Fundamentals of Plasma Physics 2025, 19 - Derivation of Fluid Equations
Since the cumulative grazing collisions dominate the cumulative large angle collisions, to see the effects of collisions on the Vlasov equations for plasma, approximate collisions with a sequence of abrupt large scattering angle encounters. The detailed phase-space trajectories characterized by a collision between two particles features an abrupt jump in v-coordinate. This can be described by simultaneous creation and annihilation operators with identical position but different velocity coords.
The collective moments of the distribution function can be obtained as integrals over the Vlasov equation, resulting in partial differential equations of the n-th moment. From the 0-th moment for example one recovers continuity. Multiplying the integrand by powers of the velocity vector adds the orders to the n-th moments. The first moment becomes the a normalized frictional drag due to interspecies collisions, etc. This will never result in a closed system of differential equations, so the chain must be closed artificially.
The most likely state of the system is the one with the highest entropy. Collisions to this state will scramble the positions into a new micro-state, which is most likely a member of the class of microscopic states with the highest entropy. Any randomization of the collision will then statistically evolve the system toward maximum entropy. While this happens, conservation relation will need to remain satisfied. In case of particle number conservation or energy conservation, this would block the actual maximum entropy state. As a variational problem:
δ S - λ1δN - λ2δ(N⟨E⟩) = 0.
For N = V∫ fdv, then N⟨E⟩ = V∫ mv2/2 f(v) dv
which defines f as the Maxwellian distribution function. Locally, plasma can be assumed Maxwellian.
2-Fluid equations can then be summarized by continuity for each species, their e.o.m. and equations of state depending on adiabatic or isothermal regime, along with the known Maxwell equations arising from the Maxwell distribution.
From the magnetohydrodynamic perspective, the same overall approach can be used. This yields an MHD continuity equation and e.o.m. equations of state, but also an Ohm's law. For invariant magnetic fluxes, the magnetic field lines must convect with the velocity, so they are "frozen" into the plasma and move with it. This is the central assumption to ideal MHD, but might be invalidated through electron inertia, electron pressure, or Hall terms.
Collisionless Vlasov equilibria, combined with Poisson's equation becomes a model for the potential in steady-state transition region between a plasma and conducting wall, called a "sheath". It's non-neutral and has a width at Debye-length scale. The exact profile introduces a numerical problem. A Langmuir probe (or metal wall) terminating the plasma that is biased negative wrt. the plasma, requires the potential be measured relative to the plasma potential. The bias potential is shielded out by the plasma within the sheath. The relative potential varies within the sheath tends toward the difference between probe and plasma at the probe, and to 0 for values much larger than Debye length. Only electrons with sufficiently high energy can be picked up by the probe. Ultimately,
