L4 · Chapter 7

Why equal free energies can give different reaction speeds

Sandbox Physics · 2026-10-28

Free energy describes equilibrium occupancy. Dynamics determines how long reaching a region takes. Increasing friction fourfold with the corresponding thermal noise preserves the equilibrium distribution and multiplies mean arrival times by four. To describe a route, ask another question: from a complete configuration, what is the probability of reaching product first?

Slow the entire clock

Keep the potential and bath temperature fixed. Use the same reflecting domain and initial distribution. Increasing isotropic friction from $\zeta$ to $\lambda\zeta$ divides both drift and diffusion by $\lambda$ . The second process has the first process’s path distribution with time slowed by that factor.

Worked example: an initial preparation has mean first arrival time $12$ in model units. Fourfold friction makes it $48$ in the continuous model. Both processes retain the same Boltzmann density and projected free energy. Finite-step simulations require the corresponding convergence checks.

Thus even an exactly known free energy cannot uniquely specify a rate. Anisotropic mobility can additionally change the preferred channel, generally defeating a single overall time rescaling.

X_t^{(\lambda\zeta)}\overset{\mathrm{law}}{=}X_{t/\lambda}^{(\zeta)},\qquad \mathbb E[\tau_B^{(\lambda\zeta)}]=\lambda\,\mathbb E[\tau_B^{(\zeta)}]

Both processes use the same initial distribution. This is equality of path distributions under a time change. Independent seeds retain random fluctuations.

Original clock-scaling schematic. The equilibrium curves coincide. For the same overdamped model, fourfold uniform friction stretches an entire first-passage distribution by four in time. Curve shapes here are illustrative; the scaling law is exact for the stated continuous process.

Ask which region is reached first

Define disjoint reactant and product sets $A$ and $B$ , with first hitting times $\tau_A,\tau_B$ . The forward committor $q(\mathbf r)$ is the probability of reaching $B$ first from the complete configuration $\mathbf r$ . It is zero in reactant and one in product.

For constant isotropic mobility, the backward generator is $\mathcal L=(-\nabla U\cdot\nabla+k_{\mathrm B}T\nabla^2)/\zeta$ . Averaging the future committor over a very short step gives $\mathcal Lq=0$ . A reflecting outer boundary has zero normal derivative. Uniform friction cancels from this zero equation and leaves the committor unchanged.

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

A committor belongs to specified dynamics and specified sets. Changing sets or relative mobilities generally changes the answer.

Source: Weinan E and Eric Vanden-Eijnden · Mauro Lapelosa and Cameron F. Abrams

An analytic check on a line

Consider free diffusion on $[0,L]$ with absorbing endpoints. They are reactant and product. The equation $Dq\prime\prime=0$ integrates to a line; endpoint conditions give $q(x)=x/L$ . Independent launches from $x=L/4$ should reach the right endpoint first about one quarter of the time, with binomial scatter.

The mean time $T_{\mathrm{exit}}(x)$ to either endpoint solves $DT_{\mathrm{exit}}\prime\prime=-1$ with zero endpoint values, giving $x(L-x)/(2D)$ . The midpoint committor remains one half as diffusion changes, but the mean exit time changes.

q(x)=\frac{x}{L},\qquad T_{\mathrm{exit}}(x)=\frac{x(L-x)}{2D}

This original analytic limit has absorbing endpoints. The two-dimensional experiment reflects at its outer box and stops first-passage paths on the defined target sets.

Equal projected positions can have different futures

In a coupled landscape, configurations with equal $x$ but different $y$ can have different $q(x,y)$ . A broad committor distribution at the proposed transition coordinate signals missing predictive information.

Launch independent unbiased trajectories from candidate transition configurations and count product-first outcomes. Retain the number that reach neither set before a finite cutoff: these are censored outcomes, not automatic failures. Launch-count uncertainty, starting-position uncertainty and numerical discretization are separate errors.

Path sampling treats a whole transition as a sample

Equilibrium sampling asks how often a position occurs. Path sampling studies complete reactive segments, their channels and their returns. A shooting move selects a configuration within a path and generates new dynamics; a complete algorithm also accepts or rejects according to the correct path probability.

The browser experiment shows unbiased first-passage paths and calculates the committor with a backward equation. The next section samples entire endpoint-conditioned paths in a separate offline discrete model. Umbrella, metadynamics and path methods can exchange starting configurations or candidate coordinates; their statistical objects must match the equilibrium or dynamical question.

A rate also needs an initial ensemble, normalization and time definition. The inverse mean first-passage time from a fixed point need not equal the reaction rate constant of an equilibrated reactant ensemble. The experiment starts from a fixed configuration and reports arrivals within a finite horizon, retaining non-arrivals as censored paths.

