Kinetic equation is the full description of plasma in the phase space of position and velocity. Fluid equations can be derived from it.

Boltzmann equations (Collisional plasma)

It all start from the derivation of distribution function f(r⃗,v⃗,t)f(\vec{r},\vec{v},t). The total differential of the distribution is the so called Boltzmann equation [Stix_1992_book][Colonna_2016_book].

dfdt=∇rf⋅dr⃗dt+∇vf⋅dv⃗dt+∂f∂t\frac{df}{dt} = \nabla_{r} f\cdot\frac{d \vec{r}}{d t} + \nabla_{v}f\cdot\frac{d \vec{v}}{d t} + \frac{\partial f}{\partial t}

If we mark the whole term dfdt\frac{df}{dt} as the collisional term dfdt∣c\frac{df}{dt}|_c, and consider the Newton’s second law: md2r⃗dt2=F⃗⇒dv⃗dt=d2r⃗dt2=F⃗mm\frac{d^2 \vec{r}}{d t^2} = \vec{F} \Rightarrow \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}=\frac{\vec{F}}{m}. Then the whold Boltzmann equation becomes:

v⃗⋅∇rf+F⃗m⋅∇vf+∂f∂t=dfdt∣c\vec{v}\cdot\nabla_{r}f + \frac{\vec{F}}{m}\cdot\nabla_{v}f +\frac{\partial f}{\partial t} = \frac{df}{dt}|_c

We further consider the force upon a charged particle in plasma, it shall be: F⃗=q(E⃗+v⃗×B⃗)\vec{F} = q(\vec{E} + \vec{v}\times\vec{B}). Then the Boltzmann equation becomes:

v⃗⋅∇rf+qm(E⃗+v⃗×B⃗)⋅∇vf+∂f∂t=dfdt∣c\vec{v}\cdot\nabla_{r}f + \frac{q}{m}(\vec{E}+\vec{v}\times\vec{B})\cdot\nabla_{v}f +\frac{\partial f}{\partial t} = \frac{df}{dt}|_c

This shall be the complete Boltzmann equation in the plasma with collisional term.

Vlasov equation (Collisionless)

Valsov equation only apply to the high temperature collisionless plasma. Which is to say the effect of short range collision is neglected, where dfdt∣c=0\frac{df}{dt}|_c = 0. It starts from the derivative of distribution function, then we get:

∂f∂t+v⃗⋅∇rf+a⃗⋅∇vf=0\frac{\partial f}{\partial t} + \vec{v}\cdot\nabla_r f + \vec{a}\cdot\nabla_v f = 0

This is the expression of famous Valsov equation [wikipedia_Vlasov_equation], where a⃗=qm(E⃗+v⃗×B⃗)\vec{a} = \frac{q}{m}(\vec{E} + \vec{v}\times\vec{B}).

Reference

[wikipedia_Vlasov_equation] https://en.wikipedia.org/wiki/Vlasov_equation

[Stix_1992_book] T.H. Stix, Waves in Plasmas, 1992, Springer-Verlag, New York.

[Colonna_2016_book] G. Colonna and A. D’Angola, Plasma Modeling Methods and Applications, 2016, IOP Publishing, Bristol, UK.