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.

Equilibrium-density change $0$

Friction clock ratio $4$

Right-first probability $0.334$

Arrivals within reference horizon $23\,/\,24$

Change the friction

Gold-path friction

\gamma

4

Probe horizontal coordinate

x_0

0

Probe hidden coordinate

y_0

0.9

Thermal energy

k_{\rm B}T

1

Drag the white probe across the landscape. Its chance of reaching the right basin first changes with both coordinates.

Basins, clocks and numerical limits

$A=\{x\le-0.75\},\qquad B=\{x\ge0.75\}$

All first-passage paths begin at the left minimum. The white probe is a separate initial-condition question for the committor. Cyan uses unit friction. Gold replays the same Brownian realizations with a stretched physical clock. This pairing isolates friction; it does not double the number of independent paths.

$\rho_{\rm eq}\propto e^{-U/(k_{\rm B}T)},\quad a=b=3,\quad c=1.5,\quad d=0$

The two-dimensional potential and reflecting square match the projection experiment. Height is clipped above 9.8 energy units. A 45-point grid in each direction approximates the committor; the absorbing basin edge is resolved to one grid spacing. The finite-step Langevin record is a numerical approximation.

Cyan shading is a pointwise 95% Wilson interval for the finite path count. The finite observation horizon leaves some arrival times censored.

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

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 misconceptionA biased trajectory crosses the barrier often, so its crossings directly reveal the real reaction rate.

Better mental modelA 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

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.
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.
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.

Same Free Energy, Different Clock

How long, and which basin first?

Equilibrium tells you where, dynamics tells you when

Core mathematical model

The overdamped process

The equilibrium density

The chance of arriving on the right first

Common difficulties

Many crossings under a bias

Run the experiment

Watch equal landscapes with different clocks

Keep the horizontal coordinate fixed

Respect the observation horizon

Related experiments and articles

From Boltzmann probabilities to reaction paths