Kinetic Theory Distribution Functions and Equilibrium
A A Agbormbai
Department of Aeronautics
Imperial College of Science, Technology and Medicine
Prince Consort Road
London SW7 2BY, England
Abstract. This paper clarifies the issue of distribution functions in kinetic theory and statistical mechanics by describing
the variety of distributions and their interrelationships. It differentiates between: (a) single particle distributions and Nparticle distributions (b) phase space distributions and velocity distributions (c) ordinary distributions and generalised
distributions (d) phase space distributions and contracted space distributions (e) flux distributions and density distributions. It also discusses the issue of equilibrium distributions in the phase space and in the contracted space, for single
particles and for TV-particles. This work is fundamental to any consistent analytic formulation of rarefied gas dynamics.
INTRODUCTION
In my exposition of reciprocity theory for molecular collisions and gas surface interactions (Refs. 1 to 6) I have onployed a variety of distribution functions defined in a variety of spaces. Some of these distribution functions are
readily transformed into one another. Nevertheless, it is easy to confuse the different types of distributions. Hence
the need for a clarification among these types and for a discussion of the interrelationships among the different
types. In fact, kinetic theory distributions are the cornerstone of any analytic discussion of rarefied gas dynamics.
Yet in view of their varied types these distribution functions cause a great deal of confusion in theoretical formulations of gas dynamical phenomena. It is important to know what makes each distribution function unique and what
interrelationships exist among the distribution functions. In this paper I introduce and discuss the properties and i nterrelationships among the variety of distribution functions that appear in kinetic theory and statistical mechanics. I
demonstrate the relationships between:
•
•
•
•
•
single particle distributions and Af-particle distributions
phase space distributions and velocity distributions
ordinary distributions and generalised distributions
phase space distributions and contracted space distributions
flux distributions and density distributions
Finally I consider the issue of equilibrium distributions in the phase space and in the contracted space. I formulate
expressions for the single particle distribution and for the A/-particle distribution.
KINETIC THEORY DISTRIBUTIONS
Let us start by examining the differences and relationships between phase space distributions and velocity distributions. Then for molecules with excited internal states we contrast distributions defined in the phase space from di stributions defined in the contracted space. The phase space is defined by the generalised co-ordinates and velocities
of the molecules, whereas the contracted space is defined by the energy modes of the molecules.
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
For a monatomic gas the phase space is defined by the position co-ordinates r and the velocity co-ordinates c. In
rarefied gas dynamics the statistics of molecular phenomena is usually formulated within an elementary control volume dr of physical space. The statistics is formulated in terms of either the single-particle phase space distribution
function (PSDF), Fn (c, r), or the single-particle velocity distribution function (VDF), / (c). Whereas the singleparticle PSDF is non-normalised (hence the subscript ri) the single-particle VDF is normalised.
The single-particle PSDF gives the number of points (atoms) dNthat are located in an elementary extension dcdr
of phase space as:
dN = Fn (c, r)dcdr
where N is the number of atoms in the control volume dr
(1)
The single-particle VDF gives the fraction of points (atoms) that are located within the velocity element dc of phase
space as:
—— = — = f(c)dc
N
n
where
n = n(r) = —
dr
is the number density of the gas
(2)
Within the control volume dr centred on r the elementary number density is given by:
*-£dr
(3)
Combining (1) - (3) gives:
F JI (c,r) = /i(r)/(c)
(4)
This equation provides the mathematical relationship between the single-particle PSDF and the single-particle VDF.
As long as this relationship is remembered either distribution function can be used to formulate the statistics of rar efied gas processes, especially as epitomised in the Boltzmann equation.
This relationship applies even for molecules with excited internal states. If the internal co-ordinates are denoted
q/ then the relationship becomes:
Fn (4f, qz, c, r) = n(r)f(ty, qz, c)
(5)
This equation relates the single-particle generalised PSDF (GPSDF) to the single-particle generalised VDF
(GVDF).
We can derive similar relationships for JV-particle distribution functions. Consider the monatomic case. Using
molecular chaos we can express the JV-particle PSDF in terms of N single-particle PSDFs and then invoke equation
(4) to derive a relationship between the Af-particle PSDF and the JV-particle VDF, viz.:
N-l
7=0
N-l
= M HJ (TJ )fj (Cj)
7=0
N-l
N-l
= FT HJ (TJ )j"T fj (Cj)
7=0
(from equation (4))
(from expansion)
7=0
N-l
= f(N\QY[nj(rj)
(from molecular chaos)
7=0
For internally excited molecules this result becomes:
N-l
where Q, =
(7)
7=0
CONTRACTED DISTRIBUTIONS
When describing rarefied gas phenomena it is often useful to transform the description from a phase space defined
by the generalised co-ordinates and velocities to a contracted space defined by the energy modes of a gas. For gases
with internally excited states the phase space description is too detailed and too abstract. It is more useful to adopt a
contracted description that uses practical variables, which are fewer in number and are more concrete.
