H Theorem for Vibrationally-Excited Many Body Collisions
A A Agbormbai
Department of Aeronautics
Imperial College of Science, Technology and Medicine
Prince Consort Road
London SW7 2BY, England
Abstract. Whereas rarefied gas dynamics has traditionally rested on the dilute gas assumption, which supposes that the
densities are so low that only binary collisions and single-body gas surface interactions occur, expressions for many-body
collision rates and for many-body gas surface interaction (GSI) rates seem to suggest that at lower heights the dilute gas
assumption is not valid. In particular, in the pure rarefied regime, two-body GSIs and some three-body interactions occur
whereas, in the transition regime into continuum flow, four body collisions and four-body GSIs occur. In this paper I
formulate an H theorem for vibrationally excited many-body collisions. Such an exercise constitutes the final stage of
constructing reciprocity proofs for many body collisions. The first two stages were pursued during the formulation of the
many body collision models. They involved demonstrating that reciprocity models for vibrationally excited many-body
collisions satisfy detailed balance at equilibrium as well as symmetry with respect to the forward and inverse processes. I
apply a standard transformation developed in earlier works to transform the description into an intermediate space. Then I
contract the description into an operating space. Three main results are obtained: a contracted version of the generalised
velocity distribution function, a contracted version of the Boltzmann equation, and a contracted version of the collision
integral. I formulate the properties of the contracted collision integral and then I use these to formulate an H theorem for
reciprocity-based many-body inelastic collisions. Equilibrium distributions are derived from the approach. Finally, I
derive a conservation of extension (i.e. a Liouville theorem) equation in the contracted space for many-body inelastic
collisions. An H theorem proof for reciprocity models demonstrates that these models will drive a gas towards
equilibrium when used in DSMC computations.
INTRODUCTION
Previously (Refs. 1 and 2) I applied statistical modelling to vibrationally excited many body collisions involving
polyatomic gases. The statistical approach was based on reciprocity or detailed balance at equilibrium. The models
were derived from a reciprocity equation using partial exchange modelling. What remains is to construct reciprocity
proofs for these models. Reciprocity proofs involve demonstrating that reciprocity based models satisfy three laws:
reciprocity at equilibrium, symmetry with respect to the forward and inverse processes, and the H theorem. When
formulating the models for many body collisions I showed that these models satisfy detailed balance and symmetry.
Now I discuss the final aspect: satisfaction of the //theorem.
The H theorem is the statistical representation of the second law of thermodynamics, which defines the proper
direction of change of a process. It states that whenever a process is given the right to act on its own the process will
always seek new equilibrium states. In other words a non-equilibrium process will always readjust towards an
equilibrium state. The mechanism for this readjustment derives from the characteristic behaviour of microscopic
interactions. Through these interactions all the macroscopic effects of a gas manifest. For instance, pressure,
temperature and density variations are observed throughout the flow; and heat transfer, lift and drag are generated at
solid surfaces. And at all times the gas is driven towards equilibrium.
Any rigorous model of microscopic interactions must obey the H theorem as well as reciprocity at equilibrium.
These laws impose constraints that the modelling process will satisfy. The laws for microscopic interactions must be
the same whether or not a gas is in equilibrium. The presence or absence of equilibrium is attached to the relative
rates of forward and inverse interactions, not to differences in the laws of molecular processes. In non-equilibrium,
the laws of the interactions ensure that the rates of forward and inverse interactions remain unequal. Nevertheless
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
these same laws continually drive the rates towards equality at equilibrium. At equilibrium the laws ensure that the
rates remain equal.
A process that satisfies reciprocity and the //theorem is called a natural process . These are the only processes
that nature tolerates. However, it is possible to construct models that represent an unnatural process. When this
happens we say that the models have been improperly formulated. Such models do not satisfy reciprocity or the
second law of thermodynamics. When a model has all the required natural characteristics built in, the model has
greater chances of success.
THE BOLTZMANN EQUATION
Microscopic interactions are responsible for all the interesting phenomena that occur in a gas. They govern the
variations in properties across the fluid and also govern the transfer of heat across surfaces as well as the generation
of forces on these surfaces. These variations in fluid properties are often formulated in terms of statistical
distributions. Statistical distributions are also used to formulate the surface boundary condition. The Boltzmann
equation arises from the attempt to equate the variations in fluid properties to the behaviour of the microscopic
interactions that generate them.
