure power. Picture the equation-of-state surface $h(m,t)$ near the origin. Mean field draws it as the simple cubic $h=atm+bm^3$ . The scaling hypothesis says the true surface, though distorted by fluctuations, still has the property that rescaling $m\to\lambda m$ can be undone by rescaling $t\to\lambda^{1/\beta}t$ and $h\to\lambda^{\delta}h$ — a self-similarity that pins the whole surface to one curve. Cut the surface three ways — at $t=0$ (critical isotherm $\to\delta$ ), at $h=0,\ t<0$ (coexistence $\to\beta$ ), and take the slope $\partial m/\partial h|_{h=0}$ ( $\to\gamma$ ) — and because all three cuts live on the same homogeneous surface, the exponents cannot be chosen independently.

Figure

Critical isotherm h vs φ on log-log — slope δ at T_c, the sub- and super-critical branches curve away

Universality means the shape of the scaling function $f$ , not just the exponents, is shared by water near its critical point, a uniaxial ferromagnet, and a binary alloy — once you absorb two material-specific amplitude scales. That is the content of the data-collapse plot below.

Figure

Data collapse — rescaled isotherms of chemically distinct systems fall on ONE Widom–Griffiths curve

The core equations

The scaling equation of state expresses the ordering field as $m^{\delta}$ times a universal function of a single combined variable. Everything else in the article is a limit of this one relation.

h = m^{\delta}\, f\!\left(\frac{t}{m^{1/\beta}}\right) \qquad (1)

where: $h$ is the ordering field conjugate to $m$ ; $m$ the order parameter; $t=(T-T_c)/T_c$ ; $\delta$ the critical-isotherm exponent; $\beta$ the coexistence-curve exponent; $f$ the universal scaling function (fixed up to two non-universal amplitudes). Physically: one function of one variable controls all critical thermodynamics. Setting $t=0$ collapses the scaling argument to zero and leaves a pure power law, which defines $\delta$ .

h \propto |m|^{\delta}\,\mathrm{sgn}(m)\qquad (T=T_c) \qquad (2)

where: $\delta$ is the critical-isotherm exponent — mean-field $3$ ; 2D Ising $15$ ; 3D Ising $\approx4.79$ . Physically: at $T_c$ there is no field scale, only a power. Below $T_c$ at zero field the scaling function has a zero that pins the spontaneous order, defining $\beta$ .

m \propto (-t)^{\beta}\qquad (h\to0^+,\ T<T_c) \qquad (3)

where: $\beta$ — mean-field $\tfrac12$ ; 2D Ising $\tfrac18$ ; 3D Ising $\approx0.326$ . The zero-field slope of the isotherm is the susceptibility, diverging with exponent $\gamma$ .

\chi=\left.\frac{\partial m}{\partial h}\right|_{h=0}\propto |t|^{-\gamma} \qquad (4)

where: $\chi$ is the isothermal susceptibility (or fluid compressibility); $\gamma$ — mean-field $1$ ; 2D Ising $\tfrac74$ ; 3D Ising $\approx1.237$ . Because all three exponents come from the one homogeneous function, they satisfy the Widom relation — this is the article's owned structural result.

\gamma = \beta(\delta-1) \qquad (5)

where: $\gamma,\beta,\delta$ are the three equation-of-state exponents. Physically: fix any two and the third is determined. The concrete mean-field realization is the cubic Landau equation of state obtained by minimizing the $\phi^4$ free energy.

h = a\,t\,m + b\,m^{3} \qquad (6)

where: $a,b>0$ are Landau coefficients analytic in $T$ near $T_c$ . Setting $t=0$ gives $h=bm^3$ ( $\delta=3$ ); setting $h=0,\ t<0$ gives $m=\sqrt{-at/b}$ ( $\beta=\tfrac12$ ). The singular free energy itself is homogeneous, introducing the gap exponent and yielding the Rushbrooke relation.

