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We investigate the crossing of an energy barrier by a self-propelled particledescribed by a Rayleigh friction term. We reveal the existence of a sharptransition in the external force field whereby the amplitude dramaticallyincreases. This corresponds to a saddle point transition in the velocity flowphase space, as would be expected for any type of repulsive force field. We usethis approach to rationalize the results obtained by Eddi \emph{et al.}[\emph{Phys. Rev. Lett.} \textbf{102}, 240401 (2009)] who studied theinteraction between a drop propelled by its accompanying wave field and asubmarine obstacle. This wave particle entity can overcome potential barrier,suggesting the existence of a "macroscopic tunneling effect". We show that theeffect of self-propulsion is sufficiently strong to generate crossing of thehigh energy barrier. By assuming a random distribution of initial angles, wedefine a probability distribution to cross the potential barrier that matcheswith the data of Eddi \emph{et al.}. This probability is similar to the oneencountered in statistical physics for Hamiltonian systems \textit{i.e.} aBoltzmann exponential law.

Self-propulsion and crossing statistics under random initial conditions