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For freely streaming particles, a spatial Fourier perturbation of a velocity distribution evolves with factors $e^{-ikvt}$. Contributions from different velocities acquire different phases and can cancel in a density or field moment even while the fine-grained distribution retains structure. In a self-consistent plasma, resonant field–particle exchange gives Landau damping. This is distinct from collisional thermalization and from orbital phase mixing in a planetary ensemble.

Define the one-dimensional thermal widths ${\sigma }_p=\sqrt{k_{\mathrm{B}}{T}_p/m_e}$ and ${\sigma }_b=\sqrt{k_{\mathrm{B}}{T}_b/m_e}$. Normalize each velocity distribution to its electron number density:The total is $f=f_p+f_b$. It consists of a broad peak centred at zero and a much narrower displaced beam near $V$, with $V\gg {\sigma }_p\gg {\sigma }_b$. Their areas, rather than their peak heights, are $n_p$ and $n_b$. A weak narrow beam can make a positive-slope region on its low-velocity side even while contributing little to the total density.