The upper panel displays a comparison between the characteristic timescales of all relevant processes inside the bubble: advection, diffusion and energy losses for both protons and electrons. For diffusion, different types of turbulence are considered as detailed in the figure legend. The shaded area represent timescales larger of the age of the system (black line). The lower panel shows the maximum energy of electrons and protons for the Bohm (dashed) and the Kraichnan (solid) cases, obtained as the intersection between the acceleration time and the time-scale that results from the combination of escape ($T_{\mathrm{esc}}={(T_{\mathrm{adv}}^{-1}+T_{\mathrm{diff}}^{-1})}^{-1}$), energy losses (T_loss) and the age of the system T_age.