The mean shape of transition and first-passage paths

We calculate the mean shape of transition paths and first-passage paths based on the one-dimensional Fokker-Planck equation in an arbitrary free energy landscape including a general inhomogeneous diffusivity profile. The transition path ensemble is the collection of all paths that do not revisit the start position $x_A$ and that terminate when first reaching the final position $x_B$. In contrast, a first-passage path can revisit but not cross its start position $x_A$ before it terminates at $x_B$. Our theoretical framework employs the forward and backward Fokker-Planck equations as well as first-passage, passage, last-passage and transition-path time distributions, for which we derive the defining integral equations. We show that the mean time at which the transition path ensemble visits an intermediate position $x$ is equivalent to the mean first-passage time of reaching the starting position $x_A$ from $x$ without ever visiting $x_B$. The mean shape of first-passage paths is related to the mean shape of transition paths by a constant time shift. Since for large barrier height $U$ the mean first-passage time scales exponentially in $U$ while the mean transition path time scales linearly inversely in $U$, the time shift between first-passage and transition path shapes is substantial. We present explicit examples of transition path shapes for linear and harmonic potentials and illustrate our findings by trajectories generated from Brownian dynamics simulations.