---
title: "Free Energy Hides a Dimension Interactive Lab | Sandbox Physics"
description: "Why does a low potential-energy route sometimes look unlikely in a free-energy profile?"
canonical_url: "https://sandboxphysics.com/labs/projected-free-energy/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/projected-free-energy/"
---

FR15-A · Look behind the projection

# Free Energy Hides a Dimension

**Why does a low potential-energy route sometimes look unlikely in a free-energy profile?** Probability depends on how many configurations fit at each position as well as their energies. A narrow low-energy passage can carry less equilibrium weight than a wider valley. A free-energy profile sums over the hidden configurations.

Rotate a coordinate across a two-dimensional landscape. Follow thermal paths, compare free energy with the lowest potential energy, and open a slice to see the motion hidden by the projection.

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

Model assumptions

The model couples two dimensionless double-well coordinates in a reflecting square. Free energies and conditional densities come from numerical integration; moving beads replay seeded overdamped Langevin trajectories. The height display clips high energies for framing. These are synthetic model results, without molecular calibration.

Independent numerical quadrature

## One coordinate contains many places

Cyan sums the probability along each slice; dashed gold keeps only its lowest potential energy. Click the upper plot to choose a slice.

[Continue the course on free energy and rare events](https://sandboxphysics.com/stories/free-energy/)

Physics tutorial

## A profile sums over the directions you cannot see

**Background** A thermal particle explores a two-dimensional landscape. Recording only one coordinate folds many distinct positions onto the same point.

**Why it matters** A low point in potential energy says where one configuration is cheap. A free-energy profile asks how much probability all compatible configurations carry together.

Start with the essentials

- **Focus question**: What changes when the same landscape is viewed along another coordinate?
- **One-sentence intuition**: Free energy contains the energy and the number of hidden configurations.

### Core mathematical model

#### Choose the visible and hidden coordinates

$$
s=x\cos\theta+y\sin\theta,\qquad h=-x\sin\theta+y\cos\theta
$$

Both configuration coordinates are dimensionless. This is a rotation, so the area element is unchanged. The reflecting square becomes a rotated square in these coordinates.

#### Add the Boltzmann weights along a slice

$$
F(s)=-k_{\rm B}T\ln\!\int_{\Omega_s}e^{-U(s,h)/(k_{\rm B}T)}\,dh+C
$$

The integral includes every allowed hidden coordinate at the chosen position. The additive constant makes the displayed minimum zero. The gold minimum-energy curve is shifted separately.

#### Ask where the hidden coordinate lies

$$
p(h\mid s)=\frac{e^{-U(s,h)/(k_{\rm B}T)}}{\int_{\Omega_s}e^{-U(s,h')/(k_{\rm B}T)}\,dh'}
$$

Two peaks at one visible coordinate mean the projection merges two regions. A free-energy profile alone does not tell how quickly the particle moves between them.

### Common difficulties

#### A probability profile is not a speedometer

**Typical misconception** Once the free-energy barrier is known, the reaction time follows directly.

**Better mental model** A rate also needs mobility and a coordinate that resolves slow motion. The kinetics experiment keeps equilibrium fixed and changes friction to show this directly.

### Run the experiment

1. 01

   #### Look through the middle

   Start with Hidden twin routes. Keep Selected slice at the center and inspect the lower distribution.

   **What to observe:** The hidden coordinate has two preferred regions. The single central value on the horizontal axis contains both.

2. 02

   #### Rotate the projection

   Drag across the terrain or move Projection angle. Compare See along the valley.

   **What to observe:** The terrain stays fixed but its projection changes. Peaks can merge or separate because different positions are added together.

3. 03

   #### Predict the effect of warming

   Compare Warm landscape with the initial preset. Keep the angle fixed when adjusting Thermal energy yourself.

   **What to observe:** Higher-energy configurations carry more weight as temperature rises. The difference between free energy and the minimum energy can change.

Physical conclusion

Inspect the hidden-coordinate distribution before trusting a one-dimensional reaction coordinate.

## Related experiments and articles

Free energy and rare events · Reading path

### From Boltzmann probabilities to reaction paths

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

1. [Article **Why the deepest potential well need not be the most populated**](https://sandboxphysics.com/stories/free-energy/boltzmann-and-projection/)
2. [Article **How metadynamics gradually fills a visited valley**](https://sandboxphysics.com/stories/free-energy/well-tempered-metadynamics/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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