---
title: "Same Free Energy, Different Clock Interactive Lab | Sandbox Physics"
description: "Can two systems have the same free energy and different reaction times? Yes."
canonical_url: "https://sandboxphysics.com/labs/equilibrium-vs-kinetics/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/equilibrium-vs-kinetics/"
---

FR15-C · Probability and passage time

# Same Free Energy, Different Clock

**Can two systems have the same free energy and different reaction times?** Yes. Increasing friction slows overdamped motion while leaving its equilibrium distribution unchanged. A free-energy profile fixes relative probabilities; reaction times also depend on mobility and on slow coordinates hidden by the projection.

Replay paired thermal paths with different friction. Move a starting probe across the landscape and find how the hidden coordinate changes its chance of reaching the right basin first.

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

Model assumptions

The browser uses a dimensionless two-coordinate potential with reflecting boundaries. Paired unbiased Langevin paths use the same random increments; isotropic friction rescales their physical clock exactly. A reversible finite-grid backward equation approximates the committor. Survival curves have finite-sample uncertainty and a finite observation horizon. These are synthetic model results.

Seeded unbiased model paths

## How long, and which basin first?

Grid residual and restricted mean time

$r_{\rm grid}<10^{-7}$

The grid residual measures convergence of the discrete equation. It does not include grid-spacing error. Restricted mean time integrates survival only to each branch’s scaled horizon.

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

Physics tutorial

## Equilibrium tells you where, dynamics tells you when

**Background** The cyan and gold particles move on the same potential at the same temperature. The gold particles experience more friction.

**Why it matters** Enhanced sampling can recover equilibrium probabilities after reweighting. The simulated clock changes when the forces or mobility change, so it needs a separate dynamical argument.

Start with the essentials

- **Focus question**: What extra information does a reaction time need?
- **One-sentence intuition**: Equal equilibrium probabilities allow different physical clocks.

### Core mathematical model

#### The overdamped process

$$
d\mathbf r=-\gamma^{-1}\nabla U\,dt+\sqrt{2k_{\rm B}T/\gamma}\,d\mathbf W_t
$$

Coordinates, energy and time use fixed reference units. Friction multiplies both the deterministic and random parts in the way required by thermal equilibrium. Increasing friction by a factor stretches every first-passage time by that factor for this isotropic model.

#### The equilibrium density

$$
\rho_{\rm eq}(x,y)=Z^{-1}e^{-U(x,y)/(k_{\rm B}T)}
$$

Friction is absent from this normalized equilibrium density. Reflecting boundaries keep probability inside the square.

#### The chance of arriving on the right first

$$
q(\mathbf r)=\Pr_{\mathbf r}(\tau_B<\tau_A),\quad\mathcal Lq=0,\quad q|_A=0,\quad q|_B=1
$$

The left basin contains points at or left of the negative threshold, and the right basin contains points at or right of the positive threshold. The browser approximates the backward generator on a reversible grid. The probe can start between the basins.

### Common difficulties

#### Many crossings under a bias

**Typical misconception** A biased trajectory crosses the barrier often, so its crossings directly reveal the real reaction rate.

**Better mental model** A bias changes the forces and therefore the path probabilities. Equilibrium weights alone do not restore physical waiting times. A rate calculation needs a valid dynamical correction and its assumptions.

### Run the experiment

1. 01

   #### Watch equal landscapes with different clocks

   Choose A slower clock, then compare the two colors at the same displayed time.

   **What to observe:** The gold paths have explored less of their paired trajectories. The equilibrium-density difference remains zero.

2. 02

   #### Keep the horizontal coordinate fixed

   Compare Probe the upper channel and Probe the lower channel. Read the right-first probability.

   **What to observe:** The horizontal coordinate is the same, yet the outcomes have different probabilities. The hidden coordinate contains information that the projected profile loses.

3. 03

   #### Respect the observation horizon

   Save the record and count trajectories with no first arrival. Compare Matching clocks with the slower preset.

   **What to observe:** Unarrived paths are right-censored at the horizon. The export reports a restricted mean time, which uses the horizon for those paths; it is not the full mean first-passage time.

Physical conclusion

Use equilibrium weights for probabilities and a specified dynamical model for clocks.

## 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 **How much independent information is in ten thousand frames**](https://sandboxphysics.com/stories/free-energy/trajectories-and-correlation/)
2. [Article **Why equal free energies can give different reaction speeds**](https://sandboxphysics.com/stories/free-energy/committor-and-rates/)

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)