Page 318 - Intro to Tensor Calculus

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• The particle density per unit volume

+∞
Z ZZZ
~
~
n = n(~r, t)= dτ V f(~r, V, t)= f(~r, V, t)dV x dV y dV z (2.5.84)
−∞
where ρ = nm is the mass density.
• The mean velocity
ZZ
+∞ Z
1
~
~
~
~
V 1 = V = V 1 f(~r, V 1 ,t)dV 1x dV 1y dV 1z
n
−∞
~
For any quantity Q = Q(V 1 ) define the barred quantity
+∞
Z ZZZ
1 1
~
~
~
~
Q = Q(~r, t)= Q(V )f(~r, V, t) dτ V = Q(V )f(~r, V, t)dV x dV y dV z . (2.5.85)
n(~r, t) n
−∞
~ F
~
Further, assume that is independent of V , then the moment of equation (2.5.83) produces the result
m
3 3
∂ X ∂ X F i ∂φ
nφ + nV 1i φ − n =0 (2.5.86)
∂t ∂x i m ∂V 1i
i=1 i=1
known as the Maxwell transfer equation. The first term in equation (2.5.86) follows from the integrals
Z ~ Z
∂f(~r, V 1 ,t) ~ ∂ ~ ~ ∂
φ(V 1 )dτ V 1 = f(~r, V 1 ,t)φ(V 1 ) dτ V 1 = (nφ) (2.5.87)
∂t ∂t ∂t
where differentiation and integration have been interchanged. The second term in equation (2.5.86) follows
from the integral
3
Z Z
X ∂f
~ ~
V 1 ∇ ~r fφ(V 1 )dτ V 1 = V 1i i φdτ V 1
∂x
i=1
3 Z
X ∂
= V 1i φf dτ V 1 (2.5.88)
∂x i
i=1
3
X ∂
= nV 1i φ .
∂x i
i=1
The third term in equation (2.5.86) is obtained from the following integral where integration by parts is
employed
Z ~ Z 3
F X F i ∂f
∇ ~ V 1 fφ dτ V 1 = φdτ V 1
m m ∂V 1i
i=1
+∞
ZZZ 3
X F i ∂f
= φ dV 1x dV 1y dV 1y
m ∂V 1i
i=1 (2.5.89)
−∞
Z
∂ F i
= − φ fdτ V 1
∂V 1i m

∂ F i F i ∂φ
= −n φ = −
∂V 1i m m ∂V 1i