Schematic depicting a projection of short trajectory swarm data (gray) from an initial point x 0 (black) to specific coordinates q 1 and q 2 at later times.
For a reaction coordinate whose dynamics follow a one-dimensional Smoluchowski equation, swarms of trajectories from different individual configurations on each isosurface will drift and diffuse similarly. Therefore, for each isosurface, the projection of the individual swarms should each resemble the combined projection of all swarms (depicted to the right of the free energy surface). In the case depicted above, the individual swarms behave differently from each other, and therefore differently from their combined projection.
If the dynamics of q follow a one-dimensional Smoluchowski equation, dynamical self-consistency should apply at all isosurfaces of q. (Left) For a poor reaction coordinate q(x) the value of q alone is not a good predictor of drift or diffusion from the configuration x. (Right) If q shows dynamical self-consistency for all isosurfaces of q from reactant A to product B, then q is an accurate reaction coordinate.
The solid curves are contours of the actual free energy landscape βF(q 1, q 2) with a saddle point at the round dot. The coordinate q 1 has been used as the initial coordinate, i.e., λ(x) = q 1(x). The dotted contours are curves of constant Λ. Note that Λ is peaked where the original landscape had a saddle point.
The free energy landscape for a model of nucleation where the nucleus size can change along either fast (n F ) and slow (n S ) mobility directions.
The free energy as a function of the initial coordinate λ = n F + n S .
Histograms of endpoints of 1000 trajectories 2.0τ after initiation. Each figure shows nine swarms initiated at nine different points on the free energy landscape. In (a) the anisotropy is s = 1.0, and in (b) the anisotropy is s = 0.1. Differences in the way the swarms drift are difficult to visually discern, but the dynamical self-consistency test can detect differences.
The trial reaction coordinate q can be represented by the angle θ between the direction of progress along q and the n F -axis.
exp[−Λ] weighted Kullback-Leibler divergences for different trial reaction coordinates (represented by θ) and for different values of the diffusion anisotropy, s = 1.0, 0.3, 0.1, 0.03.
Illustrating three mechanistic regimes that prevail for different degrees of diffusion anisotropy. When s ≈ 1, the diffusion tensor is isotropic and most pathways follow the minimum free energy path. When s < 1, but not extremely small, a two step nucleation mechanism prevails with initial motion along the slow coordinate before escape in the n F -direction. Finally, when s is extremely small the Berezhkovskii-Zitserman (BZ) regime prevails. In the BZ-regime, trajectories can escape in the n F -direction with the n S degree of freedom frozen.
Narrow tube type free energy landscape as a function of fast (n F ) and slow (n S ) coordinates. The saddle point is at (n F , n S ) = (20, 20).
Projections of the free energy onto different initial coordinates. (a) The initial coordinate was λ = n S . (b) The initial coordinate was λ = n F .
⟨ΔKL[q]⟩Λ for different coordinates (represented by the angle θ as shown in Figure 8 ). (a) The initial coordinate was λ = n S . (b) The initial coordinate was λ = n F . In both cases the optimal coordinate (the minimum) rotates toward n S as the diffusion tensor becomes more anisotropic.
Contour plots of (a) the EVB potential V(x, y) with the minimum energy path between the reactant and product minima, (b) the potential Λ(x, y) constructed with λ chosen as the end-to-end direction along the minimum energy path, and (c) the potential Λ(x, y) constructed with the ideal energy gap coordinate for λ. Contour spacings are 5kBT in all plots.
⟨ΔKL[q]⟩Λ for different linear trial coordinates q(x, y) represented by the angle θ. The initial coordinate λ(x, y) gives a hysteretic free energy Fλ(λ) if sampled imperfectly. Coordinates similar to λ clearly do not have dynamical self-consistency, but other coordinates are not clearly distinguished in ⟨ΔKL[q]⟩Λ.
⟨ΔKL[q]⟩Λ for different linear trial coordinates q(x, y) represented by the angle θ. The initial coordinate λ(x, y) is the vertical energy gap between diabatic states of the EVB model.
Comparison between reaction coordinates from the dynamical self-consistency test and from KLBS theory. Coordinates identified by dynamical self-consistency are summarized for two different initial coordinates. For narrow tube potential energy landscapes the dynamical self-consistency test can correctly identify accurate reaction coordinates even for an inaccurate initial coordinate λ.
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