The Equilibrium Distribution and its Properties
In a previous lecture we derived the H theorem for a single species only, but it can be
shown that the derivation also holds for mixtures of (non-reactive) species, for which the
appropriate definition is,
�
�
�
L
L �J
3
H =
nj ln fj
=
fj ln fj d w
j
species
species
If an equilibrium is reached (dH/dt = 0), H will then have attained its minimum value,
consistent with the constraints that are implicit to the H theorem. These constraints are
(a) Conservation of the number of particles of each species (p.u. volume).
(b) Conservation of the overall momentum density (but species can exchange momentum,
so it is not conserved species by species).
(c) Conservation of overall kinetic energy density (again, not for each species).
We therefore impose the following constraints:
LJ 1
w )d3 w (one equation)
E =
mi w2 fi (w
2
i
LJ
Pw =
mi wf
w i (w
w )d3 w (three equations)
ni =
J
i
fi (w
w )d3 w
(one equation per species)
We adjoin Lagrange multipliers and minimize the functional:
LJ
i
LJ 1
LJ
2
3
mi w fi d w + wγ ·
mi wf
w i d3 w
fi (ln fi )d w +
α i fi d w + β
2
i
i
i
�
�
J
L
1
=
fi ln fi + αi + β mi w2 + wγ · mi w
w d3 w
2
i
3
L
J
3
Notice that a single Lagrange multiplier β is associated with the total sum of energies, and
also a single vector wγ is associated with the total sum of momenta; this is in fact the origin
of the eventual fact that wui = wu and Ti = T . Differentiation relative to fi gives then,
1
w +1 = 0
ln fi + αi + β mi w2 + wγ · mi w
2
1
−(1+αi ) −m
−β 2 mi w
fi = e −
e− i�γ ··w�−
−(1+αi )+
= e−
mi γ 2
2β
1
e−
2
mi β
γ 2
(w
� + �β
)
2
where we have completed the square in the exponent.
The constants αi , wγ and β will now be determined from the constraint equations; before
doing the detailed algebra however, one can readily see that the mean velocities and the
temperatures must indeed be common to all species. For species i,
1
wui =
ni
J
where the common factor e
ζw = w
w + ~�βγ :
mi β
J
J
γ 2
� ~�β
) 3
w
~
wf
w i d3 w
we
w − 2 (w+
dw
wf
w id w = J
=
m
β
J
γ
i
� β )2 3
~
f i d3 w
e− 2 (w+
dw
3
−(1+αi )+
−
wui =
J
mi γ 2
2β
has been dropped in the ratio. Change variable to
J
w − m2i β ζ 2 d3 ζ − �~γ e− m2i β ζ 2 d3 ζ
ζe
−w
−γ
β
=
J − mi β ζ 2
β
e 2 d3 ζ
since the first integration vanishes by symmetry. This result is independent of i, and so
wui = wu, the same for all i.
w − wu),
Similarly, once we know wu = − �~βγ (and recall wc = w
3
1 � �
1
kTi = mi c2 i =
2
2
ni
and changing now to wy =
V
J
mi
2
mi (w
w − wu)2 3
fi d w =
2
mi β
(w
w
2
− wu),
1
β
3
kTi =
2
J
J
mi β
2
� ~
�
~
w−u
d3 w
(w
w − wu)2 e− 2 (w−u)
J − mi β (w−u)
�
� 2 3
~ ~
dw
e 2 w−u
2
−
d3 y
y 2 e−y
J
−y 2 d3 y
e−
The ratio of integrals turns out to be 32 , showing that,
1
kβ
Ti =
again independent of i. So with,
β=
1
kT
and
wγ = −
wu
kT
we have,
ni =
then,
J
−(1+αi )+
fi (w)
w = �e−
��
fi d3 w = K
J
mi u2
2 kT
K
e−
mi (w−u)
�
�
w
~ u
~ 2
2kT
ni = K
�
J∞ d3 w
2kT
mi
0
2
�3/2
−
�e
mi (w−u)
�
�
~ u
~ 2
w
2kT
with
2
−y
e−
4πy 2 dy
mi
(w
w − wu) = y
2kT
and noticing that,
y2 = t
�
2k T
ni = K
mi
�3/2
4π
2
Solving for K,
J∞
fi (w
w ) = ni
�
2πkT
t e dt = K
mi
0
|� ��
{z √ �}
K = ni
Therefore we find,
1 1/2
t dt
2
dy =
�
1/2 −t
Γ( 32 )= 12
π
�
�3/2
mi
2πkT
mi
2πkT
�3/2
e−
�
3/2
mi (w
�
~ −u
�
~ )2
2kT
which is the Maxwellian or Equilibrium distribution function.
