Link to ADS entry
The author goes to great lengths to explain the linear advection and diffusion terms in the Boltzmann equation (equation 2.11), but is content with a vague "hand waving" explanation for the last term on the right hand side
1
3
(
∇
⋅
U
)
p
∂
f
∂
p
{\displaystyle \frac{1}{3} \left( \nabla \cdot \mathbf{U} \right) p \frac{\partial f}{\partial p}}
This term guarantees that the number of particles is conserved
N
=
∫
d
3
x
∫
d
3
p
⋅
f
{\displaystyle N = \int d^3 x \int d^3 p \cdot f }
∂
N
∂
t
=
∫
d
3
x
∫
d
3
p
⋅
∂
f
∂
t
=
{\displaystyle \frac{\partial N}{\partial t} = \int d^3 x \int d^3 p \cdot \frac{\partial f}{\partial t} = }
=
∫
d
3
x
∫
d
3
p
⋅
(
−
U
⋅
∇
f
+
∇
(
κ
∇
f
)
+
1
3
∇
⋅
U
p
∂
f
∂
p
)
=
{\displaystyle = \int d^3 x \int d^3 p \cdot \left( -\mathbf{U} \cdot \nabla f + \nabla \left( \kappa \nabla f \right) + \frac{1}{3} \nabla \cdot U p \frac{\partial f}{\partial p} \right) = }
=
4
π
∫
d
3
x
∫
p
2
d
p
⋅
(
−
U
⋅
∇
f
+
1
3
∇
⋅
U
p
∂
f
∂
p
)
=
{\displaystyle = 4 \pi \int d^3 x \int p^2 dp \cdot \left( -\mathbf{U} \cdot \nabla f + \frac{1}{3} \nabla \cdot U p \frac{\partial f}{\partial p} \right) = }
=
4
π
∫
d
3
x
∫
d
p
⋅
(
−
p
2
U
⋅
∇
f
−
1
3
∇
⋅
U
(
∂
∂
p
p
3
)
f
)
=
{\displaystyle = 4 \pi \int d^3 x \int dp \cdot \left( -p^2 \mathbf{U} \cdot \nabla f - \frac{1}{3} \nabla \cdot U \left(\frac{\partial}{\partial p}p^3\right) f \right) = }
=
4
π
∫
d
3
x
∫
d
p
⋅
(
−
p
2
U
⋅
∇
f
−
∇
⋅
U
p
2
f
)
=
{\displaystyle = 4 \pi \int d^3 x \int dp \cdot \left( -p^2 \mathbf{U} \cdot \nabla f - \nabla \cdot U p^2 f \right) = }
=
−
4
π
∫
d
3
x
∫
p
2
d
p
⋅
(
U
⋅
∇
f
+
∇
⋅
U
f
)
=
{\displaystyle = - 4 \pi \int d^3 x \int p^2 dp \cdot \left( \mathbf{U} \cdot \nabla f + \nabla \cdot U f \right) = }
=
−
4
π
∫
d
3
x
∫
p
2
d
p
⋅
(
∇
⋅
(
U
f
)
)
=
0
{\displaystyle = - 4 \pi \int d^3 x \int p^2 dp \cdot \left( \nabla \cdot \left( \mathbf{U} f\right) \right) = 0 }
but it is not unique (there may be different terms that restore conservation of particle number). This particular form has to do with adiabatic cooling , which relates the expansion to change in momentum
d
p
d
t
=
−
1
3
p
∇
⋅
U
{\displaystyle \frac{dp}{dt} = -\frac{1}{3} p \nabla \cdot \mathbf{U} }
Hence the change in the density distribution function is
(
∂
f
∂
t
)
a
d
=
∂
f
∂
p
d
p
d
t
=
−
1
3
(
∇
⋅
U
)
p
∂
f
∂
p
{\displaystyle \left( \frac{\partial f}{\partial t} \right)_{ad} = \frac{\partial f}{\partial p} \frac{dp}{dt} = -\frac{1}{3}\left(\nabla \cdot \mathbf{U} \right)p\frac{\partial f}{\partial p} }
