Skeleton Notation for Reciprocity Modelling of Rarefied
Gas Processes
A A Agbormbai
Department of Computing
City of London College, Whitechapel Road, London E1 1DU, UK.
ABSTRACT
In modelling rarefied gas interactions it is useful to employ a phenomenological description based on reciprocity
theory. Such an approach takes the detailed balance or reciprocity principle as the foundation stone for its mathematical development. A problem with reciprocity modelling is the mathematical complexity of the approach, which
arises from the complexity of the processes being modelled. Thus although the approach derives from elementary
principles the mathematical complexity of its presentation tends to hide the simplicity of its underlying concepts. To
correct this problem it is necessary to devise a notational system, which hides the mathematical complexity while
exposing the simplicity of the underlying principles. The notational system is called skeleton notation because it represents only in bare-bone or condensed form the mathematical exposition. Skeleton notation embraces two other
notational systems: bracket notation and transformation notation. Transformation notation is especially useful in
cases of extreme complexity. First skeleton notation is described and then it is applied to a number of reciprocity
modelling techniques as well as to the construction of reciprocity proofs.
INTRODUCTION
The mathematical details of reciprocity modelling were unravelled in Ref. 1, which discusses reciprocity modelling techniques. Similar mathematical details were utilised in Ref. 2 in the construction of reciprocity proofs for
binary collisions. The extension of these methods into many body interactions raises the need for a condensed
notational system that simplifies the presentation. This is because many body interactions are intrinsically complex. In previous works (Refs. 3 – 8) discussions of many body interactions have focussed on describing a number of stochastic models for the interaction without necessarily discussing the methods for formulating these
models. In this article a new notational system is discussed for reciprocity modelling and then the notation is
used to condense a number of examples of reciprocity proofs and of reciprocity modelling methods. Future work
will employ the notational system in constructing reciprocity proofs for many body interactions.
The paper begins by describing the rules of skeleton notation and then both bracket notation and transformation notation are described for representing elementary transformations. Skeleton notation is then applied
to Gamma-Beta calculus, to the explicit modelling of vibrationally excited single-body gas surface interactions,
and finally to reciprocity proofs for binary inelastic collisions.
SKELETON NOTATION
The rules are as follows:
1) Ignore all elementary differential terms from the reciprocity equation, i.e. differential terms are implied by
the presence of the probability densities for the variates whose differentials are ignored.
2) Since the reciprocity equation is symmetrical between the forward and inverse processes use only one side
to represent the whole equation. When the forward process is implied this condensed reciprocity equation
will be expressed in terms of single primes, and when the inverse process is implied the condensed recipCP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
rocity equation will be expressed in terms of double primes. Note that the LHS of the reciprocity equation
expresses the forward process while the RHS expresses the inverse process.
3) When writing down the elementary transformations express the fact that this is being done by preceding the
presentation with the symbol T:
4) When writing down the final overall transformations express the fact that this is being done by preceding
the presentation with the symbol O:
5) When writing down the final overall inverse transformations express the fact that this is being done by preceding the presentation with the symbol I:
6) When writing down the reciprocity equation express the fact that this is being done by preceding the presentation with the symbol R:
7) For Gamma distributions write:
[
]
Ga ( x1 γ 1 ); ( x2 γ 2 ); ... = Ga (x1 γ 1 ) Ga (x2 γ 2 ) ...
Ga (x µ ) =
1
x µ −1e − x , ( µ > 0, 0 ≤ x ≤ ∞ )
Γ( µ )
is the Gamma distribution for x with parameter µ , Γ( µ ) is the gamma function.
where
8) For Beta distributions write:
[
]
β ( x1 µ1 , λ1 ); ( x2 µ 2 , λ2 ); ... = β (x1 µ1 , λ1 ) β (x2 µ 2 , λ2 ) ...
where : β (x µ1 , µ 2 ) =
1
x µ1 −1 (1 − x ) µ 2 −1 ,
B ( µ1 , µ 2 )
( µ1 > 0, µ2 > 0, 0 ≤ x ≤ 1) is
the Beta distribution with parameters µ1 and µ 2 , B ( µ1 , µ 2 ) is the complete beta function.
9) For Normal distributions write:
[
]
N ( x1 µ1 ,σ 1 ); ( x2 µ2 ,σ 2 ); ... = N (x1 µ1 ,σ 1 ) N (x2 µ2 ,σ 2 )...
10) For the elementary transformations use either bracket notation or transformation notation, as described below.
