多粒子在周期随机加速度场中的扩散极限
Diffusion limit for many particles in a periodic stochastic acceleration field
摘要：
任意数量的$\cN$粒子在多个独立波场（具有强空间相关性）中的一维运动被表示为由两个维纳过程驱动的二阶随机微分方程系统。在粒子质量为零的极限${\mathfrak{m}} \to 0$，或等效的大噪声强度，我们表明，所有的$N$粒子的动量弱收敛到$N$独立的布朗运动，这种收敛保持即使是周期性的噪声。这证明了扩散方程通常应用于唯一随机力场中的粒子族是正确的。证明依赖于两个粒子的相对速度在标度极限的遍历性。
英文摘要：
The one-dimensional motion of any number $\cN$ of particles in the field of many independent waves (with strong spatial correlation) is formulated as a second-order system of stochastic differential equations, driven by two Wiener processes. In the limit of vanishing particle mass ${\mathfrak{m}} \to 0$, or equivalently of large noise intensity, we show that the momenta of all $N$ particles converge weakly to $N$ independent Brownian motions, and this convergence holds even if the noise is periodic. This justifies the usual application of the diffusion equation to a family of particles in a unique stochastic force field. The proof rests on the ergodic properties of the relative velocity of two particles in the scaling limit.
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