Comparison of the particle energy distribution (x) at various times predicted by the Fokker–Planck equation (solid and dashed curves) and by direct numerical integrations (bar histograms). The blue curves show the analytical solution of the linearized (approximate) Fokker–Planck equation (Equation (49)), while the red curves correspond to the numerical solution of the full Fokker–Planck equation (Equation (46)). Particles are initialized with ﹩{{ \mathcal A }}_{0}=2.49﹩, e₀ = 0.634, and i₀ = 15﹩\mathop{.}\limits^{\unicode{x000b0}}﹩8, corresponding to U_∞ = 0.5 or ﹩{ \mathcal T }=2.75﹩. The planet has mass M_p = 1 × 10⁻⁴ at a_p = 1 au, and its scattering timescale t_S = 450 kyr is computed using Equation (58). The final panel shows the surviving fraction f_survive as a function of time, with the blue dashed curve given by Equation (51).