随机系统中的小惯量和有限惯量：矩和累积量形式
Small and Finite Inertia in Stochastic Systems: Moment and Cumulant Formalisms
摘要：
我们分析了两种方法来消除随机系统中的快速变量（速度）：矩和累积量形式主义。通过这些方法，我们得到相应的Smoluchovski型方程，其中只包含坐标/相位变量。用累积量和动量表示的速度绝热消去法需要前三个元素。然而，对于小惯性的情况下，修正后的Smoluchowski方程在矩方面需要五个元素，而在累积量方面，相同的前三个元素就足够了。与基于Hermite函数中速度分布的展开的方法相比，所考虑的方法具有相当的效率，但对于活性布朗粒子的情况不需要单独的数学准备，其中必须构造新的本征函数基础而不是Hermite基础。
英文摘要：
We analyze two approaches to elimination of a fast variable (velocity) in stochastic systems: moment and cumulant formalisms. With these approaches, we obtain the corresponding Smoluchovski-type equations, which contain only the coordinate/phase variable. The adiabatic elimination of velocity in terms of cumulants and moments requires the first three elements. However, for the case of small inertia, the corrected Smoluchowski equation in terms of moments requires five elements, while in terms of cumulants the same first three elements are sufficient. Compared to the method based on the expansion of the velocity distribution in Hermite functions, the considered approaches have comparable efficiency, but do not require individual mathematical preparation for the case of active Brownian particles, where one has to construct a new basis of eigenfunctions instead of the Hermite ones.
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