We consider an active Brownian particle in a $d$-dimensional harmonic trap, in the presence of translational diffusion. While the Fokker-Planck equation can not in general be solved to obtain a closed form solution of the joint distribution of positions and orientations, as we show, it can be utilized to evaluate the exact time dependence of all moments, using a Laplace transform approach. We present explicit calculation of several such moments at arbitrary times and their evolution to the steady state. In particular we compute the kurtosis of the displacement, a quantity which clearly shows the difference of the active steady state properties from the equilibrium Gaussian form. We find that it increases with activity to asymptotic saturation, but varies non-monotonically with the trap-stiffness, thereby capturing a recently observed active- to- passive re-entrant behavior.

中文翻译：

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谐波陷阱中的活性布朗粒子：矩的精确计算和重入跃迁

我们在存在平移扩散的情况下考虑 $d$ 维谐波陷阱中的活性布朗粒子。虽然 Fokker-Planck 方程通常无法求解以获得位置和方向的联合分布的封闭形式解，但正如我们所展示的，它可以用于评估所有矩的精确时间依赖性，使用拉普拉斯变换方法. 我们在任意时间给出了几个这样的时刻的明确计算以及它们向稳定状态的演变。特别地，我们计算了位移的峰度，该量清楚地显示了活动稳态特性与平衡高斯形式的差异。我们发现它随着活动增加到渐近饱和，但随着陷阱刚度非单调变化，