Freezing transition in the barrier crossing rate of a diffusing particle

We study the decay rate $\theta(a)$ that characterizes the late time exponential decay of the first-passage probability density $F_a(t|0) \sim e^{-\theta(a)\, t}$ of a diffusing particle in a one dimensional confining potential $U(x)$, starting from the origin, to a position located at $a>0$. For general confining potential $U(x)$ we show that $\theta(a)$, a measure of the barrier (located at $a$) crossing rate, has three distinct behaviors as a function of $a$, depending on the tail of $U(x)$ as $x\to -\infty$. In particular, for potentials behaving as $U(x)\sim |x|$ when $x\to -\infty$, we show that a novel freezing transition occurs at a critical value $a=a_c$, i.e, $\theta(a)$ increases monotonically as $a$ decreases till $a_c$, and for $a \le a_c$ it freezes to $\theta (a)=\theta(a_c)$. Our results are established using a general mapping to a quantum problem and by exact solution in three representative cases, supported by numerical simulations. We show that the freezing transition occurs when in the associated quantum problem, the gap between the ground state (bound) and the continuum of scattering states vanishes.