7.5 The Liouville and Boltzmann Equations
405
For atoms there are five fundamental linearly independent dynamical quantities
for which ψ vanishes, so that there are no collisional contributions to Eq. (7.5.54):
these are mass, represented by ψ = 1, kinetic energy, for which ψ =
1
2 mv 2 , and
linear momentum, with ψ = m 1 v. These quantities are referred to as the collisional
invariants. There are no other linearly independent collisional invariants. When
ψ(v) is a collisional invariant, the Boltzmann equation gives rise to a conservation
law of the general form
∂
∂t
(nψ ne ) + ∇ · (nvψ ne ) = 0 .
(7.5.55a)
For ψ = 1, Eq. (7.5.53) gives the well-known hydrodynamic equation of continuity for the number density n, viz.,
∂n
∂t
+ ∇ · (nv 0 ) = 0 ,
(7.5.55b)
with the nonequilibrium average v 0 ≡ ≡v ne representing the macroscopic flow
velocity, or stream velocity, of the flowing gas. We may also write the equation
of continuity in terms of the mass density ρ = nm as
∂ρ
∂t
+ ∇ · (ρv 0 ) =
dρ
dt
+ ρ∇ · v 0 = 0 ,
in which the operator
d
dt
≡
∂
∂t
+ v 0 · ∇ denotes the rate of change for an observer
moving with the fluid, and is called the substantial derivative in the field of fluid
dynamics. The condition ∇ · v 0 is synonymous with the incompressibility condition
dρ
dt
= 0.
The linear momentum conservation law is
∂
∂t
(ρv 0 ) + ∇ · (ρvv ne ) = 0 ,
(7.5.56a)
and is called the equation of motion of the fluid in hydrodynamics. This equation is
more commonly written in the form
∂
∂t
(ρv 0 ) = −∇ · (ρv 0 v 0 + P) ,
(7.5.56b)
with the dyadic ρv 0 v 0 representing the convective flux of momentum through the
gas; the pressure tensor P is defined in terms of the nonequilibrium average by
P = ρVV ne ;
(7.5.57)
the motion of an individual atom relative to the stream velocity v 0 is represented by
V ≡ V(r, t) = v − v 0 (r, t) ,
(7.5.58)