标题： 一维小惯性随机动力学的运动学描述 显示英文标题

标题： Kinetic description of one-dimensional stochastic dynamics with small inertia

摘要： 我们研究用于描述小惯性随机动力学的单变量方法。 我们处理的基本模型描述了在存在线性耗散项情况下具有有效惯性的被动布朗粒子和相位元件（相位振荡器、旋转器、超导约瑟夫森结）以及在非线性耗散情况下的主动布朗粒子。 消除快速变量（速度）将系统状态的表征简化为一个变量，并以四种表示形式进行表述：矩、累积量、埃尔米特函数基和最后一种的正式累积量变体。 这种消除在存在线性耗散情况下的过阻尼极限和主动布朗粒子的过活跃极限中提供了严格的数学描述。 对于前者，我们推导出一个低维方程组，该方程组将Ott-Antonsen假设推广到具有小有效惯性的系统。 在后者的情况下，我们推导出一个带有强制漂移项和一维有效扩散的福克-普朗克型方程，其中标准的二维/三维机制是不可能的。 在四种考虑的表示形式中，截断的方程链被证明对于小有限惯性的数值模拟是有用的。 显示英文摘要

摘要： We study single-variable approaches for describing stochastic dynamics with small inertia. The basic models we deal with describe passive Brownian particles and phase elements (phase oscillators, rotators, superconducting Josephson junctions) with an effective inertia in the case of a linear dissipation term and active Brownian particles in the case of a nonlinear dissipation. Elimination of a fast variable (velocity) reduces the characterization of the system state to a single variable and is formulated in four representations: moments, cumulants, the basis of Hermite functions, and the formal cumulant variant of the last. This elimination provides rigorous mathematical description for the overdamped limit in the case of linear dissipation and the overactive limit of active Brownian particles. For the former, we derive a low-dimensional equation system which generalizes the Ott-Antonsen Ansatz to systems with small effective inertia. In the latter case, we derive a Fokker-Planck-type equation with a forced drift term and an effective diffusion in one dimension, where the standard two-/three-dimensional mechanism is impossible. In the four considered representations, truncated equation chains are demonstrated to be utilitary for numerical simulation for a small finite inertia.