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arXiv 1908.10624cond-mat.stat-mechcond-mat.soft
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二维活动布朗粒子的长时间位置分布

Long time position distribution of an active Brownian particle in two dimensions

Urna Basu, Satya N. Majumdar, Alberto Rosso, Gregory Schehr

备注：
16 pages, 7 figures. References added and typos corrected. arXiv admin note: text overlap with arXiv:1804.09027
发表信息：
Phys. Rev. E 100, 062116 (2019)
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摘要：

研究了具有速度$v_0$和旋转扩散常数$D_R$的二维单个活性布朗粒子的晚期动力学。我们表明，在晚些时候$t\gg D_R^{-1}$，而在$x$-$y$平面的位置概率分布$P(x,y,t)$接近高斯形式的峰值附近描述的典型扩散波动，它有非高斯尾部描述非典型罕见的波动时，$\sqrt{x^2+y^2}\sim v_0 t$。在这种情况下，该分布允许一个大的偏差形式，$P(x,y,t) \sim \exp\left[-t\, D_R\, Φ\left(\sqrt{x^2+y^2}/(v_0 t)\right)\right]$，其中我们计算率函数$Φ(z)$解析，也数值上使用的重要性抽样方法。我们表明，率函数$Φ(z)$，编码罕见的波动，仍然进行跟踪活动，即使在晚些时候。另一种检测晚期活动的方法是使活动粒子受到外部谐波势的影响。在这种情况下，我们表明，固定分布$P_\text{stat}(x,y)$明确地依赖于活动参数$D_R^{-1}$和经历的交叉，$D_R$增加，从一个环形的强主动限制（$D_R\to 0$）的高斯形状的强被动限制$(D_R\to \infty)$。

英文摘要：

We study the late time dynamics of a single active Brownian particle in two dimensions with speed $v_0$ and rotation diffusion constant $D_R$. We show that at late times $t\gg D_R^{-1}$, while the position probability distribution $P(x,y,t)$ in the $x$-$y$ plane approaches a Gaussian form near its peak describing the typical diffusive fluctuations, it has non-Gaussian tails describing atypical rare fluctuations when $\sqrt{x^2+y^2}\sim v_0 t$. In this regime, the distribution admits a large deviation form, $P(x,y,t) \sim \exp\left[-t\, D_R\, Φ\left(\sqrt{x^2+y^2}/(v_0 t)\right)\right]$, where we compute the rate function $Φ(z)$ analytically and also numerically using an importance sampling method. We show that the rate function $Φ(z)$, encoding the rare fluctuations, still carries the trace of activity even at late times. Another way of detecting activity at late times is to subject the active particle to an external harmonic potential. In this case we show that the stationary distribution $P_\text{stat}(x,y)$ depends explicitly on the activity parameter $D_R^{-1}$ and undergoes a crossover, as $D_R$ increases, from a ring shape in the strongly active limit ($D_R\to 0$) to a Gaussian shape in the strongly passive limit $(D_R\to \infty)$.

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