\phi_{\rm sing}(t,h)=|t|^{2-\alpha}\,\Phi\!\left(\frac{h}{|t|^{\Delta}}\right),\quad \Delta=\beta\delta=\beta+\gamma,\qquad \alpha+2\beta+\gamma=2 \qquad (7)

where: $\alpha$ is the specific-heat exponent; $\Delta=\beta\delta$ the gap exponent; $\Phi$ the free-energy scaling function.

Derivation ACritical isotherm and the exponent δ from the scaling form

The goal is to show that the homogeneous equation of state forces a pure power law on the critical isotherm and to identify the universal amplitude that appears there. We take the Widom equation of state, specialize to the critical temperature by setting the reduced temperature to zero, evaluate the scaling function at the origin, and restore the sign symmetry of the ordered phase. The absence of any field scale in the result — a single power with no additive constant — is the thermodynamic fingerprint of the scale invariance that reigns exactly at the critical point.

Step 1. On the critical isotherm the scaling argument vanishes: $x=t/m^{1/\beta}\big|_{t=0}=0$ for any $m$ . Why: the combined variable is what carries temperature dependence; at $t=0$ it is identically zero.

Step 2. Evaluate the scaling function there: $h=m^{\delta}f(0)$ , with $f(0)$ finite and positive so that $h$ increases with $m$ (stability). Why: $\partial h/\partial m>0$ is required for a stable phase, so $f(0)>0$ .

Step 3. Restore the symmetry $m\to-m,\ h\to-h$ of the symmetric phase: $h=f(0)\,|m|^{\delta}\,\mathrm{sgn}(m)$ . Why: the free energy is even in $m$ at $h=0$ , so the equation of state is odd in $m$ .

Step 4. Interpret: there is no characteristic field strength, only a power. The number $f(0)$ is universal up to the two amplitude scales, so the shape is shared across a universality class. Why: scale invariance forbids any additive or multiplicative dimensional constant beyond the two allowed amplitudes. *Verified: verify_aq-sm-critical-isotherm-equation-of-state.py checks that the mean-field $f$ (from $h=atm+bm^3$ ) gives $h=bm^3$ at $t=0$ , i.e. $\delta=3$ , $f(0)=b$ .*

Go deeper · Derivation B · 1 minMean-field exponents β, γ, δ from Landau theory and the amplitude ratio

Derivation BMean-field exponents β, γ, δ from Landau theory and the amplitude ratio

The goal is to derive all three equation-of-state exponents and the universal susceptibility amplitude ratio from a single $\phi^4$ free energy, then confirm they satisfy Widom's relation. We minimize the Landau free energy to obtain the cubic equation of state, then read the critical isotherm, the coexistence curve, and the susceptibility above and below the transition. The amplitude ratio $\Gamma_+/\Gamma_-=2$ that drops out is itself a universal number in mean-field theory, distinct from and finer than the exponents.

Step 1. Minimize $\phi=\tfrac12 atm^2+\tfrac14 bm^4-hm$ : $\partial_m\phi=atm+bm^3-h=0$ , i.e. $h=atm+bm^3$ . Why: equilibrium is the stationary point of the free energy at fixed field.

Step 2. Critical isotherm $t=0$ : $h=bm^3\Rightarrow\delta=3$ . Coexistence $h=0,\ t<0$ : $m=\pm\sqrt{-at/b}\Rightarrow\beta=\tfrac12$ . Why: each limit isolates one exponent by killing one variable.

Step 3. Susceptibility above $T_c$ : $\chi^{-1}=\partial h/\partial m|_{m=0}=at\Rightarrow\chi=(at)^{-1}\sim t^{-1}$ , so $\gamma=1$ . Why: at $m=0$ the cubic term drops, leaving the linear response $at$ .

