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.

Figure 1

(a) A schematic illustration of classical potentials $U(x)$ whose left tails as $x\to -\infty$, increase (i) faster than $|x|$ (red dotted line), (ii) as $|x|$ (blue solid line), and (iii) slower than $|x|$ (magenta dashed line). There is an absorbing barrier at $x=a$ (dot-dashed line). (b) Schematic illustrations of the corresponding quantum potentials $V(x)$ in Eq. (3), whose left tails as $x\to -\infty$, (i) diverges (red dotted line), (ii) approaches a constant (blue solid line), and (iii) tends to zero (magenta dashed line), respectively. $V(x)=\infty$ for $x\ge a$.

Figure 2

Schematic diagram of energy levels for the quantum potential $V(x)={\alpha }^2/(4D)-\alpha \delta (x)$ for $x<a$ and $V(x)=\infty$ for $x\ge a$, for different values of $a$. The (red) dots represent the ground state energy for different values of $a$ whereas the (blue) bands represent the continuum of energy levels from ${\alpha }^2/(4D)$ to $\infty$. The gap vanishes at $a=a_c=D/\alpha$.

Figure 3

(a) The (red) solid line plots the analytical $\theta (a)$ vs $a$ in Eq. (8), for the potential in Eq. (5b) in case (ii), with $\alpha =D=1$. The (blue) points represent the simulation results. The (magenta) dashed line plots the analytical expression (see [23]) of the inverse of the mean first-passage time. The vertical (gray) dashed line marks $a=a_c$. (b) Same plot as in (a) but for the case (i) in Eq. (5b) with $\mu =\alpha =D=1$ and $b=2$. (c) The (magenta) dashed line plots the inverse of the mean first-passage time for case (iii) in Eq. (5b), with $\alpha =D=\lambda =1$ and $b=c=e$. In this case $\theta (a)=0$ for all $a$. (d) The (blue) points represent simulation results for the survival probability $S_a(t|0)$ for the same potential as in (c), while the (red) dashed line represents the asymptotic decay of $S_a(t|0)$ in Eq. (13).