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Transcript
H Theorem for Many Body Collisions
A A Agbormbai
Department of Aeronautics
Imperial College of Science, Technology and Medicine
London SW7 2BY, England
Abstract. Although rarefied gas dynamics has traditionally stood 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 many-body collisions involving atoms. This 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 monatomic many-body collisions satisfy
detailed balance at equilibrium as well as symmetry with respect to the forward and inverse processes. I begin by deriving the Boltzmann equation for monatomic gases undergoing many body collisions. This leads to the formulation of an
elastic collision integral. All these descriptions are carried out in the single-particle phase space of the gas. I formulate the
properties of the elastic collision integral and then I use these to formulate an H theorem for reciprocity-based many body
collisions. Equilibrium distributions are derived from the approach. 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 dynamical and statistical modelling to many body collisions involving monatomic gases. The statistical element 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 //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
these same laws continually drive the rates towards equality at equilibrium. At equilibrium the laws ensure that the
rates remain equal.
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gain's
© 2001 American Institute of Physics 0-7354-0025-3/01/\$18.00
87
A process that satisfies reciprocity and the H 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 important.
The Boltzmann equation for monatomic gases experiencing binary collisions is:
~\T7'
or'
?J 77
o*n
?JZ7
Or
„
dt
3r
3c
.
I ) Z7 \
I / Or
n I
I /^
or' Z7
(^ dt J.oll
(^ dt
where : 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 consistof the collision integral and the gas surface interaction integral.
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:
3c
where j indicates the number of interacting atoms. For gas surface interactions j indicates the number of atoms
that are interacting with the surface.
THE COLLISION INTEGRALS
We can formulate the collision integral from first principles, as for gases undergoing binary collisions. However, it
is easier to proceed by extending the binary collision integral. The binary collision integral for monatomic gases is:
= (c,,c 2 )
coli
(molecular chaos),
where :
F^
is the two - particle phase space distribution;
Fnl,Fn2 are single -particle phase space distributions
g
is the relative speed of the particles,
b
is the impact parameter of the collision
£
is theazimuthal angle defining the collision plane
Single primes denotethe pre - collision state; double primes denote the post- collision state.
To incorporate many-body collisions choose particle 0 as the reference particle and introduce TV -1 reduced particles
for an TV-body system. The three-body collision integral is then:
= (c 0 ,c b c 2 )
*{l\db(de{ g2b2db2d£2dcldc2,
F^}
b{, b2
is the three - particle phasespace distribution;
g\-,g2 are
are the impact parameters for scattering of the reduced particles
e t , £2
are the azimuthal angles defining the pre - collision planes
tne
where :
relative speeds of the two reduced particles
Single primes denote the pre - collision state; double primes denotethe post- collision state.
The TV-body collision integral is:
Ijycy,
C = (c0,ch...,cN_{)
7=0
F^
is the TV - particle phasespace distributbn,
gj
is the relative speed of reduced particle j
bj
is the impact parameter for scattering of reduced particle j
£j
is the azimuthal angle defining the pre - collision plane for reduced particle j
Single primes denote the pre - collision state; double primes denote the post- collision state.
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.
for three -body collisions:
0(C) = 00(co) + 0i(ci) +<fe( c 2)
where
C=(c 0 ,c 1 ,c 2 )
N-l
for TV -body collisions:
0(C) = N 0y(c7-)
where
7=0
89
C = (co,c lv .., CN_I)
Consider the following moment of the collision integral, which is taken for molecule 0:
g(b{db(de{ g'2b'2db'2de'2 dc'0<fc;</c'2
For the inverse process the corresponding moment is:
Write:
de' = g'jbj db'jde'j
for each reduced particle j (an extension of the binary result)
c'2 = dcQdc\dc'2
(Liouville' s theorem)
to get:
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:
* (O) g[b(db{de{ g'2b'2db'2de'2 dc^c'i^z
- tf 3)(O)
(1)
g'tidb[de{ g'2b'2db'2de'22 <0cc2
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:
JJJ tfb - 4>o) k(3)(C') - F;(3) (C')j g'}b{db[de{ g2b'2db'2de'2
= JJJ « -tf) (Fn^\C") -F;(3)(C')) g{b[db[de{ g'2b'2db'2de'2
= JJJ (02 -fc') k'(3)(C') -^'(3)(C')) g(b(db[de{ g'2b'2db'2de'2
= | JJJ (^o - 0o + tf - 0i"+ 02 - fc") k"(3) (C*) - F'n^(C})g{b[db(de{ g'2b2db2ds2
For the TV-body problem this becomes:
90
J'=0
7=0
_y=o
J
j=i
y=o
Combining (1) and (2) we get the final result of this symmetrisation procedure as:
- F'(3) (C') g
(4)
g'2b2db2de2
g'2b'2db2d£'2
-F;(3)(C'))^XJ£; g'2b'2db'2de2
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 (4) to formulate an H theorem for three-body collisions, to confirm that reciprocity -based three-body collision models satisfy the second law of thermodynamics.
For TV-body collisions the symmetrisation yields:
A/-1
N-l
N-l
j (5)
j=0
7-0
For collision invariants (i.e. mass, momentum or energy) the following sum vanishes:
7=0
91
Therefore the moment vanishes. We can use the result of equation (5) to formulate an //theorem for TV-body collisions, to confirm that reciprocity-based TV-body collision models satisfy the second law of thermodynamics.
THE H THEOREM AND EQUILIBRIUM DISTRIBUTIONS
The entropy of the system is:
S=
—— f f f ^ o h ^ o ^ c o+const
nQJJJ
where subscript 0 denotes molecules of species Oand k is Boltzmann' s constant
Differentiating with respect to time gives:
o t
Since the readjustment towards equilibrium is caused by collisions we substitute the collision integral into this
equation to get:
For an TV-body system this is:
Now symmetrise these equations by applying (4) and (5), to get:
-dt= —
ff
6n»fJJJ
In PL'(3)
;(3)
-Fwgftdb;de;
where we have used molecular chaos:
g'2b>2db'2d£'2
( } ,
F7(3)
= FnQFnlFn2
n
For an TV-body system we get:
dS
k
-dt= —
ff
2Nn,f JJJ
2Nnn
In ^
where we have used molecular chaos:
In both these equations the integrand on the right has the standard form:
92
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. If x is less than y then both terms in the product
are negative so that the product is positive. Ifx is equal to j; 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:
dt
dt
dt
Thus, the entropy always increases or H always decreases towards equilibrium.
We can construct equilibrium distributions from this approach. At equilibrium we have:
dS _ dH _
dt~ dt ~
This gives:
orfor/Y -body:
Taking logs gives:
In F/(3) = In F;(3)
or for N -body: In F/(A° = In F^(
Invoking molecular chaos these results become:
In F^o + In Fw" + In Fw'2 = In F^0 + In F'nl + In Fw'2
and
7=0
7=0
These are conservation equations for the collision and imply that
In Fne
(where subscript e denotes equilibriu m conditions)
is a linear function of the collision invariants, i.e. of the molecular mass, momentum and energy. Proceeding in the
standard manner we get ultimately:
' m ^
. ...
———
exp ——- (c-c)
(2nkT
REFERENCES
^gbormbai A. A., Dynamical and Statistical Modelling of Many Body Collisions I: Scattering, submitted to Rarefied Gas Dynamics 22nd symp.
2
Agbormbai A. A, Dynamical and Statistical Modelling of Many Body Collisions II: Energy Exchange, submitted to Rarefied
Gas Dynamics 22nd symp.
93
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