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

Sandra Cerrai, Jan Wehr, Yichun Zhu

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)132-148
Number of pages17
JournalJournal of Statistical Physics
Volume181
Issue number1
DOIs
StatePublished - Oct 1 2020

Keywords

  • Averaging principle
  • Hamiltonian systems
  • Smoluchowski–Kramers approximation
  • Stochastic differential equations
  • Stochastic equations on graphs

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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