The derivation of the Boltzmann equation is well established for monatomic gases. These derivations have
assumed that external forces depend only on the position co-ordinate of the molecules. Also, derivations have been
made in an elementary control volume that is located far away from the surfaces. As a result only the effects of bulk
collisions are included explicitly in the equation. The effects of gas surface collisions are only accounted for
implicitly as a surface boundary condition on solutions of the Boltzmann equation. This situation is fine as long as
the Boltzmann equation can be solved for realistic problems. Unfortunately, attempts to solve the equation have only
met with disappointment. Instead of approaching the equation directly researchers have developed an alternative
method which numerically simulates solutions of the equation. The technique is called the Direct Simulation Monte
Carlo (DSMC) method. The formulation of an H theorem for DSMC-based gas surface interaction models requires
that a new formulation of the Boltzmann equation be given which incorporates the effects of gas surface interactions
directly in the formalism. This is the line of approach that we shall pursue.
We can choose to derive the many-body Boltzmann equation from first principles, as for gases experiencing only
binary collisions. However, since this elementary approach is well established for binary collisions, we choose to
begin our derivation from the two-body Boltzmann equation. We locate our elementary control volume on a solid
surface so that both molecular collisions and gas surface interactions are significant.
The Boltzmann equation for monatomic gases experiencing binary collisions is:
dt
Fn
is the non- normalised single particle distribution
r
is the position vector of a particle, c is the velocity of a particle
The terms on the RHS are the collision integral and the gas surface interaction integral
Using this result we can write down directly the Boltzmann equation for polyatomic gases experiencing only binary
collisions. This is:
*
3q
3q
dt
coll
dt
gsi
where q is the generalised co - ordinate vector comprising both position and internal co - ordinates
In a straightforward way we can generalise this equation to incorporate many body interactions. We simply include
additional interaction integrals on the RHS to account for each type of many-body encounter that occurs in the gas.
The above equation becomes:
rlF
\
—3f
where j indicates the number of interacting atoms.
(3)
IsiJ
For gas surface interactions j indicates the number of atoms that are interacting with the surface.
I have expressed the different versions of the Boltzmann equation in terms of the single-particle distribution
function, but we can introduce the velocity distribution function by writing:
where n is the number density of the gas. The Boltzmann equation is an equation of continuity in the single particle
phase space.
THE COLLISION INTEGRALS
For vibrationally excited three body collisions we can formulate the collision integral from first principles to get:
x - v , ~\\
[v °t
MrM
"r
ycoiij
= JJJfe3)(^Q'.Qv)-^(3)(c'>Q^Qv)K(c;,Q;,Q;|c:,Q';,Q;)dc:jQ;jQ;
(4>
x g{b{db{ds{ g2b2db2d£2 dc[dqfrldqvldc2dqr2dqfv2
For TV-bodies the collision integral is:
d(nf)}
",Q;,Q;)-F^>(C-,Q;,Q;))^(C;,Q;,Q;|C:,Q;,Q^C:^;JQ;
THE CONTRACTED GENERALISED VELOCITY DISTRIBUTION FUNCTION
Given the velocity distribution function in the single-particle phase space we want to find the velocity distribution
function in the contracted space. The velocity and internal energies define this space. We denote the phase space
distribution function by/and the contracted space distribution function by/ We want to relate these distributions so
that we can use the result as a basis for deriving a contracted version of the Boltzmann equation.
Given the velocity distribution in the phase space we can obtain the velocity distribution in the contracted space
by first transforming the distribution into the intermediate space and then contracting the result by integrating over
extraneous variables. We use the PhaSIS transformation:
/
-m.—i
l=2,...,vr
n=l
where er is the rotational energy,y = (y2,..., yv ) are the extraneous variables, vr is the
number of rotational degrees of freedom, / counts through the degrees of freedom, and b = (Z^,..., bVr) are constants.
(5)
(6)
= 2,...,Vv
m=\
where ev is the vibrational energy, z = (z2,..., zv ) are the extraneous variables, vv is the
number of vibrational degrees of freedom, / counts through the degrees of freedom, andb = (Z^,..., bv ) are constants.
