---
title: "Coarse Graining and Renormalization Flow Interactive Lab | Sandbox Physics"
description: "Why can a coarser field keep the same long-distance predictions? Integrating out short-scale variables changes the parameters of the remaining field."
canonical_url: "https://sandboxphysics.com/labs/renormalization-flow/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/renormalization-flow/"
last_modified: "2026-10-27"
---

F002 · What survives a change of scale?

# Coarse Graining and Renormalization Flow

**Why can a coarser field keep the same long-distance predictions?** Integrating out short-scale variables changes the parameters of the remaining field. Matching a few long-wavelength quantities can preserve other low-energy predictions, with an error that grows when the coarse model omits important terms.

Keep every second field value and integrate out the sites between them. Match two long-wavelength modes, then test a third that was left out of the match.

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

Euclidean samples and one-loop flow use separate models.

MATCH TWO MODES, TEST ANOTHER

## How much of the spectrum survives?

Exact marginal Dashed: local approximation Gold point: held-out mode

Which modes determine the match?

The zero mode and the first nonzero mode determine the two parameters. The next mode tests the covariance prediction. The full exact marginal is retained in the exported trajectory.

### What the twelve field samples estimate

Sample mean and one standard error Exact covariance Dashed: matched local covariance

Compare gold estimates with the exact covariance.

How the sample uncertainty is calculated

Each point averages Fourier-mode power from the same twelve retained configurations shown in the sample gallery. Whiskers use the sample variance across these independent configurations. Matching uses exact modes zero and one; the sampled points show finite-record uncertainty. That sampling uncertainty and the exact held-out truncation error above are different.

A SEPARATE INTERACTING THEORY

## One direction approaches the Wilson–Fisher point

Thick cyan: tuned critical trajectory Wilson–Fisher fixed point Drag the gold initial point

Drag gold to compare nearby flow trajectories.

Flow interpretation and critical exponents

The thick cyan curve numerically traces the stable direction of Tong’s one-loop equations (section 3.5, Figure 30), with arrows toward the purple point. Thin curves show nearby starting conditions; the small epsilon belongs to this separate model near four dimensions.

$\nu=\frac12+\frac{\epsilon}{12}+O(\epsilon^2)$ $\nu_{\mathrm{leading}}=0.508$

$\gamma=1+\frac{\epsilon}{6}+O(\epsilon^2)$ $\gamma_{\mathrm{leading}}=1.02$

Model equations, source reading and limits

The main model is a periodic one-dimensional Euclidean Gaussian field with an optional higher-derivative term. Samples are independent configurations, not a movie in physical time. A separate one-loop scalar flow explores dimensions close to four.

Lengths use a fixed reference unit; field amplitudes are scaled so that the Euclidean action is dimensionless. The 64-site periodic ring has positive mass and nonnegative bending stiffness. Exact elimination uses a Schur complement. A two-parameter nearest-neighbor action then matches the first two modes. With zero bending stiffness this one-dimensional Gaussian family closes exactly under decimation.

$$
r=\mu^2/\Lambda^2,\quad u=9\widetilde g/(2\pi^2),\quad \frac{dr}{ds}=2r+\frac{u}{3(1+r)},\quad \frac{du}{ds}=\epsilon u-\frac{u^2}{(1+r)^2}
$$

The separate flow keeps only the one-loop scalar mass and quartic coupling in dimensions slightly below four. The plot shows a small-coupling window. The code finds the fixed point of these truncated equations; exponent labels retain only first order in epsilon. No three-dimensional precision exponent or nonperturbative continuum limit is inferred.

[Tong: chapter 3, section 3.5, printed pages 90–91 and Figure 30 (equations and figure read)](https://davidtong.org/pdfs/teaching/statistical-field-theory/sft3.pdf)

[Wilson & Fisher, PRL 28, 240 (1972): the published abstract’s first-order susceptibility exponent (abstract read)](https://doi.org/10.1103/PhysRevLett.28.240)

[Wilson’s 1982 Nobel lecture: the physical meaning of changes of scale](https://www.nobelprize.org/uploads/2018/06/wilson-lecture.pdf)

Physics tutorial

## Coarse graining preserves a distribution on the retained sites

**Background** A long wave passes through many lattice sites. We can ask whether a smaller set of sites contains enough information to predict that wave.

**Why it matters** This is a concrete version of renormalization: eliminate variables, determine the effective parameters, and test a prediction that was not used to determine them.

Start with the essentials

- **Focus question**: How far can two matched long-wavelength modes predict a third?
- **One-sentence intuition**: Integrating out a variable preserves the probability distribution of the variables left behind; replacing the resulting action by a shorter formula introduces an approximation.

### Core mathematical model

#### A positive Gaussian lattice action

$$
S=\frac a2\sum_i\left[m^2\phi_i^2+\left(\frac{\phi_{i+1}-\phi_i}{a}\right)^2+\kappa\left(\frac{\phi_{i+1}-2\phi_i+\phi_{i-1}}{a^2}\right)^2\right]
$$

The ring has 64 sites. Length and field amplitude use fixed reference units, and the Euclidean weight is the exponential of minus the action.

#### Integrating out the eliminated sites

$$
Q_{\mathrm{eff}}=Q_{rr}-Q_{re}Q_{ee}^{-1}Q_{er},\qquad C_{rr}=Q_{\mathrm{eff}}^{-1}
$$

The retained sites keep exactly their original covariance. The field values on the gold row are copied from the corresponding sites of the turquoise row.

#### A two-parameter local approximation

$$
A_{\mathrm{local}}(q)=c_0+4c_1\sin^2(q/2),\qquad q_n=2\pi n/N_{\mathrm{coarse}}
$$

The modes $n=0,1$ determine the two coefficients. The mode $n=2$ is left out and tests the resulting prediction.

### Common difficulties

#### Decimation is not averaging

**Typical misconception** The gold field is the average of each block.

**Better mental model** This experiment retains every second, fourth or eighth site. The probability of the retained values includes the integral over all eliminated values.

#### Two matching points do not validate a curve

**Typical misconception** The fitted spectrum passes through two points, so the coarse model is correct.

**Better mental model** Those points fixed its parameters. Check a third mode and then inspect the shorter-wavelength part of the spectrum.

### Run the experiment

1. 01

   #### Keep the large features

   Choose “Long waves survive” and drag across the field.

   **What to observe:** The gold sites keep the same values as their turquoise counterparts. The fitted short formula stays closest at long wavelengths.

2. 02

   #### Find exact local closure

   Choose “Exact local closure”.

   **What to observe:** With no higher-derivative term, decimating this one-dimensional nearest-neighbor Gaussian chain gives another nearest-neighbor Gaussian chain. The held-out error falls to rounding precision.

3. 03

   #### Test the approximation

   Choose “Too coarse for the prediction”, then compare the gold held-out point with the dashed spectrum.

   **What to observe:** The third mode was not used for matching. Its covariance error measures a real consequence of truncating the effective action.

Physical conclusion

Stable low-energy predictions come from matching and testing an effective theory at a stated resolution.

## 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 **Changing resolution while keeping predictions**](https://sandboxphysics.com/learn/field-theory/keeping-low-energy-physics/)
2. [Article **Reading a Wilson–Fisher flow diagram**](https://sandboxphysics.com/learn/field-theory/wilson-fisher-and-universality/)
3. [Lab **Finite-size Scaling**](https://sandboxphysics.com/labs/finite-size-scaling/)

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)