Dynamic Scaling and the Critical Exponent z

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome back to our study of advanced topics in critical phenomena.

In the last lesson, we took a crucial step from the static, equilibrium world into the realm of dynamics. We introduced the concept of critical slowing down, the dramatic divergence of a system's relaxation time ( $τ_{c}$ ) as it approaches a critical point. We saw that this divergence is governed by a new, independent dynamic critical exponent, $z$ , through the scaling relation $τ_{c} \sim ξ^{z}$ , where $ξ$ is the diverging correlation length.

Today, our goal is to formalize and generalize this idea. We will state the dynamic scaling hypothesis and define the dynamic critical exponent $z$ in its full context. This hypothesis provides a powerful, unified framework for describing how all time- and space-dependent quantities behave near a critical point, much like the static scaling hypothesis does for equilibrium properties.

1. The Dynamic Scaling Hypothesis as a Homogeneity Principle

At its core, scaling theory is built on the idea of self-similarity. Near a critical point, the system looks statistically the same at all length scales, provided we also scale other quantities appropriately. The correlation length $ξ$ is the only relevant length scale.

The dynamic scaling hypothesis extends this principle to include time. It posits that near criticality, there is a unique relationship between the scaling of space and time. If we rescale all lengths by a factor $b$ , we must rescale time by a factor $b^{z}$ to preserve the system's statistical properties. The exponent $z$ is precisely the dynamic critical exponent from our last lesson.

This idea can be expressed formally as a homogeneity condition. Let's explore this.

Dynamic scaling - Grokipedia

The following resource formally defines the scaling hypothesis in dynamics based on this principle of homogeneity. It's the most fundamental way to state the hypothesis.

Please read the section titled 'The scaling hypothesis'. Focus on how a physical quantity's invariance under a simultaneous rescaling of space (x → bx) and time (t → b^z t) leads to a functional equation and, ultimately, a scaling 'ansatz' or form. This formalizes the idea of dynamic self-similarity.

As the reading explains, a dynamic quantity $f (x, t)$ that obeys this scaling must satisfy a relation of the form:

f (b x, b^{z} t) = b^{- θ} f (x, t)

where $θ$ is the scaling dimension of the quantity $f$ . For the order parameter correlation function, $C (x, t) = ⟨ S (x, t) S (0, 0)⟩$ , the scaling dimension is twice that of the order parameter field, so $θ = 2 Δ_{S} = d - 2 + η$ .

This homogeneity principle is the cornerstone of dynamic scaling. All the specific scaling forms we will now look at are direct consequences of it.

2. The Scaling Forms for Correlation and Response

From the homogeneity principle, we can derive the explicit scaling forms for key observables. These forms tell us how a function depends on its variables ( $x, t, ξ$ ) near the critical point. They are the practical expression of the dynamic scaling hypothesis.

[PDF] Phase Transitions and Scaling in Systems Far From Equilibrium

The following text by Uwe C. Täuber presents the concrete mathematical statements of the dynamic scaling hypothesis for the correlation function and the dynamic susceptibility.

Please read the first two paragraphs of Section 2.1, 'Dynamic Scaling Hypothesis and Relaxational Critical Dynamics'. Pay close attention to equations (9) and (11). These are the standard forms used to express the hypothesis.

Let's break down the key equations from the resource you just read. The dynamic scaling hypothesis for the time-dependent correlation function $C (τ, x, t)$ and the dynamic susceptibility $χ (τ, q, ω)$ can be stated as:

Correlation Function (Real Space):
$C (τ, x, t) = ∣ x ∣^{- (d - 2 + η)} C_{\pm} (\frac{∣ x ∣}{ξ}, \frac{t}{τ _{c}})$
Here, $τ = (T - T_{c}) / T_{c}$ is the reduced temperature, $ξ \sim ∣ τ ∣^{- ν}$ is the correlation length, and $τ_{c} \sim ξ^{z} \sim ∣ τ ∣^{- z ν}$ is the characteristic time. $C_{\pm}$ is a universal scaling function (the $\pm$ refers to above/below $T_{c}$ ). This equation tells us that if we measure the correlation, rescale it by the factor $∣ x ∣^{- (d - 2 + η)}$ , and express distance in units of $ξ$ and time in units of $τ_{c}$ , the result is a universal function, independent of the specific material or temperature (as long as it's near $T_{c}$ ).
Dynamic Susceptibility (Fourier Space):
$χ (τ, q, ω) = ∣ q ∣^{- (2 - η)} \overset{χ}{^}_{\pm} (∣ q ∣ ξ, ω τ_{c})$
This is the Fourier-space equivalent. Since your background includes field theory, you'll recognize the utility of this form. The prefactor $∣ q ∣^{- (2 - η)}$ is the scaling of the susceptibility at $T_{c}$ . The arguments of the universal function $\overset{χ}{^}_{\pm}$ show that the relevant variables are wavevector measured in units of $1/ ξ$ and frequency measured in units of $1/ τ_{c}$ .

