Critical Exponent Scaling Relations

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the seventh lesson in our course on the Statistical Mechanics of Phase Transitions.

In the previous lesson, we introduced the scaling hypothesis. We postulated that near a critical point, the singular parts of the free energy and correlation function take the form of generalized homogeneous functions. This powerful assumption elegantly explains phenomena like data collapse and suggests a deep, underlying connection between the various critical exponents.

Today, we will put that hypothesis to work. Starting from the scaling forms for the free energy and correlation functions, we will systematically derive the four famous scaling relations: the Rushbrooke, Widom, Fisher, and Josephson identities. These relations are a cornerstone of modern critical phenomena theory, as they mathematically prove that the six standard critical exponents are not independent. In fact, they reduce the description of a universality class to just two independent exponents.

From Hypothesis to Identities

Let's begin by recalling the two central statements of the scaling hypothesis from our last lesson:

The singular part of the free energy density, $f_{s in g}$ , is a generalized homogeneous function of the reduced temperature $t$ and the ordering field $h$ : $f_{s in g} (t, h) = ∣ t ∣^{2 - α} g_{f} (\frac{h}{∣ t ∣ ^{Δ}})$
The correlation length $ξ$ and correlation function $G (r)$ also obey scaling forms. For our purposes, the key assumptions are:
- The free energy singularity is driven by the divergence of the correlation length: $f_{s in g} \propto ξ^{- d}$ .
- The correlation function at $h = 0$ scales as $G (r, t) = r^{- (d - 2 + η)} G (r / ξ)$ .

Our task is to use these forms to derive relations between the exponents $α, β, γ, δ, ν,$ and $η$ .

Thermodynamic Scaling Relations: Rushbrooke and Widom

The first two relations, named after Rushbrooke and Widom, connect the thermodynamic exponents ( $α, β, γ, δ$ ). They can be derived directly from the scaling form of the free energy. The following video segment provides an excellent walkthrough of these derivations.

6. The Scaling Hypothesis Part 1

In this segment from the MIT OpenCourseWare lecture by Prof. Mehran Kardar, he systematically derives the Rushbrooke and Widom identities by taking derivatives of the assumed scaling form of the free energy. He also verifies the Rushbrooke identity using experimental data.

Please watch from 28:00 to 48:11. Focus on how each thermodynamic quantity (magnetization, susceptibility) is related to a derivative of the free energy, and how this leads to a relation between their corresponding exponents.

Let's summarize and formalize the steps from the video. We start with the free energy scaling form:

f_{s in g} (t, h) = ∣ t ∣^{2 - α} g_{f} (\frac{h}{∣ t ∣ ^{Δ}})

1. Magnetization $m$ :

The magnetization is $m = - \frac{\partial f _{s in g}}{\partial h}$ . Applying the chain rule:

m (t, h) = - ∣ t ∣^{2 - α} \cdot \frac{1}{∣ t ∣ ^{Δ}} \cdot g_{f}^{'} (\frac{h}{∣ t ∣ ^{Δ}}) = - ∣ t ∣^{2 - α - Δ} g_{f}^{'} (\frac{h}{∣ t ∣ ^{Δ}})

To find $β$ : We look at the spontaneous magnetization at $h = 0$ . Assuming $g_{f}^{'} (0)$ is a non-zero constant, we get $m (t, 0) \sim ∣ t ∣^{2 - α - Δ}$ . By definition, $m (t, 0) \sim ∣ t ∣^{β}$ , so we find our first relation: $β = 2 - α - Δ$
To find $δ$ : We look at the critical isotherm $t = 0$ . As we discussed in the previous lesson, for the expression to be independent of $t$ as $t \to 0$ , the scaling function must behave as a power law for large arguments. This leads to the relation (as you can re-derive from the video's logic): $δ = \frac{Δ}{2 - α - Δ} = \frac{Δ}{β}$

2. Susceptibility $χ$ :

The susceptibility is $χ = \frac{\partial m}{\partial h}$ . Differentiating our expression for $m$ :

χ (t, h) = - ∣ t ∣^{2 - α - Δ} \cdot \frac{1}{∣ t ∣ ^{Δ}} \cdot g_{f}^{''} (\frac{h}{∣ t ∣ ^{Δ}}) = - ∣ t ∣^{2 - α - 2Δ} g_{f}^{''} (\frac{h}{∣ t ∣ ^{Δ}})

To find $γ$ : We examine the zero-field limit $h = 0$ . Assuming $g_{f}^{''} (0)$ is a non-zero constant, we have $χ (t, 0) \sim ∣ t ∣^{2 - α - 2Δ}$ . By definition, $χ (t, 0) \sim ∣ t ∣^{- γ}$ , which gives: $- γ = 2 - α - 2Δ ⟹ γ = 2Δ + α - 2$

3. Eliminating the Gap Exponent $Δ$ :

We now have three equations relating the five exponents $α, β, γ, δ, Δ$ . We can eliminate the phenomenological gap exponent $Δ$ to find relations solely among the standard critical exponents.

