We study the first-passage time distribution (FPTD) $F\left(t_f\right|x_0,L)$ for a freely diffusing particle starting at $x_0$ in one dimension, to a target located at $L$, averaged over the initial position $x_0$ drawn from a normalized distribution $(1/\sigma )g(x_0/\sigma )$ of finite width $\sigma$. We show the averaged FPTD undergoes a sharp dynamical phase transition from a two-peak structure for $b=L/\sigma >b_c$ to a single-peak structure for $b<b_c$. This transition is generated by the competition of two characteristic timescales ${\sigma }^2/D$ and $L^2/D$, where $D$ is the diffusion coefficient. A very good agreement is found between theoretical predictions and experimental results obtained with a Brownian bead whose diffusion is initialized by an optical trap which determines the initial distribution $g(x_0/\sigma )$. We show that this transition is robust: It is present for all initial conditions with a finite $\sigma$, in all dimensions, and also exists for more general stochastic processes going beyond free diffusion.

中文翻译：

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布朗运动首通过概率中的动态相变

我们研究首次通过时间分布（FPTD） $F\left({吨}_f\right|{\times }_0,升)$ 对于一个自由扩散的粒子，开始于 ${\times }_0$ 在一维中，目标位于 $升$, 平均在初始位置 ${\times }_0$ 从归一化分布中提取 $(1/\sigma )克({\times }_0/\sigma )$ 有限宽度 $\sigma$. 我们展示了平均 FPTD 从两个峰结构经历了急剧的动态相变$乙=升/\sigma >{乙}_{丙}$ 为单峰结构 $乙<{乙}_{丙}$. 这种转变是由两个特征时间尺度的竞争产生的${\sigma }^2/D$ 和 ${升}^2/D$，哪里 $D$是扩散系数。在理论预测和使用布朗珠获得的实验结果之间发现了非常好的一致性，该珠的扩散由确定初始分布的光阱初始化$克({\times }_0/\sigma )$. 我们表明这种转变是稳健的：它适用于所有初始条件，并具有有限的$\sigma$，在所有维度中，并且也存在于超出自由扩散的更一般的随机过程。