Step 4. Susceptibility below $T_c$ : at $m^2=-at/b$ , $\chi^{-1}=at+3bm^2=at-3at=-2at\Rightarrow\chi=\tfrac12(-at)^{-1}$ . Same $\gamma=1$ , but amplitude ratio $\Gamma_+/\Gamma_-=2$ . Why: the ordered-phase stiffness is larger by exactly the factor from the cubic term.

Step 5. Check Widom: $\beta(\delta-1)=\tfrac12(3-1)=1=\gamma$ . Why: the three exponents come from one free energy, so the relation must hold identically. *Verified: verify_aq-sm-critical-isotherm-equation-of-state.py derives δ=3, β=½, γ=1 symbolically from h=atm+bm³, confirms Γ₊/Γ₋=2, and checks Widom γ=β(δ−1).*

Go deeper · Derivation C · 1 minFree-energy scaling, the gap exponent, and the Rushbrooke relation

Derivation CFree-energy scaling, the gap exponent, and the Rushbrooke relation

The goal is to derive the equation of state from a homogeneous singular free energy and obtain the gap exponent $\Delta=\beta\delta=\beta+\gamma$ together with the Rushbrooke relation $\alpha+2\beta+\gamma=2$ . We postulate the singular free energy as a generalized homogeneous function of temperature and field, differentiate once for the order parameter and twice for the susceptibility, and match the resulting powers of $|t|$ to the definitions of the exponents. Eliminating the free scaling exponent between two of these relations produces the gap exponent, and the specific-heat exponent falls out of the overall prefactor.

Step 1. Postulate $\phi_{\rm sing}(t,h)=|t|^{2-\alpha}\Phi(h/|t|^{\Delta})$ with a single scaling function $\Phi$ . Why: this is the homogeneity assumption specialized to the free energy; the prefactor sets the specific heat.

Step 2. Order parameter: $m=-\partial\phi/\partial h\sim|t|^{2-\alpha-\Delta}\Rightarrow\beta=2-\alpha-\Delta$ . Why: each $\partial_h$ removes one power $|t|^{-\Delta}$ from the scaling argument.

Step 3. Susceptibility: $\chi=-\partial^2\phi/\partial h^2\sim|t|^{2-\alpha-2\Delta}\Rightarrow-\gamma=2-\alpha-2\Delta$ . Why: a second $\partial_h$ removes another $|t|^{-\Delta}$ .

Step 4. Add the two relations to eliminate $\alpha$ : $\beta+(-\gamma)=(2-\alpha-\Delta)+(2-\alpha-2\Delta)-\dots$ solving gives $\Delta=\beta+\gamma$ ; matching the critical isotherm forces $\Delta=\beta\delta$ , hence $\beta\delta=\beta+\gamma$ (Widom again), and the prefactor gives $\alpha+2\beta+\gamma=2$ (Rushbrooke). Why: two equations in $\alpha,\Delta$ determine the gap exponent and, through the specific-heat prefactor, Rushbrooke. *Verified: verify_aq-sm-critical-isotherm-equation-of-state.py checks Δ=β+γ=βδ and Rushbrooke α+2β+γ=2 for the mean-field set (α=0,β=½,γ=1,δ=3) and the 3D Ising set.*

Limiting cases

Case	Result	Meaning
Critical isotherm	$t=0:\ h=f(0)\|m\|^{\delta}\mathrm{sgn}(m)$	Pure power law defining $\delta$ ; $f(0)$ a finite positive constant.
Coexistence curve	$h\to0^-,\ t<0:\ m\sim(-t)^{\beta}$	$f$ has a zero $x_0<0$ giving spontaneous order; defines $\beta$ .
Susceptibility (symmetric phase)	$t>0:\ \chi=\partial m/\partial h\|_{h=0}\sim t^{-\gamma}$	Small- $m$ expansion of $f$ gives Widom $\gamma=\beta(\delta-1)$ .
Mean-field above $d_c=4$	$\delta=3,\ \beta=\tfrac12,\ \gamma=1$	Landau values; exact for $d\ge4$ where fluctuations are irrelevant.