We can transform results from the phase space to the contracted space by invoking the standard PhaSIS tran sformation that I have used several times to formulate reciprocity theory from first principles. This transformation
assumes that an energy mode can be expressed as a sum of squares of the phase space co-ordinates that define the
energy mode. To complete the transformation a set of dummy variates are introduced, but these are readily integrated out after the transformation is applied. This integration process causes a contraction of the phase space into an
operating space - alternatively called the contracted space.
Let us imagine that the gas has excited rotational and vibrational internal energies. Then we can write the PhaSIS
transformation for the rotational mode as:
= 2,...,vr
(8)
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?1?...,Z?V ) are constants.
For the vibrational mode the transformation is:
= 2,...,vv
£„ =
(9)
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, and b = (&[,...,&v ) are constants.
These transformations have the following properties (Ref. 7):
dqr =
dqr
d(er,y)
1=2
derdy,
(10)
d(er,y)
1 /-I
2' 2
where Cr is a constant, j8 (^l/Xj, /12) is the Beta distribution with parameters /^ and /12,
and ^(/^i, /^2) is the complete beta function.
•Jn-lG(z),
3(£v>z)
G( Z )=n^, r 2;
(11)
_,_, B( ^
21
u \=—————^^(l-x)^
' ,'
j-,/\
v
/
2/
~r L
(ai > 0~, ur L2 >0,~ 0:
\r i
where Cv is a constant, j8 (jc| ji^, /X 2 ) is the Beta distribution with parameters fa and fa,
and $(/^i, /^2) is the complete beta function.
In these equations I have assumed that all the internal co-ordinates and velocities of each mode have been collected
together into a vector denoted q/ . For instance qr is the vector of rotational co-ordinates and velocities while qv is
the vector of vibrational co-ordinates and velocities.
We can use these results to transform phase space distributions into contracted space distributions, and then we
can derive a relationship between the contracted versions of the GPSDFs and the contracted versions of the GVDFs.
Consider the single-particle VDF which is defined by equation (2). For internally excited gases this equation becomes:
— = /(q v ,q r ,c)Jq Jq//c
n
where q is the vector of internal co - ordinates and velocities for each mode y" (12)
We can transform the RHS in one of two ways:
a) by applying the transformation theorem of distribution calculus; this theorem is epitomised in the equation:
where / is the Jacobian of the inverse transformation x = x(y) and /ris the transformed distribution
b) by applying the Jacobian definition theorem, which is epitomised in the equation:
Note that either theorem excludes the other, because the transformation theorem (a) subsumes the Jacobian definition theorem (b) - i.e. when proving (a) we make use of (b). This means that whenever we transform any term that
has the form:
/(variate - set) d(variatel) J(variate2)A J(variateA^),
where
variate-set = (variatel, variate2,...,variateAT)
we cannot use both (a) and (b). We must use either (a) or (b).
Using (8) to (1 1) to transform (12) we get:
— = CrCv£/r-l£v7*-1 G(y)G(z) /(c,q r ,q v ) dcd£rd£vdydz
n
where
qr = qr(er,y),
qv = q v (£ v ,z),
jr = V^//,
(13)
/v = V^/
This relationship is expressed in the intermediate space defined by the energy modes and by the dummy variates of
the transformation; i.e. the dimensions of the intermediate space are ( e,, £v, y, z). The relationship also defines the
intermediate-space single-particle GVDF as:
/7(c,er,ev,y,z) = CrCvery'-evy
G(y)G(z) /(c,q r ,q v )
(14)
To contract this expression into the operating space we integrate (13) over the dummy variates y and z to get:
— = CrCverr'-Ievr""1 f(c,qr,qv)dcderdev
(15)
n
This relationship defines the contracted single-particle GVDF as:
/c(c,er,ev) = CrCve/'-1e/v"1 /(c,q r ,q v )
The contractedN-particle GVDF is:
CrjCvj£rJ7r
X/V
> where
E
r = (£rO> e rlv»,£rtf-l)>
E
v
=
(^vO^vlv^vAT-l), (17)
7=0
We can now relate the contracted GPSDFs to the contracted GVDFs and also determine relationships between
the GPSDFs and their contracted versions. Rewrite (5) as:
F w (q v ,q r ,c,r) = /i(r)/(q v ,q r ,c)
(18)
and rewrite (7) as:
N-l
W /"*• ^
n ,c,Kj
C* T*\ -—/ /*(^0/Ti
^^^l 1f^
f77(^0/TI
^ v ,v^
vv v? vnr? ^;J^
7^ 7^
n
r
HQ\
^ ^
7=0
Combining (16) and (18) gives:
Fnc(r,c,£r,£v) = n(r)fc(c,£r,ev)
(20a)
where Fnc is the contracted single - particle GPSDF, and:
Equation (20b) is an expression for the contracted single-particle GPSDF.