We could re-derive the Maxwellian limit using an alternative argument to the optimization
procedure discussed above. During our discussion of the H-theorem, we obtained,
dH
1
=
dt
4
J
dΩ
JJ
(f 0' f1'0
− f f1 ) ln
�
�
f f1
gσd3 wd3 w1 ≤ 0
0'
0
'
f f1
and the equality (equilibrium) can only be true if,
w
for all w,
w w
w 1, Ω
f '0 f1'0 = f f1
Hence the quantity,
ln f (w)
w + ln f (w
w 1)
is conserved in a collision between particles with velocities w,
w w
w 1 . This is an additive quantity.
What other additive quantities are conserved? The list is short; assuming zero momentum,
they are:
(a) From number conservation, any constant quantity (the quantity 1, for instance)
(b) From energy conservation, the quantity 12 mw2
Hence ln f must be a linear combination of these:
1
ln f = ln c1 − c3 mw2
2
and therefore,
c3
f = c1 e− 2 mw
2
If there is non-zero momentum, we should include it and write,
1
w − c3 mw2
ln f = ln c1 + wc2 · mw
2
3
The values of c1 and c3 (and wc2 , if needed) come from imposing normalization such that,
J
J
J
1
3
3
3
wf
w d w = nwu
mw2 f d3 w = n kT
fd w = n
2
2
The result, as with the minimization method, is,
�
�
3/2
m
f (w)
w = n
2πkT
e−
m(w−u)
�
~ �
~ 2
2kT
This method can be generalized to multi-species situations, although in that case, since
there are several kinds of collisions, there will be more than one necessary conditions like
f '0 f1'0 = f f1 , and some care must be exercised with terms arising from unlike particles.
Characteristic energies and velocities for a Maxwellian distribution
In a frame in which the gas is at rest (wu = 0), the mean vector velocity is zero. More
generally, h(w
w − wui)s = 0, for any species s.
We generally define wcs = w
w − wus , the velocity of a particle with regard to the mean of its
species. This is sometimes called the “diffusion velocity”, but care must be taken not to
confuse it with wc = w
w − wu, where wu is the mean mass velocity of all the species present.
We see from the definition that h(wcs i)s ≡ 0 , but h(wcs i) ≡ wus − wu, which, in a non-equilibrium
situation, can be non-zero.
An important velocity magnitude is cs ≡ h(cs i)s , where the magnitude, and not the vector, is
involved. For a Maxwellian,
1
cs =
n�s
�
J
�
ms
cs�
n�s
2πkTs
�
32
ms c2
s
e
− 2kTs d3 cs
Since only |w
|cs | appears, use spherical coordinates, where d3 cs = 4πc2s dcs
cs =
J
∞
cs
0
Define,
ms cs2
x =
2kTs
2
→
cs =
�
�
ms
2πkTs
32
�
�
21
2kTs
ms
x
ms c2
s
e− 2kTs 4πc2s dcs
and
dcs =
�
2kTs
ms
�
12
dx
The integral can be evaluated by changing x
2 = t, x3 dx = 12 tdt, and its value is 12 .
So,
�
�
1
2 2kTs 2
cs = √
π
ms
4
cs =
r
8 kTs
π
ms
V
(c2s )s . This can be calculated
Another important velocity is the RM S velocity, or cRM S =
more easily, in fact, for any distribution, because,
(cs2 ) ≡
2 3
kTs
ms 2
→
cRM S =
3
kTs
ms
Sometimes the distribution of interest is where particles are classified by either velocity
magnitude or by energies. Looking at the first of these, we define a different distribution
(assumed to be isotropic) by,
h(cs )dcs ≡ f (cs )d3 cs = f (cs )4πc2s dcs
or
32
ms
2πkTs
h(cs ) = 4πc2s ns
→
h = 4πc2s f
2
mcs
e− 2kTs
The most probable velocity magnitude follows from,
d ln h
2
mcs
=
−
= 0
dcs
kTs
cs
(cs )most probable =
2
kTs
ms
h(cs )
cs
2kTs
ms
The other (related) definition is when particles are grouped by energies
E =
mc2s
2
→
cs =
2E
ms
In this case,
3
g(E)dE ≡ f d cs =
and so,
dcs = √
and
f 4πcs2 dcs
√
4 2π 1
= f 3/2 E 2 dE
m
√
4 2π 1
g(E) =
E
2 f (E)
m3/2
and for a Maxwellian,
f (E) = n
s
ms
2πkTs
5
dE
2ms E
3
2
E
e− kTs
g(E)
E
Emost probable
we obtain,
1
2ns E 2 � kTE
√
g(E) = √
e s
π (kTs ) 32
The most probable energy follows from,
d ln g
1
1
= 0
=
�
dE
2E kTs
kTs
Emost prob. =
2
,
(cs )most prob. energy =
kTs
ms
All these velocities are comparable to the speed of sound,
kTs
5 kTs
csound =
γ
=
(for a monoatomic gas)
3 ms
ms
0
√√
1
r
6
5
3
√√
2
r
3
8
⇡
π 2
c
q ss
kTss
kT
ms
MIT OpenCourseWare
http://ocw.mit.edu
16.55 Ionized Gases
Fall 2014
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