Fundamentals of Plasma Physics

44 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD

is

N=V

∫

fdv. (2.46)

The energy of an individual particle isE=mv^2 / 2 wherevis the velocity measured in
the rest frame of the center of mass of the entire collection ofNparticles. Thus, the total
kinetic energy of all the particles in this rest frame is

N〈E〉=V

∫

mv^2

2

f(v)dv (2.47)

and so the variational problem becomes

δ

∫

dv

(

flnf−λ 1 Vf−λ 2 V

mv^2

2

f

)

= 0. (2.48)

Incorporating the volumeVinto the Lagrange multipliers, and factoring out the coefficient
δfthis becomes
∫
dvδf

(

1 + lnf−λ 1 −λ 2

mv^2

2

)

= 0. (2.49)

Sinceδfis arbitrary, the integrand must vanish, giving

lnf=λ 2

mv^2

2

−λ 1 (2.50)

where the ‘1’ has been incorporated intoλ 1.
The maximum entropy distribution function of an isolated, energy and particle conserv-
ing system is therefore

f=λ 1 exp(−λ 2 mv^2 /2); (2.51)

this is called a Maxwellian distribution function. We will often assume that the plasma is
locally Maxwellian so thatλ 1 =λ 1 (x,t)andλ 2 =λ(x,t).We define the temperature to
be

κTσ(x,t) =

1

λ 2 (x,t)

(2.52)

where Boltzmann’s factorκallows temperature to be measured in various units. The nor-
malization factor is set to be

λ 1 (x,t) =n(x,t)

(

mσ

2 πκTσ(x,t)

)N/ 2

(2.53)

whereNis the dimensionality ( 1 , 2 ,or3)so that

∫

fσ(x,v,t)dNv=nσ(x,t).Because the
kinetic energy of individual particles was defined in terms of velocities measured in the rest
frame of the center of mass of the complete system of particles, if this center of mass is
moving in the lab frame with a velocityuσ, then in the lab frame the Maxwellian will have
the form

fσ(x,v,t) =nσ

(

mσ

2 πκTσ

)N/ 2

exp(−mσ(v−uσ)^2 / 2 κTσ). (2.54)