Combining (17) and (19) gives:
N-l
Fw(f}(R,C,Er,Ev) = /c(Ar)(C,Er,Ev)^Qw7.(ry),
where F^
is the contracted A^-particle GPSDF, and :
7=0
N-l
7=0
Equation (21b) is an expression for the contracted N-particle GPSDF.
(2 la)
GAS SURFACE DISTRIBUTIONS
The relationships for gas surface interactions can be derived in a similar manner to the above. In fact, the only di fference is that the solid vibrational energy enters into the results. In the phase space we use solid vibrational coordinates q5 , and in the contracted space we use the solid vibrational energy es . Note that q5 incorporates both position co-ordinates and velocities. We can summarise the results as follows.
a) Relationship between single-particle PSDF and single-particle VDF (monatomic gas):
Fn(r,c,qJ = /i(r)/(c,q,)
(22)
b) Relationship between JV-particle PSDF and ^/-particle VDF (monatomic gas):
N-l
FW(R,C,q,) = /^(QqjfJ/i/r,),
where
C = (CQ,...,^),
R = (ib,...,!^)
(23)
7=0
c) Relationship between single-particle GPSDF and single-particle GVDF (polyatomic gas):
Fn(r,c,qr,qv,qs) = w(r)/(c,q r ,q v ,q,)
(24)
d) Relationship between JV-particle GPSDF and JV-particle GVDF (polyatomic gas):
F(R,C,Qr,Qv,qJ = /(C,Qr,Qv,q,)
/i/r y )
(25)
7=0
Qr = OlrO^rlv^qrAr-lX
QV = (^vQ^vl^'^vN-l^
C =
(cO>->cN-l)>
R =
( r O>-,%-l)
e) Relationship between contracted single-particle PSDF and contracted single-particle VDF (monatomic gas):
FBC(r,c,eJ) = /i(r)/c(c,eJ)
f)
(26)
Relationship between contracted single-particle GPSDF and contracted single-particle GVDF (polyatomic gas):
Fnc(r,c,£r,£v,es) = n(r)fc(c,£r,£v,£s)
(27)
g) Relationship between contracted AT-particle PSDF and contracted AT-particle VDF (monatomic gas):
N-l
FW(R,C, ej ) = fm(C9es)Ylnj(Tj)
(28)
7=0
h) Relationship between contracted ^/-particle GPSDF and contracted JV-particle GVDF (polyatomic gas):
(ry) where Er =(er0,erl,.,er^_1), Ev =(ev0,evl,.,ev^_1) (29)
7=0
i)
Relationship between single-particle PSDF and contracted single-particle PSDF (monatomic gas):
Fnc(r,c,es) = CfJ--1 F w (r,c,q s )
(30)
j)
Relationship between single-particle GPSDF and contracted single-particle GPSDF (polyatomic gas):
^ C (r,c,e r ,e v ,e 5 ) = CrC/^/'-V^V''1 ^(r,c,q r ,q v ,q s )
(31)
k) Relationship between JV-particle PSDF and contracted JV-particle PSDF (monatomic gas):
FW(R,C,O = C,e/'-XW(R,C,q,)
1)
(32)
Relationship between JV-particle GPSDF and contracted ^/-particle GPSDF (polyatomic gas):
^C^-e^-
(33)
7=0
m) Relationship between single-particle VDF and contracted single-particle VDF (monatomic gas):
fc(c,es) = CseJ--1 /(c,q,)
(34)
n) Relationship between single-particle GVDF and contracted single-particle GVDF (polyatomic gas):
fc(c,er,ev,es) = CrCvCJerr'"1e/v"1e/'"1 /(c,q r ,q v ,q 5 )
(35)
o) Relationship between JV-particle VDF and contracted AT-particle VDF (monatomic gas):
fc(N\C,£s) = CseJ*-lf(N\C,qs)
(36)
p) Relationship between JV-particle GVDF and contracted AT-particle GVDF (polyatomic gas):
rv
r7.Cv^% ^
(37)
7=0
FLUX VS. DENSITY DISTRIBUTIONS
In 1971 Nocilla, Chiado-Piat and Riganti (Ref. 8) pointed out the need for a clear distinction between flux distributions g(c), which form the basis of gas surface theory, and density distributions flc) or Fn(c, r) which form the basis
of molecular collision theory. Flux distributions are based on the number flux of molecules crossing an element of
surface which forms the base of an elementary control volume dr sitting on the surface. Density distributions are
based on the number, or number density, of molecules which are located in an elementary control volume dr . The
Boltzmann equation is usually formulated in terms of density distributions. We now wish to formulate a mathematical relationship between the flux and density distributions.
