Top: Local relaxation rate, ∂F/∂t, in action space as predicted by BL (Equation 3) for the Mestel disc (Section 2.1), computed in the centre of the action bins. The red regions correspond to an increase in the number of particles, while the blue contours correspond to a depletion. Bottom: Relaxation rate measured in N–body simulations of the same disc, averaged over 1 000 realisations with N​ = ​25​ × ​10⁶ particles each. The changes in the DF are computed at t/t_dyn​ = ​150, safely before the discs become unstable (Figure 1). The prediction and measurement are in good agreement, both in shape and amplitude. Slices in action space are illustrated in Figure 4. Collective effects play a crucial role in shaping the long-term evolution of razor-thin discs.