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arXiv 2412.03045cond-mat.stat-mech
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间歇势中布朗运动的非平衡稳态

Nonequilibrium steady state of Brownian motion in an intermittent potential

Soheli Mukherjee, Naftali R. Smith

备注：
18 pages, 4 figures
发表信息：
J. Stat. Mech. (2025) 033205
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AI总结：

研究了间歇势下布朗运动的非平衡稳态，发现典型涨落遵循玻尔兹曼分布而远尾区呈现普适行为，并揭示了一维周期系统中的一阶动态相变。

摘要：

我们计算了在以恒定速率γ开启和关闭的间歇性约束势下，布朗粒子位置的非平衡稳态分布$P_{\text{SSD}}(\boldsymbol{X})$。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，且在维度$d>1$时，我们额外假设$U(\boldsymbol{x})$是中心势。我们重点关注快速切换极限$γ\to \infty$。典型涨落遵循玻尔兹曼分布$P_{\text{SSD}}(\boldsymbol{X}) \sim e^{- U_{\text{eff}}(\boldsymbol{X}) / D}$，其有效势为$U_{\text{eff}}(\boldsymbol{X}) = U(\boldsymbol{X})/2$，其中$D$为扩散系数。然而，我们也计算了$P_{\text{SSD}}(\boldsymbol{X})$的尾部，其表现截然不同。在远尾区$|\boldsymbol{X}| \to \infty$，出现了一种普适行为$P_{\text{SSD}}\left(\boldsymbol{X}\right)\sim e^{-\sqrt{γ/D} \\, \left|\boldsymbol{X}-\boldsymbol{x}_{0}\right|}$，该行为与捕获势无关。在主阶近似下，到达位置$\boldsymbol{X}$的平均首达时间由$\sim 1/P_{\text{SSD}}(\boldsymbol{X})$给出。当$\boldsymbol{X} \simeq \boldsymbol{x}_0$时，这与阿伦尼乌斯定律（针对有效势$U_{\text{eff}}$）一致，但在其他位置则偏离该定律。我们给出了谐振势的显式结果。最后，我们将结果推广到一维周期系统。在此我们发现，在$γ\to \infty$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态相变（DPT）。该DPT在没有任何外部漂移的情况下发生。我们还计算了稳态中非零的概率流，这是系统非平衡本质的结果。

英文摘要：

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 $γ$. 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 $γ\to \infty$. 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 $P_{\text{SSD}}\left(\boldsymbol{X}\right)\sim e^{-\sqrt{γ/D} \, \left|\boldsymbol{X}-\boldsymbol{x}_{0}\right|}$ 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 $γ\to \infty$ 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.

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