For a monatomic gas the elementary number flux crossing or leaving a surface is:
dQ = nc.ef(c)dc
Where e is the unit normal vector pointing away from the surface, so that c.e is the normal component of velocity.
With the chosen normal vector pointing away from the surface it follows that incident fluxes are negative (normal
(38)
component of velocity pointing downwards) whereas reflected fluxes are positive (normal component of velocity
pointing upwards).
The total number flux crossing the surface is:
Q=[
nc.ef(c)dc
(39)
This integration is carried out over a normal velocity half space to give the total incident number flux or the total
reflected number flux. The half space is defined as follows:
Incidence :
c.e < 0,
Reflection :
c.e > 0
Therefore in gas surface interactions a molecule crosses from one half space to another as it impacts the surface.
This is because the normal velocity component reverses direction upon impact.
Define the flux distribution in the manner:
(40)
-5^
This is similar to defining the density distribution using equation (2). Using (38) we get:
(41)
This result relates the flux distribution to the density distribution within an elementary control volume dr sitting on a
surface. The density distribution is normalised in fall velocity space as follows:
J
Whereas the flux distribution is normalised in a velocity half space as follows:
J
c.e<0 or
c.e>0
For polyatomic gases equation (41) becomes:
g(c,q r ,q v ) =
c.e/(c,q r ,q v ),
or
gc(c,er,£v) =
c.e/c(c,er,£v)
EQUILIBRIUM
Equilibrium is a state that is sustained and preserved by the mechanism of molecular collisions and gas surface i nteractions. Therefore equilibrium conditions should be derivable from molecular interaction theory. In particular, it
should be possible to derive equilibrium distributions from interaction theory. In the formulation of reciprocity theory I have shown that the theoretical framework allows equilibrium distributions to be derived as a special case of
molecular interaction theory.
Equilibrium is sustained through the reciprocity principle, which states that the rates of forward and inverse i nteractions equal. This means that at equilibrium the collision integral of the Boltzmann equation vanishes. The most
(42)
general situation for deriving equilibrium distributions is that involving many body collisions and many-body gas
surface interactions. For an JV-body system the integrand of the collision integral contains the term:
Therefore a vanishing of the collision integral means that:
F^\C",Q",Q") = F^\C',Q'r,Q'v)
where subscript ^denotes equilibrium conditions
(43)
Thus, the pre-collision equilibrium distribution has the same functional form as the post-collision equilibrium distr ibution. This equation also expresses a conservation law for the collision, because the equilibrium distribution is a
function of the state of the collision as described by the generalised co-ordinates and velocities of the system. We
can introduce logs and write:
InF^CC* Q;,Q;) = lnFW(C',Q'r,Q'v)
(44)
Now invoke molecular chaos by writing:
N-l
F^(C,Q r ,Q v ) = n^(c7,qrj,qvj-)
(45)
7=0
Equation (44) becomes:
N-l
N-l
^lnF J ^(c / y ,q; y ,q / vy ) = ^lnF J ^(c^q; y ,q; y )
7=0
7=0
This conservation equation implies that
In Fne
(where subscript e denotes equilibrium)
is a linear function of the collision invariants, i.e. of the molecular mass, momentum and energy. We can therefore
write:
lnFne = am + b • me + d—mc1 + e£r + g£v, where a,b,£/,e,and g are constants and £r = £ r (q r ), £v = £ v (q v )
We can rewrite this equation as:
Fne = A exp - — (c - B)2 p exp(-££r )F exp(-G£v ),
{ 2C
J
where A, B, C, D, E, F, G are constants
This separates the equilibrium distribution into translational, rotational and vibrational components. We may thus
write:
Fnte = ^ e x p - ( c - B )
f re =Dexp(-££r),
f ve =Fexp(-G£v)
For the translational component we use the mean velocity condition, the equipartition principle, and the normalis ation condition to obtain the constants. Thus we write:
J
cFntedc = ncy
-mc2Fntedc = -nkTt,
J
J
Fntedc = n
where Tt is the translational temperature of the gas
These relations give:
C = kTt
B=c
For the rotational and vibrational energies we use the equipartition principle and the normalisation condition by
writing:
j
erfreder =kTr,
j
freder = 1,
where Tr is the rotational temperature of the gas
I
£vfved£v = kTv,
I
fvedev=l,
where Tv is the vibrational temperature of the gas
These relationships give:
l
kTr
kTv
Combining these results we get:
JH
=
a*
M|I ___
i
I 2
7 irlrT
I
A
a*
,
i exDii _ __
i c _ —^/
c) II __ 7 exDiI _
7
I
l ^* )
\
2
( m f(
where: A = n\———
\2nkTtj
7 ^"T
lki
I kl
^"T
t
)
1 }( 1 }
—— —— L
\^TrJ^kTvj
I
r
I
e
^r
I
I __ exDiI _
^"T"7 Ikl^"T7
kl
*)
e.
