Landau damping $$ f_0 = (1+\varepsilon \sum_j \cos(k_jx_j)) /\sqrt{2\pi} (\exp(-0.5|\mathbf{v}|^2)) $$ Linear Landau damping: $k_x=0.5$, $\varepsilon = 0.01$ Non-linear Landau damping: $k_x =0.5$, $\varepsilon = 0.5$