Rushbrooke Identity: Let's calculate the combination $α + 2 β + γ$ :
$α + 2 β + γ = α + 2 (2 - α - Δ) + (2Δ + α - 2)$ $= α + 4 - 2 α - 2Δ + 2Δ + α - 2 = (α - 2 α + α) + (- 2Δ + 2Δ) + (4 - 2) = 2$
This gives the Rushbrooke identity:
$α + 2 β + γ = 2$
Widom Identity: Let's start with the expression $β (δ - 1)$ . Using $δ = Δ/ β$ :
$β (δ - 1) = β (\frac{Δ}{β} - 1) = Δ - β$
Now, substitute $β = 2 - α - Δ$ :
$Δ - (2 - α - Δ) = 2Δ - 2 + α$
We recognize this as our expression for $γ$ . This gives the Widom identity:
$γ = β (δ - 1)$

Hyperscaling Relations: Josephson and Fisher

The next two relations, Josephson's and Fisher's, are known as hyperscaling relations because they explicitly involve the spatial dimension $d$ . They arise from the scaling hypothesis for correlations.

You can follow the derivations in the next part of the lecture.

6. The Scaling Hypothesis Part 1

Prof. Kardar now introduces the second part of the scaling hypothesis, which connects the free energy to the correlation length. This leads directly to the Josephson and Fisher identities.

Please watch from 48:55 to 1:08:09. Pay close attention to the two key physical arguments: (1) how the free energy density scaling relates to the correlation volume, and (2) how the susceptibility is related to the integral of the correlation function.

Let's write down the derivations. The key ideas are also clearly laid out in the lecture notes.

[PDF] Statistical Mechanics II: Lecture 6: The Scaling Hypothesis

These sections from the MIT lecture notes, 'Statistical Mechanics II', provide the written derivations of the Josephson and Fisher identities, corresponding to the video segment you just watched.

Please read Section III.B ('Divergence of the Correlation Length') and Section III.C ('Critical Correlation Functions and Self-Similarity') on pages 38-40. Focus on Eq. (III.21) for Josephson's identity and Eq. (III.24) for Fisher's identity.

1. Josephson Identity:

This identity connects the thermal exponent $α$ to the correlation length exponent $ν$ . It comes from the physical argument that the singular part of the free energy density, $f_{s in g}$ , is determined by the density of correlation "blobs" of size $ξ$ .

The density of these blobs is $1/ ξ^{d}$ .
Therefore, $f_{s in g} \sim ξ^{- d}$ .
At $h = 0$ , we substitute the defining power laws: $f_{s in g} \sim ∣ t ∣^{2 - α}$ and $ξ \sim ∣ t ∣^{- ν}$ .
$∣ t ∣^{2 - α} \sim (∣ t ∣^{- ν})^{- d} = ∣ t ∣^{d ν}$
Equating the exponents gives the Josephson identity: $2 - α = d ν$

2. Fisher Identity:

This identity connects the susceptibility exponent $γ$ to the correlation exponents $ν$ and $η$ . It relies on the fluctuation-dissipation theorem, which states that the susceptibility is the volume integral of the correlation function.

$χ \sim \int d^{d} r G (r, t)$ .
We use the scaling form for the correlation function at $h = 0$ : $G (r, t) \approx \frac{1}{r ^{d - 2 + η}} G (\frac{r}{ξ})$ .
$χ (t) \sim \int_{0}^{\infty} r^{d - 1} d r [\frac{1}{r ^{d - 2 + η}} G (\frac{r}{ξ})] \sim \int_{0}^{\infty} r^{1 - η} G (\frac{r}{ξ}) d r$ .
Change variables to $y = r / ξ$ , so $r = y ξ$ and $d r = ξ d y$ . $χ (t) \sim \int_{0}^{\infty} (y ξ)^{1 - η} G (y) (ξ d y) = ξ^{2 - η} \int_{0}^{\infty} y^{1 - η} G (y) d y$
The integral is just a universal number, so $χ (t) \sim ξ^{2 - η}$ .
Substitute the power laws $χ \sim ∣ t ∣^{- γ}$ and $ξ \sim ∣ t ∣^{- ν}$ : $∣ t ∣^{- γ} \sim (∣ t ∣^{- ν})^{2 - η} = ∣ t ∣^{- ν (2 - η)}$
Equating exponents gives the Fisher identity: $γ = (2 - η) ν$

Test your understanding!

The four scaling relations are:

$α + 2 β + γ = 2$
$γ = β (δ - 1)$
$2 - α = d ν$
$γ = (2 - η) ν$

Assume you are studying a system in $d = 3$ dimensions and have experimentally determined that $β \approx 0.33$ and $ν \approx 0.63$ . Use the scaling relations to predict the values of the other four exponents: $α, γ, δ, η$ .

