标题： 非平衡稳态下间歇势中的布朗运动 显示英文标题

标题： Nonequilibrium steady state of Brownian motion in an intermittent potential

摘要： 我们计算在间歇性约束势下布朗粒子位置的稳态分布$P_{\text{SSD}}(\boldsymbol{X})$，该势以恒定速率$\gamma$开启和关闭。 我们假设外部势$U(\boldsymbol{x})$是光滑的，并在$\boldsymbol{x} = \boldsymbol{x}_0$处具有唯一的全局最小值，在维度$d>1$下，我们另外假设$U(\boldsymbol{x})$是中心的。 我们关注快速切换极限$\gamma \to \infty$。 典型的涨落遵循玻尔兹曼分布$P_{\text{SSD}}(\boldsymbol{X}) \sim e^{- U_{\text{eff}}(\boldsymbol{X}) / D}$，具有有效势能$U_{\text{eff}}(\boldsymbol{X}) = U(\boldsymbol{X})/2$，其中$D$是扩散系数。 然而，我们还计算了$P_{\text{SSD}}(\boldsymbol{X})$的尾部，它们的行为非常不同。 在远尾部$|\boldsymbol{X}| \to \infty$，出现了一种与捕获势无关的普遍行为$P_{\text{SSD}}\left(\boldsymbol{X}\right)\sim e^{-\sqrt{\gamma/D} \, \left|\boldsymbol{X}-\boldsymbol{x}_{0}\right|}$。 到达位置$\boldsymbol{X}$的平均首达时间在主导阶上由$\sim 1/P_{\text{SSD}}(\boldsymbol{X})$给出。 这与有效势能$U_{\text{eff}}$的阿伦尼乌斯定律相符，但对于$\boldsymbol{X} \simeq \boldsymbol{x}_0$以外的情况则偏离该定律。 我们给出了谐波势能的显式结果。 最后，我们将结果扩展到周期性的一维系统。 在此我们发现，在$\gamma \to \infty$和$D \to 0$的极限下，$P_{\text{SSD}}(X)$的对数表现出奇异性，我们将其解释为一阶动力学相变（DPT）。 这种 DPT 在没有外部漂移的情况下发生。 我们还计算了稳态中由于系统非平衡性质而产生的非零概率流。 显示英文摘要

摘要： We calculate the steady state distribution $P_{\text{SSD}}(\boldsymbol{X})$ of the position of a Brownian particle under an intermittent confining potential that switches on and off with a constant rate $\gamma$. We assume the external potential $U(\boldsymbol{x})$ to be smooth and have a unique global minimum at $\boldsymbol{x} = \boldsymbol{x}_0$, and in dimension $d>1$ we additionally assume that $U(\boldsymbol{x})$ is central. We focus on the rapid-switching limit $\gamma \to \infty$. Typical fluctuations follow a Boltzmann distribution $P_{\text{SSD}}(\boldsymbol{X}) \sim e^{- U_{\text{eff}}(\boldsymbol{X}) / D}$, with an effective potential $U_{\text{eff}}(\boldsymbol{X}) = U(\boldsymbol{X})/2$, where $D$ is the diffusion coefficient. However, we also calculate the tails of $P_{\text{SSD}}(\boldsymbol{X})$ which behave very differently. In the far tails $|\boldsymbol{X}| \to \infty$, a universal behavior $P_{\text{SSD}}\left(\boldsymbol{X}\right)\sim e^{-\sqrt{\gamma/D} \, \left|\boldsymbol{X}-\boldsymbol{x}_{0}\right|}$ emerges, that is independent of the trapping potential. The mean first-passage time to reach position $\boldsymbol{X}$ is given, in the leading order, by $\sim 1/P_{\text{SSD}}(\boldsymbol{X})$. This coincides with the Arrhenius law (for the effective potential $U_{\text{eff}}$) for $\boldsymbol{X} \simeq \boldsymbol{x}_0$, but deviates from it elsewhere. We give explicit results for the harmonic potential. Finally, we extend our results to periodic one-dimensional systems. Here we find that in the limit of $\gamma \to \infty$ and $D \to 0$, the logarithm of $P_{\text{SSD}}(X)$ exhibits a singularity which we interpret as a first-order dynamical phase transition (DPT). This DPT occurs in absence of any external drift. We also calculate the nonzero probability current in the steady state that is a result of the nonequilibrium nature of the system.