An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field

• Published in: Journal of statistical physics Vol. 181; no. 1; pp. 132 - 148
• Main Authors: Cerrai, Sandra, Wehr, Jan, Zhu, Yichun
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-10-2020; Springer; Springer Nature B.V
• Subjects: Charged particles; Hamiltonian functions; Linear equations; Magnetic fields; Magnetic flux; Markov processes; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Stochastic equations on graphs; Hamiltonian systems; Stochastic differential equations; Smoluchowski–Kramers approximation; Averaging principle
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-020-02570-8

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass
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
ϵ
>
0
. We show that for all small but fixed frictions the small mass limit of
q
μ
,
ϵ
gives the solution
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
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.