Fundamentals of Plasma Physics

2.4 Two-fluid equations 45

2.4.3 Relation between pressure and Maxwellian

The scalar pressure has a simple relation to the generalized Maxwellian as seen by recasting
Eq.(2.26) as

Pσ = −

nσmσ

N

(

β

π

)N/ 2

d

dβ

∫

e−βv

′ 2

dv

= −

nσmσ

N

(

β

π

)N/ 2

d

dβ

(

β

π

)−N/ 2

= nσκTσ,

(2.55)

which is just the ideal gas law. Thus, the definitions that have been proposed forpressure
and temperature are consistent with everyday notions for these quantities.
Clearly, neither the adiabatic nor the isothermal assumption will beappropriate when
Vph∼vTσ.Thefluid description breaks down in this situation and the Vlasov description
must be used. It must also be emphasized that the distribution function is Maxwellian
only if there are sufficient collisions or some other randomizing process. Because of the
weak collisionality of a plasma, this is often not the case. In particular, since the collision
frequency scales asv−^3 ,fast particles take much longer to become Maxwellian than slow
particles. It is not at all unusual for a plasma to be in a state where the low velocity particles
have reached a Maxwellian distribution whereas the fast particles form a non-Maxwellian
“tail”.
We now summarize the two-fluid equations:

• continuity equation for each species

∂nσ

∂t

+∇·(nσuσ) = 0 (2.56)

• equation of motion for each species

nσmσ

duσ

dt

=nσqσ(E+uσ×B)−∇Pσ−Rσα (2.57)

• equation of state for each species

Regime Equation of state Name

Vph>>vTσ Pσ∼nγσ adiabatic

Vph<<vTσ Pσ=nσκTσ,Tσ=constant isothermal

• Maxwell’s equations
  ∇×E=−

∂B

∂t

(2.58)

∇×B=μ 0

∑

σ

nσqσuσ+μ 0 ε 0

∂E

∂t

(2.59)

∇·B= 0 (2.60)

∇·E=

1

ε 0

∑

σ

nσq (2.61)