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An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field
Journal of Statistical Physics ( IF 1.6 ) Pub Date : 2020-06-09 , DOI: 10.1007/s10955-020-02570-8
Sandra Cerrai , Jan Wehr , Yichun Zhu

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass $$\mu $$ μ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity $$\epsilon >0$$ ϵ > 0 . We show that for all small but fixed frictions the small mass limit of $$q_{\mu , \epsilon }$$ q μ , ϵ gives the solution $$q_\epsilon $$ q ϵ to 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 motion $$q_\epsilon $$ q ϵ and 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.

中文翻译:

存在变化磁场时 Smoluchowski-Kramers 近似的平均方法

我们研究了方程的小质量极限,该方程描述了小质量的带电粒子在力场中的平面运动,其中包含受随机项扰动的磁性分量。我们通过添加强度 $$\epsilon >0$$ ϵ > 0 的小摩擦来正则化问题。我们表明,对于所有小但固定的摩擦,$$q_{\mu , \epsilon }$$ q μ , ϵ 的小质量极限给出了解决方案 $$q_\epsilon $$ q ϵ 到随机一阶方程,包含噪声引起的漂移项。然后,通过使用 Freidlin 和 Wentzell 对哈密顿系统的经典平均定理的推广,
更新日期:2020-06-09
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