We use the Fokker Planck equation as a starting point for studying the orientational probability distribution of an Active Brownian Particle (ABP) in $(d+1)$ dimensions. This Fokker Planck equation admits an exact solution in series form which is, however, unwieldly to use because of poor convergence for short and intermediate times. A truncated version of this series is a reasonable approximation for long times. In this paper, we present an analytical closed form expression, which gives a good approximate orientational probability distribution, which is derived using saddle point methods for short times. However, it works well even for intermediate times. Thus, we have simple analytical forms for the ${\it entire}$ range of time scales for the orientational probability distribution of an ABP. Our predictions can be tested against future experiments and simulations probing orientational probability distribution of an ABP.

中文翻译：

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活性布朗粒子的定向概率分布：分析研究

我们使用 Fokker Planck 方程作为研究 $(d+1)$ 维度中活性布朗粒子 (ABP) 的定向概率分布的起点。这个福克普朗克方程允许以串联形式存在精确解，但是由于短时间和中间时间的收敛性较差，因此难以使用。这个系列的截断版本在很长一段时间内是一个合理的近似值。在本文中，我们提出了一个解析闭合形式表达式，它给出了一个很好的近似定向概率分布，这是使用鞍点方法在短时间内推导出来的。但是，它即使在中间时间也能很好地工作。因此，对于 ABP 的方向概率分布的 ${\it whole}$ 时间尺度范围，我们有简单的分析形式。