论文标题
布朗颗粒的冻结过渡限制电势
Freezing transitions of Brownian particles in confining potentials
论文作者
论文摘要
我们将平均第一通道时间（MFPT）研究为有限域中的外部电势V（x）$的一维棕色粒子的吸收目标。我们专注于外部电势限制的情况，形式为$ v（x）= k | x-x_0 |^n/n $，粒子的初始位置与$ x_0 $重合。我们首先考虑$ x = 0 $的吸收目标与$ x = c $的反射墙之间的粒子。在固定$ x_0 $的情况下，我们表明，当目标距离$ c $超过临界值时，存在非零的最佳刚度$ k _ {\ rm opt} $，将MFPT最小化。但是，当$ c $低于临界值时，最佳刚度$ k _ {\ rm opt} $消失。因此，对于任何$ n $的值，随着域大小的变化，最佳电位刚度会经历连续的“冷冻”过渡。另一方面，当反射墙被第二个吸收目标取代时，$ k _ {\ rm opt} $中的冻结过渡就会不连续。然后，$（x_0，n）$ - 平面中的相图展示了三个动态阶段和亚稳定性，其“三重”点为$（x_0/c \ simeq 0.17185 $，$ n \ simeq 0.39539）$。对于谐波或高订单电位$（n \ ge 2）$，对于任何$ x_0 $或域的大小，MFPT总是以$ k $在小$ k $下的$ k $增加。这些结果与在有限域中最佳重置下扩散的问题形成鲜明对比。
We study the mean first passage time (MFPT) to an absorbing target of a one-dimensional Brownian particle subject to an external potential $v(x)$ in a finite domain. We focus on the cases in which the external potential is confining, of the form $v(x)=k|x-x_0|^n/n$, and where the particle's initial position coincides with $x_0$. We first consider a particle between an absorbing target at $x=0$ and a reflective wall at $x=c$. At fixed $x_0$, we show that when the target distance $c$ exceeds a critical value, there exists a nonzero optimal stiffness $k_{\rm opt}$ that minimizes the MFPT to the target. However, when $c$ lies below the critical value, the optimal stiffness $k_{\rm opt}$ vanishes. Hence, for any value of $n$, the optimal potential stiffness undergoes a continuous "freezing" transition as the domain size is varied. On the other hand, when the reflective wall is replaced by a second absorbing target, the freezing transition in $k_{\rm opt}$ becomes discontinuous. The phase diagram in the $(x_0,n)$-plane then exhibits three dynamical phases and metastability, with a "triple" point at $(x_0/c\simeq 0.17185$, $n\simeq 0.39539)$. For harmonic or higher order potentials $(n\ge 2)$, the MFPT always increases with $k$ at small $k$, for any $x_0$ or domain size. These results are contrasted with problems of diffusion under optimal resetting in bounded domains.