These transformations have been shown (Ref. 3) to have the following Jacobians:
dqr =
(7)
d(er,y)
I
l
~l
2' 2
1=2
where Cr is a constant, ^xjjUj,^ is the Beta distribution with parameters
and 5(/i!,/i2) is the complete beta function.
(8)
1 /-I
( A £ l >o, A i 2 >0,
7=2
where Cv is a constant, ^(^l^j,^) is the Beta distribution with parameters
! , 1^2 ) is the complete beta function.
The transformation theorem of distribution calculus gives the intermediate space distribution in terms of the
phase space distribution as:
/
J
= J j
r vf
and
=
J,, =
(9)
The contracted space distribution is therefore:
fc = JJJ// W*
(10)
Combining equations (7) - (10) we get for the contracted velocity distribution function:
f (c 'P r jPt v}/
-/cA
= C rC vPc r /2~lpc v /2 ~l ./V
f(cc ?aH r a' H y}/
To formulate an //theorem in the contracted space we need an expression for the entropy in the contracted space. In
the phase space the entropy is:
S = -kH = -k\\\flnf
dcdqrdqv,
where k is Boltzmann' s constant
Using (11) and (7) - (8) in this expression, crossing out terms and integrating over the dummy variables gives:
flV11 1^/
s = -t
fc
In
(12)
dcderdev
CONTRACTING THE BOLTZMANN EQUATION AND THE COLLISION INTEGRAL
Far away from the surfaces the Boltzmann equation can be written:
,
dt
dt
coll
or
O o,—
/ o• )CQ
/ -t, d(flb/o)
•/ , dfob/o), •/ , dQo/o)
—-—7—— • CQ -r ———;—— • qz-Q -r ——-}—— (
dr0
dc0
dq.0
dq- 0
where q^ are the internal co-ordinates and velocities. Subscript 0 denotes molecules of species 0.
The collision integrals are:
vs -^, ^ \
n (3)
'r9Q^^^
Aon.
x g{b{db[de{ g'2b'2db'2de'2
or:
7-1
(j = 2,3,..,
We can simplify the LHS by assuming that the external force does not affect the internal energies (i.e. the external
force is non-rotary and non-oscillatory). These energies are only changed during an inelastic collision - they are not
changed between collisions. Between collisions the external force only accelerates the translational motion. With
this assumption, the rightmost term disappears because the internal acceleration is zero. The other internal term also
disappears by writing:
dfoo/o) , %0
•'
aq;0
= dWo)d£A)
•/
ae;0 aq;0 qr°
dQo/o) d<o ,qv• / ^ dQo/o) d£r0
a^ aq;0 ° a<0 jr
8(^/0) de'^ ^ 3(^0/0) de^ ^
& a<0 dt
a^
because both internal energies are constant between collisions (in the absence of a rotary or oscillatory external
force).
For the RHS we invoke molecular chaos or statistical independence by writing:
= Fn0FnlFn2 = « 0 «i "2/0/1/2
or
Fn
=
Fnj =
j=0
njfj
j=0
Then transform the whole equation into the intermediate space by invoking the transformation relationships (5) - (9)
to get:
d (KO//O)
T
~
c
JJJ (f'M'nW ~ //0//1//2 )gfo'<2\'<fei g^2^2^2 *>'|P')^P' dt\d£'rlde'vldy\dz\ dc'2de'r2de'v2dy'2dz'2
r r r r r r
f P' P'' P'
P' \ '
>
yr _ JrQJr\Jr2J vOJvljv2 ^ fc rO fc rl fc r
jff jt jt jff jf jff
* // ^
J
rO J rl J r2 J vO J vl J v2
I fc rO fc rl fc r
.
2
P' P' ^\ ' 2
For the TV-body problem we get:
N-l
7=0
where : JF = •
7=0
N-l
P=
, y Ar _ 1 ;z 0 ,z 1 ,
7=0
dzN_l9
Note that for the sake of simplicity I have ignored the summation taken over all possible types of TV-body interaction
(i.e. 1-body, 2-body, etc.). We can now contract this intermediate space Boltzmann equation by multiplying the
equation by:
integrating over all dummy variates and applying equation (10). Note that the reflected elementary variates required
to contract the reflected distributions derive from the term dP" ' . Note also that the dummy Beta variates vary
between 0 and 1 . We also write:
Rc s^S'eJi/ET = Re p ^ P P 7 ' = G(^)d&'9 where S = (et, E.),
et is the vector of translational energies for the
reduced particles, Ez- is the vector of internal energies of the molecules, s is the correlatio n variate vector and G
is the set of correlatio n densities.
