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Unformatted text preview: 3. Kinetic versus fluid description of plasmas Reading: Shu, Vol.II, Ch.2 3.1 Boltzmann’s equation Having established a means to describe the statistical distribution of particles in the six- dimensional phase space, the distribution function f ( ~x,~p, t ), we are now looking for a way to quantitatively follow the changes imposed on the distribution function by interactions. It appears useful to distinguish between two kinds of changes. Collisions will abruptly change the momentum of particles, whereas interactions with, e.g., electromagnetic fields lead to con- tinuous changes that can be describes as a convection of a particle in phase space. d~x dt = ~v = ~p m (3 . 1) d~p dt = q ~ E ( ~x, t ) + 1 c ~v × ~ B ( ~x, t ) (3 . 2) The collisions are better described as a catastrophic loss or sink term for the state incoming particle and a source term for the outgoing particle. Then the total rate of change of the distribution function should be df dt = sources- sinks = f c = ∂f ∂t + ˙ ~x ∂f ∂~x + ˙ ~p ∂f ∂~p = ∂f ∂t + ~v ∂f ∂~x + q " ~ E ( ~x, t ) + ~v c × ~ B ( ~x, t ) # ∂f ∂~p (3 . 3) This equation is known as Boltzmann’s equation. Very often in the astrophysical context we can neglect the collision term f c and set it to zero, thus deriving Vlasov’s equation. In the non-relativistic regime one may use the velocity instead of momentum as second coordinate, so the distribution function would be differential in velocity, not momentum. The Boltzmann equation would then write as f c ( ~v ) = ∂f ( ~x,~v ) ∂t + ~v ∂f ( ~x,~v ) ∂~x + q m " ~ E ( ~x, t ) + ~v c × ~ B ( ~x, t ) # ∂f ( ~x,~v ) ∂~v The electromagnetic fields that govern the force term in Boltzmann’s equation are generated by the collective motion of all the other particles and possibly charges and currents external to the system of interest. The fields derive from the charge and current densities in the plasma, which we define to be ρ e = X j q j Z d 3 v f j ( ~x,~v, t ) (3 . 4) ~ j = X j q j Z d 3 v ~v f j ( ~x,~v, t ) (3 . 5) 1 The four Maxwell equations therefore read ~ ∇ · ~ B = 0 (3 . 6) ~ ∇ × ~ B = 1 c ∂ ~ E ∂t + 4 π c ~ j ext + 4 π c ~ j int = 1 c ∂ ~ E ∂t + 4 π c ~ j ext + 4 π c X j q j Z d 3 v ~v f j (3 . 7) ~ ∇ · ~ E = 4 π ρ ext + 4 π X j q j Z d 3 v f j (3 . 8) ~ ∇ × ~ E =- 1 c ∂ ~ B ∂t (3 . 9) We note the nonlinear coupling between the electromagnetic fields and the particles. The kineticWe note the nonlinear coupling between the electromagnetic fields and the particles....
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