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 - 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 -