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\section{Conclusion}  Finally, we point out that the investigation of the low velocity limit of stopping power is important for the understanding of the non-adiabatic coupling between ions and electrons \cite{Caro_2015} and also for modeling dissipative molecular dynamics \cite{Caro_1989,Duffy_2006}.   In simulations of radiation events the final state is precisely controlled by dissipation in the late stages when ions move slowly but still non-adibatically \cite{Zarkadoula_2014}.  In summary, in this paper we have reported the $S_\text{e}$ of protons in copper in a very wide range of velocities.  TDDFT-based electron dynamics is capable of capturing most of the physics in the different ranges, starting from non-linear screening effects, electron-hole excitations and production of plasmons.   We disentangled channeling and off-channeling effects and observe a collapse of the two curves at low velocities; and identified five regimes