Boltzmann Equation
Velocity distribution functions of particles
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Derivation of Boltzmann Equation
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Ludwig Eduard Boltzmann (February 20, 1844 - September 5, 1906), an
Austrian physicist famous for the invention of statistical mechanics. Born
in Vienna, Austria-Hungary, he committed suicide in 1906 by hanging
himself while on holiday in Duino near Trieste in Italy.
Distribution Function (probability
density function)
Random variable y is distributed with the probability
density function f(y) if for any interval [a b] the
probability of a<y<b is equal to
b
P=∫ f  y dy
a
f(y) is always non-negative
∫ f  ydy=1

Velocity space
Axes u,v,w in velocity space
have the same directions as
axes x,y,z in physical
space.
Each molecule can be
represented in velocity space
by the point defined by its
velocity vector v with
components (u,v,w)
dv
dv
dw duu
v
v
w
Velocity distribution function
Consider a sample of gas that is homogeneous in physical space
and contains N identical molecules. Velocity distribution function
is defined by
d N =Nf  v d u d v d w
(1)
where dN is the number of molecules in the sample with velocity
components (ui,vi,wi) such that
u<ui<u+du, v<vi<v+dv, w<wi<w+dw
dv = dudvdw is a volume element in the velocity space. Consequently,
dN is the number of molecules in velocity space element dv.
Functional statement if often omitted, so f(v) is designated as f
Phase space distribution function
Macroscopic properties of the flow are functions of position and time, so
the distribution function depends on position and time as well as velocity.
At any instant, each monoatomic molecule can be described by a point in
6-dimensional phase space (x,y,z,u,v,w). In the distribution function
for phase space the total number of molecules N in formula (1)
should be replaced by the number of molecules in the physical space
element, which is equal to ndxdydz = ndr.
The number of molecules in the 6-D volume element of phase space is
therefore written as
dN=nf(v,r)dvdr
(2)
Multi-particle distribution function
At any instant, a complete system of N monoatomic molecules can be
represented by a point in the 6N dimensional phase space. In terms of
multi-particle distribution function the probability of finding the system
in the volume element dv1dv2...dvNdr1dr2...drN is written as
F(N)(v1,r1,v2,r2,...vN,rN,t)dv1dv2...dvNdr1dr2...drN
where F(N) is N-particle distribution function.
A reduced distribution function for R of N molecules is defined by
∞
F R  v 1 , r 1 , v 2 , r 2 , ... v R , r R ,t = ∫ F  N d v R1 ...d v N d r R1 d r N
−∞
Single particle distribution function F(1)(v1,r1,t) is obtained by setting R=1:
∞
F  v1, r 1, t = ∫ F
1
−∞
N
d v 2 ... d v N d r 2 d r N
Physically, F(1)(v1,r1,t)drdv is the probability that the first molecule is located
in (r,r+dr) space element with velocity in the (v,v+dv) interval, or in drdv
phase space element. Since the molecules are indistinguishable, total
number of molecules in drdv phase space element is equal to
dN=NF(1)drdv
Recall equation (2):
dN=nf(v,r)dvdr. It follows that
NF(1)=nf(v,r)
Two particle distribution function is obtained by setting R=2.
In a dilute gas, it is assumed that the probability of finding a pair of
molecules in a particular state is the product of the probabilities of
finding the individual molecules in the two corresponding one-particle
configurations. Therefore
F(2)(v1,r1,v2,r2,t)=F(1)(v1,r1,t)F(1)(v2,r2,t)
Equation (3) represents the principle of molecular chaos.
(3)
Boltzmann Equation Assumptions
1.The density is sufficiently low so that only binary
collisions need be considered
2.Molecular chaos
3.The spatial dependence of gas properties is
sufficiently slow (distribution function is constant
over the interaction region)
4.Collisions can be thought of as being
instantaneous
At a particular instant, the number of molecules in the phase space element
is equal to nf(v,r)dvdr. If the location and shape of the element does not
vary with time, the rate of change of the number of molecules in the
element is
∂ nf d v d r
∂t
There are several processes that contribute to the change in the number
of molecules within dvdr.
Because of assumption (3) we can regard v as constant within dr, and
think of dv as being located at r. Phase space element can be represented
as separate volume elements in physical space and velocity space.
Velocity space
Physical space
er
Collisions
(iii)
v(i)
dv
dSc
ec
dr
F(ii)
i.Molecules moving with velocity v leave physical space element dr.
ii.Molecules attain speed in the range (v,v+dv) as a result of external
force per unit mass (acceleration) F (a).
iii.Scattering of molecules in and out of dv due to collisions. Because
of assumption (4), collisions change only velocity, but not the position
of the molecule.
Because v= ṙ and a= v̇ , processesi and iiare mathematically similar
Process (i)
Number density of class v molecules within dr is nfdv. The number flux
of molecules across surface element dSr is equal to
nfdv(v•erdSr)
where dSr is the surface element of dr and er is the unit vector normal to
the surface and pointing out of volume element dr. Total number flux
through the surface dSr is equal to
−∫ nf v⋅e r dS r d v
Sr
After applying Gauss theorem, surface integral can be replaced by
volume integral:
−∫ ∇⋅nf vd d r d v
dr
Or, after taking constants out of integral and performing integration:
−∇⋅ nf vd r d v
Since ∇⋅nf v=∇ nf ⋅vnf  ∇⋅v ,and  ∇⋅v =0 because v is constant,
we can finally write the net rate of change of number of molecules because
of process (i) as
∂nf 
−v⋅
d vdr
∂r
Process (ii)
We have already established the similarity between processes (i) and (ii), so
we can write the rate of change of number of molecules due to process (ii) as
∂nf 
−F⋅
d vd r
∂v
Process (iii)
The change in the number of molecules in dvdr phase space element
happens due to

Collisions involving molecules with velocity in (v,v+dv) range

Collisions that produce molecules with the velocity in (v,v+dv) range
Consider the first type of collisions:
v,v1→v*,v1*
One molecule of class v moving with relative velocity vr with respect to the
molecules of class v1 and scattering molecules into elementary solid angle
r
nf
d
v
d

/ d ¿d

dΩsuffers
collisions
per unit time. There are nfdrdv
such molecules in the phase space element, so the total number of such
collisions in the phase space element per unit time is
1
¿
r
2
n ff
d
¿d d v 1 d v d r
¿
d
Here we used the principle of molecular chaos to independently write
down the number of the molecules of classes v and v1.
Similar formula can be written down for the collisions of the second type:
r
*
2
n f
d
*
*
f *
d  d v 1 d v d r
d
*
*
¿
Because collisions of the first and the second kinds are symmetric, we can
write:
*
d
d
*
*
∣
d  d v 1 d v ∣=∣
d d v1 d v∣
d
d
so the net rate of increase in the number of molecules of class v in the
phase space element dvdr due to the two type of collisions can be written
as
d
n  f f − ff 1  vr
d  d v1 d v d r
d
2
*
*
1
To get the total rate of increase in the number of molecules of class v we
need to integrate over all possible collision partners of class v1 and over all
possible elementary solid angles dΩ:
∞ 4
d
∫ ∫ n  f f − ff 1 v r d  d  d v 1 d v d r
−∞ 0
2
*
*
1
Now let's combine the terms describing all three processes and
cancel out dvdr to obtain Boltzmann equation:
∞ 4
d
2
* *
∂ nf v⋅ ∂ nf F⋅ ∂ nf =
n  f f 1 − ff 1  v r
d  d v1
∫
∫
∂t
∂r
∂v
d
−∞ 0