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.

Projection angle $0^{\circ}$

Central free-energy rise $3.84$

Hidden-coordinate spread $0.943$

Selected coordinate $0$

Turn the coordinate

Projection angle

\theta

0

Selected slice

s

0

Thermal energy

k_{\rm B}T

1

Coupling of coordinates

c

1.5

Drag across the landscape to turn the cyan direction. Move the white slice to reveal its hidden distribution.

Model, units and reading the scene

$U=a(x^2-1)^2+b(y^2-1)^2+cxy+dx$

$a=b=3,\quad d=0,\quad -1.8\le x,y\le1.8$ defines the initial landscape. The tilted preset changes the linear tilt. Coordinates and energies use fixed reference units; Thermal energy is measured in that energy unit.

Height follows potential energy up to a display ceiling. Thermal beads are configuration samples, not molecules moving through real space. A midpoint line quadrature computes the curves independently of the paths. The free-energy profile and the minimum-energy curve each have their own zero.

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

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 misconceptionOnce the free-energy barrier is known, the reaction time follows directly.

Better mental modelA 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

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.
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.
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.

Free Energy Hides a Dimension

One coordinate contains many places

A profile sums over the directions you cannot see

Core mathematical model

Choose the visible and hidden coordinates

Add the Boltzmann weights along a slice

Ask where the hidden coordinate lies

Common difficulties

A probability profile is not a speedometer

Run the experiment

Look through the middle

Rotate the projection

Predict the effect of warming

Related experiments and articles

From Boltzmann probabilities to reaction paths