An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field
An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field
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DOI：
10.1007/s10955-020-02570-8
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发表时间：
2020-06
影响因子：
1.6
通讯作者：
S. Cerrai;J. Wehr;Yichun Zhu
中科院分区：
文献类型：
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作者：
S. Cerrai;J. Wehr;Yichun Zhu
本文研究了描述小质量带电粒子在含有磁分量的力场中受随机项扰动的平面运动方程的小质量极限。我们通过增加一个小的摩擦强度来正则化这个问题。我们表明，对于所有的小，但固定的摩擦的小质量极限的随机一阶方程的解决方案，包含噪声引起的漂移项。然后，利用Freidlin和Wentzell关于Hamilton系统的经典平均定理的推广，我们取运动慢分量的极限，证明了它弱收敛于图上的马氏过程，该图是通过识别磁场强度函数水平集的相同连通分量中的所有点而得到的。
We study the small mass limit of the equation describing planar motion of a charged particle of a small massin a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity. We show that for all small but fixed frictions the small mass limit ofgives the solutionto a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motionand we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.