Finite-Size Scaling for Thermodynamic Observables

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the final lesson in our module on computational methods for studying phase transitions.

In our previous lessons, we've built a powerful computational workflow. We can simulate systems like the Ising model efficiently near a critical point using the Wolff algorithm, and we can then use histogram reweighting to extract thermodynamic properties as a continuous function of temperature from a single simulation run. This allows us to precisely locate, for example, the peak of the magnetic susceptibility $χ_{L} (T)$ for a system of finite linear size $L$ .

However, a true phase transition, with its characteristic power-law divergences and non-analyticities, is a phenomenon of the thermodynamic limit, where $L \to \infty$ . Our simulations are always performed on finite systems. How do we bridge this crucial gap? How can we use results from finite systems of size $L = 16, 32, 64$ to deduce the universal critical exponents that describe the infinite system?

This is the central question we will answer today. Our learning outcome is to state the finite-size scaling ansatz for thermodynamic observables like magnetization and susceptibility. This ansatz provides the theoretical framework for extrapolating finite-size simulation data to the thermodynamic limit, forming the cornerstone of modern computational studies of critical phenomena.

1. The Challenge of Finite Size

The defining feature of a critical point is the divergence of the correlation length, $ξ$ . In an infinite system near its critical temperature $T_{c}$ , the correlation length behaves as:

ξ \sim ∣ t ∣^{- ν}

where $t = (T - T_{c}) / T_{c}$ is the reduced temperature and $ν$ is a universal critical exponent.

However, in a computer simulation on a lattice of size $L^{d}$ with periodic boundary conditions, the correlation length cannot grow indefinitely. The maximum possible correlation distance is limited by the system size itself. When $ξ$ becomes comparable to $L$ , its growth is saturated. This finite size acts as a cutoff, "rounding off" the sharp divergences and singularities of the infinite system.

The magnetic susceptibility $χ$ , which diverges at $T_{c}$ in the infinite system, exhibits a finite peak at a size-dependent "pseudocritical" temperature $T_{c} (L)$ .
The magnetization $m$ , which is strictly zero for $T > T_{c}$ (in zero field), shows a smooth crossover to a small but non-zero value in a finite system.

Finite-size scaling theory provides a way to understand and quantify these effects. It posits that near a critical point, the behavior of any physical quantity is governed by the ratio of the two relevant length scales in the problem: the system size $L$ and the bulk correlation length $ξ$ .

2. The Finite-Size Scaling Ansatz

The finite-size scaling (FSS) ansatz is a direct generalization of the scaling hypothesis you've encountered for bulk systems. It treats the inverse system size, $L^{- 1}$ , as another scaling field, analogous to the reduced temperature $t$ and external field $h$ , which drives the system away from the critical point ( $t = 0, h = 0, L^{- 1} = 0$ ).

A very clear and concise derivation of this ansatz is presented in the following resource.

A short review of critical scaling in the Ising model

This document provides a formal and elegant derivation of the finite-size scaling ansatz, starting from the scaling hypothesis for the free energy. It is an excellent summary of the core theoretical idea.

Please read the section titled 'Finite Size Scaling' (page 2). Focus on how the scaling law for the free energy, Eq. (1), is generalized to Eq. (15) to include the system size L. Then, follow the derivation of the scaling form for the magnetization, Eq. (18).

Let's unpack the key results from this reading. The FSS ansatz begins by postulating a scaling form for the singular part of the free energy density, $f_{s in g}$ , in a system of size $L$ :

f_{s in g} (t, h, L) = b^{- d} f_{s in g} (t b^{y_{t}}, h b^{y_{h}}, L^{- 1} b)

Here, $b$ is an arbitrary length rescaling factor, $d$ is the spatial dimension, and $y_{t} = 1/ ν$ and $y_{h} = (β + γ) / ν$ are the thermal and magnetic scaling exponents. The scaling exponent for $L^{- 1}$ is simply 1.

Since $b$ is arbitrary, we can make the clever choice $b = L$ . This eliminates one of the arguments in the scaling function and leads to a more practical form:

f_{s in g} (t, h, L) = L^{- d} f_{s in g} (t L^{1/ ν}, h L^{y_{h}}, 1) = L^{- d} F (t L^{1/ ν}, h L^{y_{h}})

where $F$ is a universal scaling function (depending only on the universality class).

From this fundamental statement, we can derive the FSS ansatz for any thermodynamic observable by taking derivatives.

For Magnetization ( $m$ ):

Taking a derivative with respect to $h$ , we get the magnetization $m = - \partial f / \partial h$ . This leads to:

m (t, h, L) = L^{- d} \cdot L^{y_{h}} F_{h} (t L^{1/ ν}, h L^{y_{h}}) = L^{y_{h} - d} M (t L^{1/ ν}, h L^{y_{h}})

Using the scaling relation $β = (d - y_{h}) ν$ , we find that $y_{h} - d = - β / ν$ . This gives the finite-size scaling ansatz for magnetization:

m (t, h, L) = L^{- β / ν} M (t L^{1/ ν}, h L^{y_{h}})

where $M$ is another universal scaling function.

For Susceptibility ( $χ$ ):

Taking a second derivative with respect to $h$ , we get the susceptibility $χ = \partial m / \partial h$ . This brings down another factor of $L^{y_{h}}$ :

χ (t, h, L) = L^{2 y_{h} - d} X (t L^{1/ ν}, h L^{y_{h}})

Using the scaling relation $γ = (2 y_{h} - d) ν$ , we have $2 y_{h} - d = γ / ν$ . This yields the finite-size scaling ansatz for susceptibility:

χ (t, h, L) = L^{γ / ν} X (t L^{1/ ν}, h L^{y_{h}})

These two equations for $m$ and $χ$ (often considered at $h = 0$ ) are the central statements of this lesson. They express how observables measured in a finite system relate to the universal critical exponents of the infinite system.

