TY - JOUR

T1 - An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field

AU - Cerrai, Sandra

AU - Wehr, Jan

AU - Zhu, Yichun

N1 - Funding Information:
S. Cerrai: Partially supported by the NSF Grants DMS 1407615 and DMS 1712934. J. Wehr: Partially supported by the NSF Grants DMS 1615045 and DMS 1911358.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - 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 qμ,ϵ gives the solution qϵ 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 qϵ 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.

AB - 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 qμ,ϵ gives the solution qϵ 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 qϵ 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.

KW - Averaging principle

KW - Hamiltonian systems

KW - Smoluchowski–Kramers approximation

KW - Stochastic differential equations

KW - Stochastic equations on graphs

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U2 - 10.1007/s10955-020-02570-8

DO - 10.1007/s10955-020-02570-8

M3 - Article

AN - SCOPUS:85086167218

SN - 0022-4715

VL - 181

SP - 132

EP - 148

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 1

ER -