Source: Weinan E and Eric Vanden-Eijnden

Sample complete bridges on a discrete landscape

Place the same potential on a $13\times13$ grid. A lazy nearest-neighbor Metropolis chain has stationary distribution $\pi$ proportional to grid Boltzmann weights and transition matrix $P$ satisfying detailed balance. Fix $M=160$ steps, starting in $A$ and ending in $B$ . Intermediate returns and repeated product visits are allowed. This is a fixed-length endpoint-bridge ensemble.

An update chooses a time slice uniformly and retains its grid state. Generate the future forward and, by reversibility, the past backward. Accept a new path when its endpoints qualify; otherwise retain the entire old path. Rejected-state repeats contribute to averages under the probability below. Using only accepted paths would change their weights.

An independent forward/backward recursion sums over all possible paths and gives positive hidden-coordinate probability $0.490867$ at the midpoint. Four seeds retaining sixty thousand updates each give mean $0.479$ and between-seed standard deviation $0.0337$ , with acceptance around six percent. Empirical effective sizes are only about $166$ to $212$ per chain. Successive paths remain strongly correlated.

Offline task: run the accompanying sampler, retain rejected proposals and compare midpoint populations across seeds with the recursion reference. Explain why uncertainty follows correlations between path updates. These Metropolis steps have no physical Langevin-time calibration, so a bridge probability does not directly provide a reaction rate.

\mathcal P_{AB}(z_{0:M})\propto\pi(z_0)\,\mathbf 1_A(z_0)\!\left[\prod_{j=0}^{M-1}P(z_j,z_{j+1})\right]\!\mathbf 1_B(z_M)

The initial stationary weight and all transition probabilities determine the path weight. Endpoint indicators impose the target conditions without a first-passage restriction.

Source: Sandbox Physics · Sandbox Physics

Try the reasoning in an experiment

Keep the landscape and change the dynamics

Compare the default equilibrium reference and unbiased paths. Predict which quantities stay fixed when friction changes and which times should scale.
Change uniform friction and inspect arrival times and the committor. Then hold the horizontal probe position fixed and move its hidden coordinate to compare product-first probabilities.
Choose equal horizontal positions with different hidden coordinates. Compare product-first probabilities and explain the projection’s missing information.

Exercises and explanations

For free diffusion, $L=2$ and $D=0.5$ . Starting at the midpoint, what are the right-first probability and mean exit time?
Read the explanation
The probability is $1/2$ . Mean exit time is $1(2-1)/(2\times0.5)=1$ model time unit. Halving diffusion doubles this time and preserves the probability.
A low projected free-energy barrier accompanies two committor groups at the same projected position. What should be checked?
Read the explanation
Look for different hidden channels and slow exchange between them. Add a coordinate separating the groups, then compare conditional distributions and launch outcomes. Channel motion and launch statistics supply the missing dynamical information.
A bridge visits product, returns to reactant and eventually ends in product. Does it belong to this ensemble?
Read the explanation
Yes. This ensemble fixes only duration and endpoints. Stopping on the first product visit would change both path lengths and target probabilities, requiring a new sampler and reference.

Sources and reproduced scope

Worked examples and schematic figures are original teaching material. The numerical research slice compares specified estimator equations on our seeded two-coordinate records with independent quadrature and PyMBAR. The final chapter reads Figure 1 of the MBAR paper and states which experimental data remain outside that calculation.

Model definitions, scripts and numerical records

Sandbox Physics, fixed-length reversible path-ensemble model and validation (2026)
Original discrete endpoint-bridge target, reversible kept-slice shooting derivation, independent path-sum reference and recorded limitations. The computational clock is distinct from the browser Langevin clock.
Sandbox Physics, reproducible endpoint-bridge benchmark script (2026)
Runs four frozen seeds with accepted and rejected proposals retained, calculates serial-correlation diagnostics and writes path-sampling-summary.json and path-sampling-raw.json.
Weinan E and Eric Vanden-Eijnden, Towards a theory of transition paths, J. Stat. Phys. 123, 503–523 (2006)
Published transition-path framework and forward-committor definition, checked against the authors’ institutional publication record and the explicit low-dimensional formulation in the next source.
Mauro Lapelosa and Cameron F. Abrams, Transition-Path Theory Calculations on Non-Uniform Meshes in Two and Three Dimensions using Finite Elements, Comput. Phys. Commun. 184, 2310–2315 (2013)
Transition-Path Theory Calculations on Non-Uniform Meshes in Two and Three Dimensions using Finite Elements (2013), Sections on the backward Kolmogorov problem and reactive currents. Used for the equation and boundary-condition discussion, not for copied mesh data or figures.