In both forms, the central role of the dynamic critical exponent $z$ is to connect the scaling of time to the scaling of space via the relation $τ_{c} \sim ξ^{z}$ .

3. Physical Consequences and the Definition of $z$

The dynamic scaling hypothesis is not just a mathematical formalism; it has profound physical consequences. One of the most important arises when we consider the system exactly at the critical point, where $ξ \to \infty$ .

At $T = T_{c}$ , the dependence on $ξ$ in the scaling functions must vanish. Let's look at the characteristic frequency scale from Equation (9) in the resource LINK:

ω_{c} (τ, q) = ∣ q ∣^{z} \overset{ω}{^}_{\pm} (∣ q ∣ ξ)

As $τ \to 0$ , $ξ \to \infty$ . For the expression to remain well-behaved, the scaling function $\overset{ω}{^}_{\pm} (y)$ must approach a constant as its argument $y \to \infty$ . This leaves us with a remarkable prediction for the "dispersion relation" at criticality:

ω_{c} (0, q) \sim ∣ q ∣^{z}

This relation connects the characteristic frequency (or inverse lifetime) of a fluctuation with wavevector $q$ directly to $z$ . This provides a direct way to measure the dynamic exponent.

So, we can now offer a more complete definition:

The dynamic critical exponent, $z$ , is the universal exponent that determines the relationship between the scaling of time and space at a critical point. It is defined by the critical dispersion relation $ω_{c} \sim q^{z}$ and the scaling of the characteristic relaxation time with the correlation length, $τ_{c} \sim ξ^{z}$ .

The value of $z$ depends on the dynamic universality class, which, as we saw last time, is determined by symmetries, dimension, and crucial conservation laws. For example:

Model A (non-conserved order parameter, e.g., Ising magnet): $z \approx 2 + cη$ .
Model B (conserved order parameter, e.g., binary alloy): $z \approx 4 - η$ .
Model J (isotropic ferromagnet with conserved magnetization): $z = (d + 2 - η) /2$ .

This shows that systems with identical static critical behavior (same $η$ ) can have completely different dynamic exponents.

Test your understanding!

Let's use the dynamic scaling hypothesis for the susceptibility to check for consistency with static scaling. The static susceptibility is defined as $χ_{s t a t i c} (q) = χ (q, ω = 0)$ .

Starting from the scaling form $χ (q, ω) = ∣ q ∣^{- (2 - η)} \overset{χ}{^} (∣ q ∣ ξ, ω ξ^{z})$ , derive the scaling form for $χ_{s t a t i c} (q)$ . Does it match the known result from static scaling theory, $χ_{s t a t i c} (q) \sim ∣ q ∣^{- (2 - η)} g (∣ q ∣ ξ)$ ?

Show answer

Yes, it is consistent. To find the static susceptibility, we simply set $ω = 0$ in the dynamic scaling form:

χ_{s t a t i c} (q) = χ (q, ω = 0) = ∣ q ∣^{- (2 - η)} \overset{χ}{^} (∣ q ∣ ξ, 0 \cdot ξ^{z})

χ_{s t a t i c} (q) = ∣ q ∣^{- (2 - η)} \overset{χ}{^} (∣ q ∣ ξ, 0)

Let's define a new function $g (y) = \overset{χ}{^} (y, 0)$ . This new function only depends on the first argument of $\overset{χ}{^}$ . Substituting this in, we get:

χ_{s t a t i c} (q) \sim ∣ q ∣^{- (2 - η)} g (∣ q ∣ ξ)

This is precisely the correct scaling form for the static, wavevector-dependent susceptibility. The dynamic scaling hypothesis correctly reduces to the static hypothesis in the zero-frequency limit.

Conclusion

Today we have formalized the ideas from our last lesson into the powerful framework of the dynamic scaling hypothesis. This is the central pillar of the theory of dynamic critical phenomena.

Key Takeaways:

The dynamic scaling hypothesis posits that near a critical point, a system is self-similar under a combined rescaling of space ( $r \to b r$ ) and time ( $t \to b^{z} t$ ).
This principle leads to universal scaling forms for dynamic quantities like the correlation function $C (x, t)$ and susceptibility $χ (q, ω)$ .
The dynamic critical exponent $z$ is formally defined as the exponent relating the scaling of time and space. It manifests in the divergence of the relaxation time $τ_{c} \sim ξ^{z}$ and the critical dispersion relation $ω_{c} \sim q^{z}$ .
The dynamic scaling hypothesis correctly contains static scaling as its $ω \to 0$ limit.

Preview of the Next Lesson:

We have now established the key concepts of both static and dynamic scaling. We've seen how exponents like $ν, η, z$ describe critical behavior. We've also encountered several other exponents ( $α, β, γ, δ$ ). In the next module, we will step back and look at the bigger picture, starting with the Mermin-Wagner theorem, which places a fundamental constraint on the very existence of phase transitions by proving that continuous symmetries cannot be spontaneously broken in low dimensions ( $d \leq 2$ ). This will provide an important context for our subsequent study of the 2D XY model.