Tensors for Physics

12.3 Averages Over Velocity Distributions 219

In local thermal equilibrium, at temperatures and densities, where quantum effects
play no role,fis equal to the local Maxwell distributionfM

fM(c)≡n

(

m

2 πkBT

) 3 / 2

exp

(

−

mc^2

2 kBT

)

, (12.83)

wheremis the mass of a particle.
It is convenient to introduce a dimensionless velocity variableV,cf.(12.51),
now via

Vμ=

√

m

2 kBT

cμ=

√

m

2 kBT

(vμ−〈vμ〉), (12.84)

which implies

V^2 =

mc^2

2 kBT

, (12.85)

and to use the velocity distributionF=F(V), linked withf(c), such that

f(c)d^3 c=F(V)d^3 V.

Averages are now evaluated according to

〈ψ〉=

1

n

∫

ψ(V)F(V)d^3 V, (12.86)

which is mathematically identical to (12.53).
In thermal equilibrium,Fis equal to the local MaxwellianFM

FM(V)≡nπ−^3 /^2 exp(−V^2 ). (12.87)

Averages evaluated with this Maxwell velocity distribution function are denoted by
〈...〉M,viz.

〈ψ〉M≡π−^3 /^2

∫

ψ(V)exp(−V^2 )d^3 V=

1

n

∫

ψ(V)FM(V)d^3 V. (12.88)

The results given above for〈...〉 0 , in particular (12.56)–(12.60), apply also to the
averages〈...〉M, evaluated with the local Maxwell distribution function.
Similar to (12.65), the full distribution function is now written as

F(V)=FM(V)( 1 +Φ), (12.89)

whereΦ(t,r,V)characterizes the deviation ofFfrom the local MaxwellianFM.
The expansion ofΦis formally similar to (12.66), with one fundamental difference: