Non-Markovian work fluctuation theorem in crossed electric and magnetic fields

The validity of the transient work fluctuation theorem for a charged Brownian harmonic oscillator embedded in a non-Markovian heat bath and under the action of crossed electric and magnetic fields is investigated. The aforementioned theorem is verified to be valid within the context of the generalized Langevin equation with an arbitrary memory kernel and arbitrary dragging in the potential minimum. The fluctuation-dissipation relation of the second kind is assumed to be valid and shows that the non-Markovian stochastic dynamics associated with the particle, in the absence of the external time-dependent electric field, reaches an equilibrium state, as is precisely demanded by such a relation. The Jarzynski equality in this problem is also analyzed.