e
I
v
{
v
I
^"T7 I
klv
I
I — A^4 exDi
_
)
e
\
m,
r
v
— = —— + ——
+ ——,
kT kTt kTr kTv
I
\
? Tlr ? Tlv/ I
IrT
hi
)
I
I
_,
ef =—(c —c)
* 2
We can derive the contracted equilibrium distribution from this result by using (20b). The constants Cr and Cv can be
determined by using the normalisation condition on each of the rotational and vibrational components of the resul ting distribution. The final contracted distribution is:
T7
m
_ ,J
I
PYJ
CA
P
2nkTt
m
1
where:
AA= n\( ———
\(
m
(»
^!m^
2kTt
1
^\2 I
*")
—————-—
Y
1
,,7r-l™J
(kTr^r(7r)
1
£
r
kTr
1
—————— —
\2nkTt} [(*rr)^r(yr)J[(*rv)^r(yv)J
£
—
£
—t
}
1
^7v-l^J
(kTvY*r(7v)
£r
_
£v
£
v
kTv
m
KT kit kir kiv
Both these equations give the single particle equilibrium distribution in the phase space and in the contracted
space. Using molecular chaos (45) we can reconstruct the JV-particle equilibrium distribution from these results as
follows:
Phase space:
/ /i ,r\
(4o)
N-l
rrij
-—— (c, -c)
O/^T 7
V
-/
2
/
l
t •
IrT
r
\
IrT
g
kT
K1
t -
——exp -——
IrT
r
where: A =
i
——exp -^-
N-\f
^y
^"
*!
kT
, KI
V
% , ^ , £yj
c
c
^T7
Ki
j=0\K1t
c
IrT
r
(48)
lrT
>
IrT
K1
v ^
Contracted space:
TH;
2
exp|-^^(c,-e)
F|
VJ
>
2W,
where:
(49)
9
^ = y(—+—+—
TrT
K1
^\
j=Q\
IrT
K1
t
IfT
K1
r
TrT
K1
m, -^
^=y(c-c)2
v
CONCLUSIONS
We have elucidated and clarified the different types of distribution functions that appear in kinetic theory and stati stical mechanics, and also demonstrated their interrelationships. These results help to avoid confusion when carrying
out theoretical formulations of gas dynamical phenomena. We presented results for both single particle and Afparticle distributions. We also showed the relationship between density and flux distributions. This distinction is
fundamental to analytic formulations of gas surface interactions. Finally, we examined and clarified the problem of
equilibrium distributions in kinetic theory and statistical mechanics. This work completes my analysis of many body
collisions, and sets the stage for the discussion of many body gas surface interactions.
REFERENCES
^gbormbai A. A., PhD Thesis, Imp. Coll., Univ. London (1988).
2
Agboraibai A. A., Dynamical and Statistical Modelling of Many Body Collisions II: Energy Exchange , submitted to Rarefied
Gas Dynamics 22nd symp.
3
Agbormbai A. A., Reciprocity Theory of Vibrationally Excited Many Body Collisions , submitted to Rarefied Gas Dynamics 22nd
symp.
4
Agbormbai A. A., Reciprocity Modelling of Vibrationally Excited Four Body Collisions , submitted to Rarefied Gas Dynamics
22nd symp.
5
Agbormbai A. A., H Theorem for Many Body Collisions, submitted to Rarefied Gas Dynamics 22nd symp.
6
Agbormbai A. A., H Theorem for Vibrationally Excited Many Body Collisions , submitted to Rarefied Gas Dynamics 22nd symp.
73
Pullin D. L, Phys. Fluids, 21, p209 (1978).
8
Nocilla S, Chiado-Piat M G, and Riganti R, Rarefied Gas Dynamics, 7th symposium, vol. 1, p!57 (1971).