Rc is the contracted transition probability for internal-translational energy exchange. The resulting Boltzmann
equation is:
C
rrr
JJJ (frtJc\faW
W =\
~ fd)fcifc2)gibidb{d£{
•'^'.P'~ ^'2~l(e' e e 2 y2'1
ff
ff
ff
^° ^ ^
,
g'2b'2db'2d£2 G(^)ds dc[d£rld£vl dc'2d£r2d£v2,
or (for TV - bodies):
v
(N-l
}
/2-i f^-
1
^
IK
IK
7=0
IK
IK
7=0
N-l
where : W =
N-l
This is the contracted Boltzmann equation and the integral on the right is the contracted collision integral. The
equation applies for polyatomic gases and incorporates inelastic collisions.
PROPERTIES OF THE COLLISION INTEGRAL
Consider any microscopic property 0that depends on the state of the collision such that the value at any point is
given by the sum of values for each molecule in the collision, i.e.
= 0o(co) +0i (ci)
for three - body collisions :
+
02( C 2)
where
C = (Co,c 1? c 2 )
N-l
for N - body collisions :
0(C)
=
c/)
where
C = (c0,clv.., CN_{)
7=0
Consider the following moment of the collision integral, which is taken for molecule 0:
xdc'0de'r0de'v0
KJ:F-KMMj>[db\te(
dc\de'rlde'vl dc'2de'r2de'v2
g'2b'2db'2de'2 G(sO<fc'
For the inverse process the corresponding moment is:
-W2 JJJc«><o.<o)ta/;i/;2 - Mfcjr')gK<toi<ie; giwwz G(s>s"
^de^de^Q dc[de'ide^i (k^de^de^
(13)
where : JT* =
Write:
for each reduced particle j (an extension of the binary result)
(the
forward and inverse transition probabilities are identical)
In the phase space, Liouville's theorem writes:
dC'dQ'rdQ'v =dC"dQ"dQ"
Transforming this to the intermediate space and contracting the result gives:
crQ
where
crl
cr2
c
jr =VrA,
p' 7v ~ V
£
£
vl
v2
»
v\
vc"
£
v2
(14)
For the TV-body collision we get:
N-l
j=0
j=0
These equations can be rewritten:
(15)
This equation defines the conservation of extension in the contracted space, i.e. Liouville's theorem in the
contracted space. Substituting this equation and (14) into the inverse-process moment equation (13) we get:
o'(cs,^.*;>)(^^
f
f
xdc'0de'r0de'v0 dC\de rlde vl dt'2de'r2de'v2
Since the forward and inverse processes are symmetrical, we expect the forward and inverse moments to be equal. If
they are equal then each must equal the average of the two. Therefore we write:
{ g'2b'2db'2de'2
(16)
~ fcofcifc2
)
dc[d£'rld£'vl dc2defr2dev2
This relationship can also be written for 0in terms of subscripts 1 and 2.
