FR15-B · Windows into a rare crossing

Umbrella Sampling and MBAR

When do overlapping biased trajectories reveal the equilibrium free energy? Each window must sample its own equilibrium distribution and share configurations with its neighbors. MBAR then removes the known biases to estimate equilibrium probabilities. A hidden trapped coordinate can still spoil the result, even when the windows overlap.

Move an umbrella window across a two-dimensional landscape. Combine seeded trajectories, compare the recovered free energy with numerical integration, then leave a gap or trap the hidden coordinate.

Recovered right-side population $0.563$

Block-bootstrap standard error $0.0585$

Correlation-adjusted sample estimate $55.4$

Weakest neighboring overlap $0.319$

Connect the windows

Thermal energy

k_{\mathrm B}T

1

Window stiffness

\kappa

12

Selected window

i

4

Window center

c_i

0

Drag a colored diamond to move its window. Glowing dots replay the saved simulated trajectories.

Sampling and a new random seed

Each normal window contains two independent replicas starting on opposite sides of the hidden coordinate. The hidden-trap preset starts one replica in the upper valley and raises its vertical barrier.

$\Delta t=0.001,\quad N_{\rm burn}=8000,\quad s=20,\quad \gamma=1$

The saved record includes the seed of every replica.

Potential, boundaries and uncertainty

$U=3(x^2-1)^2+b(y^2-1)^2+1.5xy$

Coordinates are dimensionless and confined to a reflecting square with half-width 1.8. The ordinary vertical barrier coefficient is three; the hidden trap uses fourteen. Thermal energy is measured in a fixed energy unit.

The free-energy curve is a weighted histogram with forty-eight bins. Twelve circular-block bootstrap replicas estimate uncertainty in the right-half-plane population. Each block stays within one independent trajectory. This estimate describes fluctuations inside the states visited; it cannot detect an absent valley.

The overlap warning uses a documented empirical threshold. Passing it is a sampling diagnostic, not proof that each window has equilibrated. The integration reference is separate from every trajectory and fit.

Model assumptions

The data are overdamped Langevin simulations on a dimensionless two-coordinate potential with reflecting boundaries. Harmonic biases act on the horizontal coordinate. MBAR assumes each window samples its equilibrium distribution; visible overlap along one coordinate cannot establish that. A small block-bootstrap error can miss unsampled states. Biased trajectory time is not a reaction clock.

Seeded simulations · independent integration reference

What survives after removing the bias?

Connected windows compare the reconstruction with a separate reference.

Correlation, held-out region and fit diagnostics

N=7000,\quad p_{\rm R}^{\rm ref}=0.500

The central eight bins define a predeclared validation interval. Its independent integration reference is withheld from fitting; all acquisition samples enter MBAR. The record saves the recovered probability and absolute error there. The effective sample estimate divides the Kish weight count by the largest coordinate-based statistical inefficiency across replicas; it is a coordinate-based heuristic diagnostic, not a variance guarantee for every observable.

The export includes all coordinates, acquisition times, bias energies, seeds, burn-in, sample stride, normalized target weights, overlap matrix and bootstrap replicas. Every record here is simulated.

Continue the course on free energy and rare events

Start with the essentials

Focus question: When do overlapping biased trajectories reveal the equilibrium free energy?
One-sentence intuition: Overlapping equilibrated windows can share their relative statistical weights.

Core mathematical model

A known harmonic bias

B_i(x)=\frac{\kappa}{2}(x-c_i)^2,\qquad u_i=\frac{U+B_i}{k_{\rm B}T}

The window center and stiffness set its bias energy. The reduced potential divides the total energy by the thermal energy.

MBAR combines the saved configurations

\widehat f_i=-\ln\sum_n\frac{e^{-u_i(z_n)}}{\sum_jN_j e^{\widehat f_j-u_j(z_n)}}

The sum uses every saved configuration. Each state has a sample count and an unknown normalization. Setting the first reduced free energy to zero fixes an arbitrary common constant.

Return to the unbiased distribution

w_n\propto\frac{e^{-U(z_n)/(k_{\rm B}T)}}{\sum_jN_j e^{\widehat f_j-u_j(z_n)}},\qquad F(x)=-k_{\rm B}T\ln p(x)+C

The weights form a histogram of the horizontal coordinate. The pale reference comes from separate two-dimensional integration; it never enters the MBAR fit.

Common difficulties

Many saved frames can repeat little information

Typical misconceptionEvery stored frame counts as a new independent observation.

Better mental modelNearby frames share the trajectory’s history. The diagnostic estimates autocorrelation in both coordinates; the bootstrap draws contiguous blocks within each independent replica.

Run the experiment

01
Follow a shared crossing
Watch A connected crossing. Select a window or drag its diamond, then compare the cyan reconstruction with the pale integration reference.
What to observe: Neighboring colors visit some of the same configurations. The orange pooled histogram still shows the bias; MBAR changes the relative weights.
02
Remove the bridge
Choose A missing bridge. Look at the overlap matrix and the two separated clouds.
What to observe: The empirical overlap graph splits. A single global reconstruction is withheld because the record does not reliably connect the relative normalization of the two groups.
03
Test the hidden direction
Choose The hidden trap. Compare the vertical coverage with the reconstructed population and its error bar.
What to observe: The windows overlap horizontally but remain in the upper hidden valley. Resampling those same blocks can produce a small error bar around a biased answer.

Umbrella Sampling and MBAR

What survives after removing the bias?