Relations among Unidirectional Fluxes at Equilibrium, Committors, and First Passage and Transition Path Times

For multidimensional diffusive dynamics, we algebraically derive remarkable analytical expressions that relate the mean first passage and transition path times between two dividing surfaces with the number of unidirectional transitions per unit time (fluxes) at equilibrium between the two surfaces and the committor (the probability of reaching one surface before the other). In one dimension, such relationships can be easily derived because analytical expressions for all the above-mentioned quantities can be found. This is not possible in higher dimensions, and at first sight, the problem seems much harder. We circumvent the difficulty that the equations determining the mean first passage and transition path times cannot be solved analytically by multiplying these equations by the committor, integrating both sides and finally using the divergence theorem. A byproduct of our derivation is an analytical expression for the starting point distribution over which individual first passage and transition path times must be averaged. It turns out that this distribution is not the Boltzmann one, but it has a simple physical interpretation.