Dynamical phase transition in the first-passage probability of a Brownian motion

We study the first-passage time distribution (FPTD) F(t(f)vertical bar x(0), L) for a freely diffusing particle starting at x(0) in one dimension, to a target located at L, averaged over the initial position x(0) drawn from a normalized distribution (1/sigma) g(x(0)/sigma) of finite width sigma. We show the averaged FPTD undergoes a sharp dynamical phase transition from a two-peak structure for b = L/sigma > b(c) to a single-peak structure for b < b(c). This transition is generated by the competition of two characteristic timescales sigma(2)/D and L-2/D, where D is the diffusion coefficient. A very good agreement is found between theoretical predictions and experimental results obtained with a Brownian bead whose diffusion is initialized by an optical trap which determines the initial distribution g(x(0)/sigma). We show that this transition is robust: It is present for all initial conditions with a finite sigma, in all dimensions, and also exists for more general stochastic processes going beyond free diffusion.