扩散粒子越垒率的冻结相变
Freezing transition in the barrier crossing rate of a diffusing particle
摘要：
我们研究了衰减率$θ(a)$，它表征了扩散粒子在一维约束势$U(x)$中的首次通过概率密度$F_a(t|0) \sim e^{-θ(a)\, t}$的晚期指数衰减，从原点开始，到位于$a>0$的位置。对于一般的限制势$U(x)$，我们表明，$θ(a)$，一个措施的障碍（位于$a$）的交叉率，有三个不同的行为作为$a$的函数，取决于尾部的$U(x)$作为$x\to -\infty$。特别是，对于表现为$U(x)\sim |x|$的潜力时，$x\to -\infty$，我们表明，一个新的冻结过渡发生在一个临界值$a=a_c$，即$θ(a)$单调增加，$a$减少，直到$a_c$，和$a \le a_c$冻结到$θ(a)=θ(a_c)$。我们的结果是建立使用一般映射到一个量子问题，并在三个代表性的情况下，由数值模拟支持的精确解。我们表明，当在相关的量子问题中，基态（束缚）和散射态连续谱之间的间隙消失时，就会发生冻结跃迁。
英文摘要：
We study the decay rate $θ(a)$ that characterizes the late time exponential decay of the first-passage probability density $F_a(t|0) \sim e^{-θ(a)\, t}$ 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 $θ(a)$, 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, $θ(a)$ increases monotonically as $a$ decreases till $a_c$, and for $a \le a_c$ it freezes to $θ(a)=θ(a_c)$. 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.
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