The collision is symmetrical with respect to each participating molecule. This is because the collision is the
single union between the molecules. It is the single entity that unites them. It matters not whether we see the
collision as molecule 0 colliding with molecules 1 and 2 or molecule 1 colliding with molecules 0 and 2 or molecule
2 colliding with molecules 0 and 1, the properties of the collision are the same. Therefore, we can interchange
indices to write:
g'2b'2db'2de'2 G(S')dS'dC'0de'r0de'v0 </«',<, dc'2de'r2de'v2
JJJ (0o - 0o) (fA&fcir-
{ g'2b2db'2de'2 G(s')<k' dt'Qde'r(>de'M dt[de'rlde'vl ck'2de'r2de'v2
JJJ
c
c
[de{ g'2b'2db'2de'2 G(s')Js'c/Coc/<0^ dt[d£'rlde'vl dc'2de'r2de'v2
c
~ f'M'c2 }g[b{db{d£{ g2b2db2d£2 G(S^'
= - 11 I (0o ~ 00 + 01 ~ 0l"+ 02 ~ 0D (//O
3 JJJ
Xtfco^ e r0^ e vO rfCl^ferl^ e vl dc'2d£'r2d£'v2
For the TV-body problem this becomes:
#-i
#-i
N-l
N-l
7=0
7=0
7=1
7=0
AT-1
7=0
N-l
N-l
7=1
^V-l
(17)
N-l
N-l
7=0
7=0
7=0
7=0
ww n* xw=
N-l
7=1
7=0
>
TV-1
TV-1
7=0
7=0
7=0
7=1
N-l
Combining (16) and (17) we get the final result of this symmetrisation procedure as:
'2b'2db2de'2 G(s)ds'dc'0de'r0de'v0dclde'rlde'vl dc'2de'r2de'v2
g'2b'2db'2de2 G(s')ds dc0de'r0
dc'2de'r2de'v2
If 0 is a collision invariant (i.e. mass, momentum or energy) the following sum vanishes:
Therefore the moment vanishes. We can use the result of equation (18) to formulate an H theorem for three-body
inelastic collisions, thus confirming that reciprocity-based three-body collision models satisfy the second law of
thermodynamic s .
For TV-body collisions the symmetrisation yields:
IN
N-l
N-l
Af-1
7=0
7=0
7=0
(19)
7=0
7=0
7=0
7=0
7=1
7=0
A
7=0
7
N-l
7=0
»T[/<-n>*
Af-1
Af-1
7=0
7=0
N-l
N-l
7=0
For collision invariants (i.e. mass, momentum or energy) the following sum vanishes:
Therefore the moment vanishes. We can use the result of equation (19) to formulate an H theorem for TV-body
inelastic collisions, thus confirming that reciprocity-based TV-body collision models satisfy the second law of
thermodynamic s.
THE H THEOREM AND EQUILIBRIUM DISTRIBUTIONS
The entropy of the system in the contracted space is given from (12) as:
}
+ const
Differentiating with respect to time gives:
dS
— = -— fff — (n f
dt
~
HQ JJJ dt
°
l + ln
C
-'- + const
°
Since the readjustment towards equilibrium is caused by collisions we substitute the collision integral into this
equation to get:
dS_
dt
l + ln
fcO
where : di =g[b(db(d£{
S^CQ^O^VO dt{de'ride'vi dc'2d£'r2d£'v2
For an TV-body system this is:
d
-±=-k
dt
N-l
N-l
7=0
7=0
l+ln
;=i
N-l
where : dr = <
7=1
7=0
Symmetrise these equations by applying (18) and (19). The result is:
f
dt
= ii.,-2 * f IT In
6
JJJ
For anJV-body system this is:
+ const
dS_
~dt
N-l
In
AT-1
j=0
7=1
The integrand on the right of both equations has the standard form:
(x}
(x - j)ln — = (x - j)(ln x - In y)
This integrand is always positive or zero, i.e. is non-negative. We can see this as follows. If x is greater than y then
both terms in the product are positive so that the product is positive. Ifx is less thanj then both terms in the product
are negative so that the product is positive. If x is equal to y then both terms in the product are zero so that the
product is zero. Thus, the integrand is always positive or zero. We can thus write:
dH
dH
—— >0<=> —— <0
dt
dt
dS
—
dt
Thus, the entropy always increases or H always decreases towards equilibrium. We can construct equilibrium
distributions from the approach.
At equilibrium we have:
dS __ dH
dt ~ dt ~
This gives:
f» f" f"W=
f
J cQJ c\J c2yy
f
f
J cQJc\Jc2
Proceeding in the standard manner (Ref. 3) we obtain the following equilibrium distribution:
m
fe
-
m
r
^ e x D(i - —
1
O
r-i
—. ( c --*\
c r I———————e/
exp( -——
\2nkT, \
I 2kTt
1
kTr\
REFERENCES
!
Agbormbai A. A., Reciprocity Theory of Vibrationally Excited Many Body Collisions, submitted to Rarefied Gas Dynamics 22nd
symp.
2
Agbormbai A. A., Reciprocity Modelling of Vibrationally Excited Four Body Collisions, submitted to Rarefied Gas Dynamics
22nd symp.
3
Pullin D. L, Phys. Fluids, 21, p209 (1978).