论文标题
在扩散粒子的屏障交叉速率中冻结过渡
Freezing transition in the barrier crossing rate of a diffusing particle
论文作者
论文摘要
我们研究了衰减率$θ（a）$，该$ thum $ f_a（t | 0）（t | 0）\ sim e^{ - θ（a）\，t} $在一个有限的限制潜在的$ u（x）$的位置，从原点开始，到达$ a> 0 $ a> 0 $ a> 0 $。对于一般限制潜在的$ u（x）$，我们表明$θ（a）$，衡量屏障（位于$ a $）的越野率的量度，具有三种截然不同的行为，这是$ a $的函数，具体取决于$ u（x）$ as $ x \ to -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 $θ（a）=θ（a_c）$。我们的结果是使用量子问题的一般映射和在三种代表性情况下的精确解决方案建立的，并由数值模拟支持。我们表明，当在相关的量子问题，基态（绑定）之间的差距和散射状态的连续性消失时，冻结过渡发生。
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.