Show answer

Given: $d = 3, β = 0.33, ν = 0.63$ .

Find $α$ : Use the Josephson identity ( $2 - α = d ν$ ).
$α = 2 - d ν = 2 - 3 (0.63) = 2 - 1.89 = 0.11$ .
Find $γ$ : Use the Rushbrooke identity ( $α + 2 β + γ = 2$ ).
$γ = 2 - α - 2 β = 2 - 0.11 - 2 (0.33) = 2 - 0.11 - 0.66 = 1.23$ .
Find $δ$ : Use the Widom identity ( $γ = β (δ - 1)$ ).
$δ - 1 = γ / β = 1.23/0.33 \approx 3.73$ .
$δ \approx 4.73$ .
Find $η$ : Use the Fisher identity ( $γ = (2 - η) ν$ ).
$2 - η = γ / ν = 1.23/0.63 \approx 1.95$ .
$η = 2 - 1.95 = 0.05$ .

These predicted values ( $α \approx 0.11, γ \approx 1.23, δ \approx 4.7, η \approx 0.04 - 0.05$ ) are indeed very close to the accepted values for the 3D Ising universality class, demonstrating the predictive power of the scaling hypothesis.

Verification and Consequences

The scaling hypothesis, born from phenomenological insight, makes concrete, testable predictions. Let's verify them against the models we have discussed. The table below (from the resource LINK) summarizes the exponents for key models.

Quantity	Exponent	Mean-Field ( $d \geq 4$ )	2D Ising ( $d = 2$ )	3D Ising ( $d = 3$ )
Specific Heat	$α$	0 (jump)	0 (log)	0.11
Order Parameter	$β$	1/2	1/8	0.33
Susceptibility	$γ$	1	7/4	1.24
Critical Isotherm	$δ$	3	15	4.8
Correlation Length	$ν$	1/2	1	0.63
Correlation Function	$η$	0	1/4	0.04

Verification for the 2D Ising Model ( $d = 2$ ):

Rushbrooke: $α + 2 β + γ = 0 + 2 (1/8) + 7/4 = 1/4 + 7/4 = 8/4 = 2$ . Holds.
Widom: $β (δ - 1) = (1/8) (15 - 1) = 14/8 = 7/4 = γ$ . Holds.
Fisher: $(2 - η) ν = (2 - 1/4) (1) = 7/4 = γ$ . Holds. Note that the non-zero value of $η$ is crucial here.
Josephson: $d ν = 2 (1) = 2$ . And $2 - α = 2 - 0 = 2$ . Holds.

Verification for Mean-Field Theory ( $d \geq 4$ ):

Rushbrooke: $α + 2 β + γ = 0 + 2 (1/2) + 1 = 2$ . Holds.
Widom: $β (δ - 1) = (1/2) (3 - 1) = 1 = γ$ . Holds.
Fisher: $(2 - η) ν = (2 - 0) (1/2) = 1 = γ$ . Holds.
Josephson: $d ν = d (1/2) = d /2$ . And $2 - α = 2 - 0 = 2$ . The relation $d /2 = 2$ only holds if $d = 4$ . For $d > 4$ , where mean-field theory is also supposed to be correct, the identity fails: $d ν > 2 - α$ .

This last point is profound. The thermodynamic scaling relations (Rushbrooke, Widom, Fisher) hold even for mean-field theory. However, the hyperscaling relation (Josephson) breaks down above the upper critical dimension, which is $d_{c} = 4$ for the Ising/ $ϕ^{4}$ class. This breakdown is a deep feature of critical phenomena that we will understand properly when we study the Renormalization Group.

Conclusion

Key Takeaways:

The scaling hypothesis allows for the derivation of four key scaling relations that connect the six standard critical exponents.
The Rushbrooke ( $α + 2 β + γ = 2$ ) and Widom ( $γ = β (δ - 1)$ ) relations connect the thermodynamic exponents.
The Josephson ( $d ν = 2 - α$ ) and Fisher ( $γ = (2 - η) ν$ ) relations, known as hyperscaling relations, involve the spatial dimension $d$ and connect thermodynamic exponents to correlation exponents.
These relations prove that only two critical exponents are independent. Once two are known, the rest are determined.
The relations are satisfied by exact solutions (like the 2D Ising model) and numerical results. Mean-field theory satisfies the thermodynamic relations but violates hyperscaling for $d > 4$ .

Preview of the Next Lesson:

We have now established that diverse physical systems, from magnets to fluids, can be grouped together based on a common set of critical exponents that obey universal relations. This leads directly to our next topic: universality classes. We will formally define what a universality class is and, most importantly, explore the physical properties—such as spatial dimensionality and the symmetry of the order parameter—that determine which class a system belongs to.