X distribution with a unit variance compared against the 1D velocity u_L (normalized by std(u_L)) distribution on a small scale, r, from an N-body simulation. Vertical axis is plotted in the logarithmic scale (log₁₀). The X distribution with α = 1.33 and matches the simulated velocity distribution for small separation r, where both particle pairs are likely to reside in the same halo and different pairs can be from different halos. The Gaussian core and exponential wings can be clearly identified. For comparison, velocity distribution of all DM particles from the same simulation is also presented (black line). System on large scale is not fully virialized, which leads to the deviation from the predicted X distribution.