3. Applications in Data Analysis

The FSS ansatz is not just a theoretical curiosity; it is the primary tool for analyzing simulation data near criticality. It predicts two key behaviors that we can look for in our data.

a) Power-Law Scaling at $T_{c}$

Let's set $t = 0$ , placing us exactly at the critical temperature of the infinite system. The scaling functions $M$ and $X$ become constants ( $M (0, 0)$ and $X (0, 0)$ ). The ansatz then predicts a simple power-law dependence on the system size $L$ :
\begin{align*} \label{eq:1}
m(T_c, L) &\propto L^{-\beta/\nu} \
\chi(T_c, L) &\propto L^{\gamma/\nu}
\end{align*}
This is an incredibly powerful result. If we can independently find $T_{c}$ , we can perform simulations on various system sizes $L$ and measure the magnetization and susceptibility exactly at that temperature. A log-log plot of $m (T_{c}, L)$ vs. $L$ should yield a straight line with slope $- β / ν$ , and a similar plot for $χ (T_{c}, L)$ should yield a line with slope $γ / ν$ . This allows for a direct measurement of the critical exponent ratios.

Phase transitions and finite size scaling

This resource provides a practical guide on how to apply FSS, showing how to determine pseudocritical points and extract exponent ratios from the scaling of maxima.

Please read the text from the beginning up to the end of section 'Justifying FSS: order parameter probability distributions' (pages 5-20). This provides a complementary, more phenomenological perspective. Focus on: (1) How the location of the susceptibility maximum, eta_c(V), is argued to scale with L. (2) How the maximum value, \chi_{max}, scales with L. (3) The derivation of the scaling for the root mean square magnetization at T_c.

b) Data Collapse

The FSS ansatz can be rearranged. For susceptibility (at $h = 0$ ):

χ L^{- γ / ν} = X (t L^{1/ ν})

This equation is remarkable. It says that if we take our simulation data for $χ$ from many different system sizes $L$ and temperatures $T$ , and plot the scaled quantity $χ L^{- γ / ν}$ on the y-axis against the scaled quantity $t L^{1/ ν}$ on the x-axis, all the data points should "collapse" onto a single, universal curve—the scaling function $X (x)$ .

This technique, known as data collapse, is a stringent test of the scaling hypothesis and a primary method for determining critical exponents. In practice, one treats $T_{c}$ , $γ$ , and $ν$ as fitting parameters and adjusts them until the best data collapse is achieved.

Finite-Size Scaling of Susceptibility in the 2D Ising Model — Monte Carlo results for the magnetic susceptibility $\chi$ of the 2D Ising model. **(a)** Raw data for different system sizes $L$. The peaks become higher and sharper as $L$ increases. **(b)** The same data, but plotted using scaled variables according to the FSS ansatz. All curves collapse onto a single universal function, beautifully confirming the theory.

Test your understanding!

You are trying to perform a data collapse for the magnetization of the 3D Ising model. You plot $m L^{β / ν}$ versus $(T - T_{c}) L^{1/ ν}$ using data from simulations on systems of size $L = 8, 16, 32, 64$ . You find that the curves for $L = 32$ and $L = 64$ collapse well, but the data for $L = 8$ and $L = 16$ form separate curves. What is the most likely physical reason for this?

Show answer

The most likely reason is that the smaller systems ( $L = 8, 16$ ) are not yet in the asymptotic scaling regime. The finite-size scaling ansatz is strictly valid only in the limit of large $L$ and small $t$ . Smaller systems often exhibit deviations from this ideal behavior due to "corrections to scaling", which are higher-order terms that become negligible for larger $L$ . The good collapse for $L = 32, 64$ suggests that your values for the exponents and $T_{c}$ are likely correct, and these larger systems are properly in the scaling regime.

Conclusion

Today we have established the crucial theoretical link between our finite-size computational results and the universal physics of the thermodynamic limit. The finite-size scaling ansatz is the foundation upon which the modern numerical study of critical phenomena is built.

Key Takeaways:

Phase transitions are sharp only in the thermodynamic limit. Finite system size $L$ rounds off singularities.
The finite-size scaling (FSS) ansatz systematizes this rounding by treating $1/ L$ as a scaling field.
The ansatz for an observable $Q$ that diverges as $∣ t ∣^{- ρ}$ takes the form $Q (t, L) \approx L^{ρ / ν} F (t L^{1/ ν})$ .
For magnetization and susceptibility at $h = 0$ , this becomes:
- $m (t, L) \approx L^{- β / ν} M (t L^{1/ ν})$
- $χ (t, L) \approx L^{γ / ν} X (t L^{1/ ν})$
This ansatz leads to two powerful analysis techniques:
1. Power-law fitting of observables at $T_{c}$ as a function of $L$ to find exponent ratios like $β / ν$ and $γ / ν$ .
2. Data collapse, where plotting scaled variables reveals a single universal curve, confirming the theory and allowing for precise determination of exponents.

Preview of the Next Lesson:

While we can find $T_{c}$ by looking for the best data collapse, this can be a complex multi-parameter fit. It would be useful to have a method that gives a more direct and precise estimate of $T_{c}$ . One such method involves a special quantity constructed to be nearly independent of system size at the critical point. In our next and final lesson of this course, we will learn to use the Binder cumulant method to accurately locate the critical temperature from simulations on different system sizes.