Worked examples

Example 1 — check Widom's relation across three universality classes. Verify $\gamma=\beta(\delta-1)$ using the standard exponent values.

Step 1. Mean field: $\beta=\tfrac12$ , $\delta=3$ , so $\beta(\delta-1)=\tfrac12\cdot2=1$ . This equals $\gamma=1$ — consistent.

Step 2. 2D Ising (exact): $\beta=\tfrac18$ , $\delta=15$ , so $\beta(\delta-1)=\tfrac18\cdot14=\tfrac{14}{8}=1.75$ . This equals $\gamma=\tfrac74$ — consistent.

Step 3. 3D Ising: $\beta=0.326$ , $\delta=4.789$ , so $\beta(\delta-1)=0.326\cdot3.789=1.235$ . This matches the tabulated $\gamma\approx1.237$ to within rounding. One relation, three classes.

Example 2 — mean-field susceptibility amplitude ratio. Show that in Landau theory the susceptibility is twice as stiff below $T_c$ as above, i.e. $\Gamma_+/\Gamma_-=2$ .

Step 1. Above $T_c$ ( $m=0$ ): $\chi_+^{-1}=at$ , so $\chi_+=\Gamma_+ t^{-1}$ with $\Gamma_+=1/a$ .

Step 2. Below $T_c$ : at $m^2=-at/b=a|t|/b$ , the inverse susceptibility is $\chi_-^{-1}=at+3bm^2=a t+3b(a|t|/b)=at+3a|t|=-a|t|+3a|t|=2a|t|$ . So $\chi_-=\Gamma_-|t|^{-1}$ with $\Gamma_-=1/(2a)$ .

Step 3. The ratio is $\Gamma_+/\Gamma_-=(1/a)/(1/2a)=\mathbf{2}$ . This universal number is finer than the exponent $\gamma=1$ and is itself a prediction of the universality class (the mean-field value; measured ratios in 3D Ising are $\approx4.7$ ).

Origins · 1 mincritical isotherm and how this was discovered

critical isotherm

the T = T_c cut of the equation of state · The critical isotherm is the curve h(m) traced out exactly at the critical temperature, where the ordering field and order parameter obey the pure power law h ∝

History

1873

— van der Waals' equation of state gives the fluid critical point and the mean-field δ = 3

1895

— Pierre Curie measures the ferromagnetic critical isotherm and the Curie temperature

1937

— Landau formulates the general theory of continuous transitions from a symmetric free energy

1944

— Onsager's 2D Ising solution implies the non-classical δ = 15

1965

— Widom proposes the scaling equation of state h = m^δ f(t/m^{1/β})

1967

— Griffiths derives the scaling function form and the Widom–Griffiths analyticity conditions

1972

— Wilson and Fisher compute the ε-expansion exponents, giving 3D Ising δ ≈ 4.8

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 12 · Stanley, Introduction to Phase Transitions and Critical Phenomena, Ch. 10–11 · Kardar, Statistical Physics of Fields, Ch. 4 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 5–7 · Landau & Lifshitz, Statistical Physics Part 1, Ch. XIV.

Provenance. The Widom scaling equation of state (Widom 1965) and the Griffiths analyticity conditions (Griffiths 1967) underpin the article; the mean-field realization follows from the Landau $\phi^4$ free energy (L&L SP1 Ch. XIV; Pathria & Beale Ch. 12). Exact/tabulated exponent values rechecked: mean-field $\beta=\tfrac12,\gamma=1,\delta=3$ ; 2D Ising $\beta=\tfrac18,\gamma=\tfrac74,\delta=15$ ; 3D Ising $\beta\approx0.326,\gamma\approx1.237,\delta\approx4.789$ . Widom $\gamma=\beta(\delta-1)$ and Rushbrooke $\alpha+2\beta+\gamma=2$ are checked against Kardar SPF Ch. 4 and Stanley Ch. 11.

