---
title: "Finite-size Scaling Interactive Lab | Sandbox Physics"
description: "Can a beautiful data collapse give the wrong critical point?"
canonical_url: "https://sandboxphysics.com/labs/finite-size-scaling/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/finite-size-scaling/"
last_modified: "2026-10-27"
---

F003 · How big is big enough?

# Finite-size Scaling

**Can a beautiful data collapse give the wrong critical point?** Small systems can hide a misplaced critical point because their curves change slowly over the fitted range. A larger size magnifies the same error. Keeping those larger samples aside gives the proposed scaling law a useful test.

Compare Gaussian fields on five lattice sizes. Try a convincing fit on the small lattices, then uncover the larger samples and see whether the prediction survives.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

This Gaussian Euclidean field has fixed zero endpoints and replays sampled configurations.

45 recorded chains · five sizes

## A prediction must work on a new size.

Small sizes used for comparison Larger sizes held aside Dashed: continuum reference

### Before rescaling

$\chi_L=\frac{a^2}{L}\sum_{i,j}\langle\phi_i\phi_j\rangle$

### After rescaling

$x=(r-r_c)L^{1/\nu},\qquad y=\chi_L/L^2$

Comparing recorded samples.

Adjust the proposed scaling law

Proposed critical point $r_c$ $0$ Correlation-length exponent $\nu$ $0.5$

The small sizes set an interpolated comparison curve. Larger sizes never enter that curve. The residual is an RMS difference in combined standard-error units; interpolation creates correlations, so it is a diagnostic rather than a significance test.

Sampling errors and raw records

Each chain stores 16,384 projections of the full field. Neighboring draws are correlated. The displayed uncertainty uses 128-draw batch means; the generator and tests also compare a doubled batch length. The model starts in its stationary Gaussian distribution.

The effective-draw readout uses the nearest saved mass for the selected size. The animated field is a separate live sample from the same model.

Saved scaled mass used for that readout: $rL^2=4$

$q=\frac{a}{\sqrt L}\sum_i\phi_i,\qquad\widehat\chi_L=\overline{q^2}$

[15 sites · raw JSON](https://sandboxphysics.com/data/field-theory/scaling/size-15.json) [23 sites · raw JSON](https://sandboxphysics.com/data/field-theory/scaling/size-23.json) [31 sites · raw JSON](https://sandboxphysics.com/data/field-theory/scaling/size-31.json) [63 sites · raw JSON](https://sandboxphysics.com/data/field-theory/scaling/size-63.json) [127 sites · raw JSON](https://sandboxphysics.com/data/field-theory/scaling/size-127.json)

Model and scientific sources

The field has fixed zero endpoints and a quadratic Euclidean action. Successive samples follow a sampling algorithm, not physical time. This exact Gaussian benchmark has a correlation-length exponent of one half; interacting systems can have different exponents.

This is a one-dimensional free Euclidean scalar field with zero Dirichlet endpoints. The spacing is one reference length unit. The action is dimensionless. At fixed positive mass, correlation length is finite; the benchmark approaches its Gaussian critical limit from nonnegative mass squared. No interacting critical exponent is estimated here.

$S=\frac12\sum_{j=0}^{N}\frac{(\phi_{j+1}-\phi_j)^2}{a}+\frac{ar}{2}\sum_{j=1}^{N}\phi_j^2,\quad L=(N+1)a$

The dashed curve solves the continuum boundary-value problem. It is separate from the recorded lattice data. Finite spacing produces quadratic corrections in spacing divided by interval length.

$\frac{\chi_L}{L^2}\longrightarrow\frac{1-2\tanh(\sqrt{x}/2)/\sqrt{x}}{x},\qquad x=rL^2$

[David Tong · Statistical Field Theory, sections 2.2–2.3](https://www.damtp.cam.ac.uk/user/tong/sft/sfthtml/S2.html)

Physics tutorial

## Test a scaling law on a size it has never seen

**Background** A finite field cannot develop an infinite correlation length. Its endpoints interrupt correlations, and the whole interval can respond together when the mass becomes small.

**Why it matters** Rescaling can reveal one common curve behind several sizes. A small error in the proposed critical point becomes much easier to see on a larger interval.

Start with the essentials

- **Focus question**: How can small-size data collapse convincingly around a wrong critical point?
- **One-sentence intuition**: A critical-point error grows when multiplied by the square of the interval length, so larger held-out sizes test a fit much more sharply.

### Core mathematical model

#### Uniform-source response from fluctuations

$$
q=\frac{a}{\sqrt L}\sum_i\phi_i,\qquad \chi_L=\langle q^2\rangle
$$

The action is quadratic and the mean field is zero. The squared projection measures the same response that a weak uniform source would produce.

#### Gaussian finite-size scaling

$$
\chi_L/L^2\longrightarrow F(rL^2),\qquad F(x)=\frac{1-2\tanh(\sqrt{x}/2)/\sqrt{x}}{x}
$$

The continuum limit keeps $x=rL^2$ fixed as the spacing becomes small compared with the interval. At zero mass, $F(0)=1/12$.

### Common difficulties

#### A neat plot is one part of the test

**Typical misconception** If the three small-size curves sit together, the chosen critical point must be right.

**Better mental model** The wrong critical point used in the preset barely shifts the small sizes. Its shift grows enough on the two larger sizes to separate their curves.

#### Successive samples share information

**Typical misconception** Sixteen thousand stored draws give sixteen thousand independent measurements.

**Better mental model** Each new field retains part of the previous field. The uncertainty uses batches longer than that memory, and the effective number of draws is smaller.

### Run the experiment

1. 01

   #### Predict a collapse

   Choose “One curve, five sizes” and compare the plots before and after rescaling.

   **What to observe:** The unscaled response grows strongly with size. Rescaling brings all five sizes close to the continuum reference.

2. 02

   #### Make a tempting mistake

   Choose “A tempting wrong fit”. Predict where the larger samples will lie.

   **What to observe:** The small-size residual remains close to one even though the proposed critical point has shifted.

3. 03

   #### Use an independent size

   Choose “Ask the larger lattices”. Compare the orange curves with your prediction.

   **What to observe:** The held-out residual rises much more than the small-size residual. The larger interval exposes the wrong critical point.

Physical conclusion

Choose the sizes used to fit before looking at the sizes used to test.

## Related experiments and articles

Fields, scales and critical phenomena · Reading path

### From field modes to critical phenomena

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Testing a convincing data collapse**](https://sandboxphysics.com/learn/field-theory/finite-size-scaling/)
2. [Article **Reading a Wilson–Fisher flow diagram**](https://sandboxphysics.com/learn/field-theory/wilson-fisher-and-universality/)
3. [Lab **Coarse Graining and Renormalization Flow**](https://sandboxphysics.com/labs/renormalization-flow/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)