Coulomb collision operators

The Coulomb force is long ranged, so a broad range of impact parameters contributes to cumulative scattering. Screening limits large impact parameters, while close encounters or quantum diffraction set an appropriate small-scale cutoff. Their ratio appears logarithmically. A large plasma parameter and a sizeable Coulomb logarithm support the weak-coupling expansion. Strongly coupled matter, large-angle-dominated scattering, and relativistic regimes require more than the elementary nonrelativistic Landau approximation.

For one species colliding with itself, the Landau form is a divergence of an integral velocity-space flux. Its tensor projects perpendicular to relative velocity. Friction and diffusion are linked by the evolving distribution; they are not arbitrary independent knobs. The kernel's geometry and the symmetric treatment of both particles make conservation possible. Retaining only test-particle drag against a frozen background discards that symmetry and transfers momentum and energy to an unmodeled bath.

The velocity divergence makes particle-number conservation follow from vanishing flux at large velocity. Momentum conservation also uses exchange symmetry between the two colliding populations. For different species, one population's collision term can change its momentum and energy; the changes in the partner population are equal and opposite. Each species' number is separately conserved by elastic collisions, while momentum and kinetic energy are conserved only after summing the appropriate pair or all species.

The full multispecies operator relaxes toward Maxwellians with a common bulk velocity and temperature, with separate densities allowed. Equal temperatures do not mean equal thermal speeds because masses differ. An isolated single-species self-collision operator cannot remove a uniform drift or change the total kinetic energy merely to reach a preferred laboratory Maxwellian. A model that forces every species toward zero drift independently may therefore violate total momentum even if it keeps the density exactly constant.

Under the Landau operator's assumptions, the summed collisional entropy production is nonnegative. The logarithm can be made dimensionless with an arbitrary fixed reference phase-space density; number conservation makes that choice irrelevant to entropy production. Equality characterizes the compatible Maxwellian equilibrium for a connected colliding mixture. This statistical irreversibility differs from ideal Vlasov transport, which rearranges fine structure without itself increasing the fine-grained entropy.

A linear Ornstein–Uhlenbeck bath operator is an instructive approximation, not the full Coulomb operator. It conserves test-particle number while relaxing mean velocity and variance toward prescribed bath values. Unless the bath response is included, the test particles alone do not conserve momentum or energy. Such a model may represent a dilute population in a large reservoir. For a closed plasma or precision transport calculation, conserving field-particle corrections and the correct velocity dependence can be essential.