Builds on: Order Parameter · Continuous Phase Transitions · Critical Exponents · Landau Theory of Phase Transitions. Leads to: Scaling Hypothesis and Scaling Relations · Universality and Universality Classes · Correlation Function.

What is the difference between the equation of state here and the scaling hypothesis article?

The scaling hypothesis article owns the general statement — that the singular free energy is a generalized homogeneous function — and derives the full set of scaling relations among all six exponents. This article specializes that idea to the equation of state $h(m,t)$ : it isolates the critical isotherm, defines $\delta$ , and derives the Widom relation $\gamma=\beta(\delta-1)$ as the equation-of-state consequence. In short, scaling hypothesis = the structural principle; critical isotherm / equation of state = its concrete thermodynamic face.

Why does the critical isotherm have zero slope at the origin exactly at T_c?

The slope at $m=0$ is $\partial h/\partial m|_{m=0}=\chi^{-1}$ . Above $T_c$ this is finite and positive; as $T\to T_c^+$ it vanishes because $\chi$ diverges. At exactly $T_c$ the leading behaviour is $h\sim m^{\delta}$ with $\delta>1$ , so $\partial h/\partial m\to0$ as $m\to0$ : the curve is horizontally tangent at the origin. Off-critical the finite $\chi^{-1}$ re-establishes a nonzero slope, which is why only the true critical isotherm shows a pure power.

How do I extract δ from experimental data without knowing T_c precisely?

This is exactly why data collapse and the Arrott plot are used. On an Arrott plot ( $h/m$ versus $m^2$ ) the mean-field isotherms are straight lines and the $T=T_c$ line passes through the origin, so you tune the assumed $T_c$ until that happens; the slope then gives the coefficients and the intercepts test $\delta=3$ . For non-classical systems a modified Arrott plot ( $(h/m)^{1/\gamma}$ vs $m^{1/\beta}$ ) linearizes the data for the correct exponent pair, and $T_c$ is where the critical isotherm passes through the origin. Getting $T_c$ slightly wrong systematically curves the isotherms, which is the standard diagnostic.

Are the amplitude ratios universal too, or only the exponents?

Both, but at different levels. The exponents are the coarsest universal data. The scaling function $f$ and dimensionless amplitude ratios (such as $\Gamma_+/\Gamma_-$ ) are also universal, but finer — they distinguish some systems that share exponents only up to two non-universal scale factors that must first be divided out. Mean-field gives $\Gamma_+/\Gamma_-=2$ ; 3D Ising gives $\approx4.7$ . Measuring amplitude ratios is a more stringent test of a universality-class assignment than exponents alone.

Does the liquid–gas critical point really share a universality class with a magnet?

Yes. Map the density difference $\rho-\rho_c$ to the magnetization $m$ and the chemical-potential offset $\mu-\mu_c$ to the field $h$ . The liquid–gas transition then has a scalar (Ising, $\mathbb{Z}_2$ -like) order parameter in three dimensions, so it falls in the 3D Ising class: measured $\delta\approx4.8$ , $\beta\approx0.33$ , $\gamma\approx1.24$ for CO $_2$ , Xe and water agree with uniaxial ferromagnets. The asymmetry of the coexistence curve (liquid and gas are not perfectly symmetric) introduces small correction-to-scaling terms but does not change the leading exponents.

What is the gap exponent Δ and where does it appear?

The gap exponent is $\Delta=\beta\delta=\beta+\gamma$ ; it is the power of $|t|$ that rescales the field in the free-energy scaling form $\phi_{\rm sing}=|t|^{2-\alpha}\Phi(h/|t|^{\Delta})$ . It sets how the field and temperature scales are locked together and appears whenever you differentiate the free energy with respect to $h$ — each derivative pulls out one factor $|t|^{-\Delta}$ . Its two expressions $\beta\delta$ and $\beta+\gamma$ being equal is precisely Widom's relation, so the gap exponent is the compact way to state that the equation-of-state exponents are not independent.