Self-propulsion and crossing statistics under random initial conditions

We investigate the crossing of an energy barrier by a self-propelled particle described by a Rayleigh friction term. We reveal the existence of a sharp transition in the external force field whereby the amplitude dramatically increases. This corresponds to a saddle point transition in the velocity flow phase space, as would be expected for any type of repulsive force field. We use this approach to rationalize the results obtained by Eddi \emph{et al.} [\emph{Phys. Rev. Lett.} \textbf{102}, 240401 (2009)] who studied the interaction between a drop propelled by its accompanying wave field and a submarine obstacle. This wave particle entity can overcome potential barrier, suggesting the existence of a "macroscopic tunneling effect". We show that the effect of self-propulsion is sufficiently strong to generate crossing of the high energy barrier. By assuming a random distribution of initial angles, we define a probability distribution to cross the potential barrier that matches with the data of Eddi \emph{et al.}. This probability is similar to the one encountered in statistical physics for Hamiltonian systems \textit{i.e.} a Boltzmann exponential law.