Bracket Notation
For each elementary transformation and its inverse write:
( x1 + x2 ; y2 ) → y1
or
(Combining)
( y1; y2 ) → ( x1 , x2 )
where : y1 = x1 + x2 ,
or
= sum of variates
x1 = y1 y2 ,
= product of variates
with Jacobian
( Decomposing)
y2 = x1 ( x1 + x2 )
(forward)
= ratio of first variate to sum of variates
~
x2 = y1 ~
y2
y2 = 1 − y2 = complement of y
(inverse)
= product of first variate and complement of second
∂ ( x1, x2 )
∂ (components)
=
= y1 = aggregate
∂( y1, y2 ) ∂(aggregate, corr. var.),
The forward and inverse transformations represent the processes of combining and decomposing energy modes
respectively. Each condensed format represents both the forward and inverse expanded forms. All variates, apart
from y2, are energy modes or energy components. The variate y2 is a correlation variate that is used to combine
or decompose the energies but also allows correlation of the pre-and post-interaction states. Note that the total
energy is separated from the correlation variate via a semicolon.
Transformation Notation
For each elementary transformation and its inverse write:
T [( x1 , x2 ) ↔ ( y1; y2 )]
Combining
T [( y1; y2 ) ↔ ( x1 , x2 )]
Decomposing
where : y1 = x1 + x2 ,
or
= sum of variates
x1 = y1 y2 ,
= product of variates
with Jacobian
y2 = x1 ( x1 + x2 )
(forward)
= ratio of first variate to sum of variates
~
x2 = y1 ~
y2
y2 = 1 − y2 = complement of y
(inverse)
= product of first variate and complement of second
∂ ( x1, x2 )
∂ (components)
=
= y1 = aggregate
∂( y1, y2 ) ∂(aggregate, corr. var.),
For multiple elementary transformations write:
T [( x11 , x21 ) ↔ ( y11; y21 ); ( x12 , x22 ) ↔ ( y12 ; y22 ); ...] = T [( x11 , x21 ) ↔ ( y11; y21 )] , T [( x12 , x22 ) ↔ ( y12 ; y22 )] , ...
Transformation notation is particularly useful in complex processes.
APPLICATION TO RECIPROCITY MODELLING
The notational system is illustrated both for Gamma-Beta calculus and for the explicit method of reciprocity
modelling. Vibrationally excited single-body gas surface interactions with free energy exchange are used as the
test-bed for elucidating the explicit method.
GAMMA-BETA CALCULUS
The fundamental tenets of Gamma-Beta calculus have been described in Ref. 1 wherein it was shown that the
calculus is concerned with two fundamental problems: decomposition of energy modes and aggregation of energy components. In this section these problems are re-examined using skeleton notation. Nothing new is added
to the contents of the calculus, but for the difference in the mathematical presentation. The presentation below
uses skeleton notation to condense the description and to focus understanding of the underlying concepts. The
result is that only a few lines are required to present the same content. Bracket notation is used for the transformations.
Decomposing Energy Modes
R:
Ga (ξ γ ) G ( s1 )
(1)
Write:
G ( s1 ) = β (s1 γ 1 , γ − γ 1 ) with γ = γ 1 + γ 2 so that γ 2 = γ − γ 1 , γ 1 and γ 2 are the
respective parameters of the two new Gamma distributions.
T:
(ξ ; s1 ) → (ξ1 ,ξ 2 )
R:
Ga (ξ1 γ 1 ) ; (ξ 2 γ − γ 1 )
[
]
Note that in writing down the final Gamma distributions, the energy component that appears in the numerator of
the s1 equation takes the first parameter of the Beta distribution. The remaining parameter is taken up by the
second energy component.
Aggregating Energy Modes
[
]
R:
Ga (ξ1 γ 1 ); (ξ 2 γ − γ 1 )
T:
(ξ1 ,ξ 2 ; s1 ) → ξ
R:
Ga (ξ γ ) β (s1 γ 1 , γ − γ 1 )
( 2)
Note that the energy component that appears at the numerator of the s1 equation assigns its parameter as the first
parameter of the Beta distribution. The other energy component assigns its parameter as the second parameter of
the Beta distribution.
EXPLICIT METHOD FOR VIBRATIONALLY EXCITED SINGLE-BODY GAS SURFACE INTERACTIONS WITH FREE EXCHANGE
The explicit method of reciprocity modelling has been discussed in detail (Ref. 1) using vibrationally excited
single-body gas surface interactions as the avenue for elucidating the method. In this section the method is reexamined using the same single-body problem but focusing only on free energy exchange. The results are the
same as before. However, the presentation using skeleton notation is markedly shorter and clearer.
[
]
Ga (ξ t′ γ t ) ; (ξ s′ γ s ); (ξ r′ γ r ) ; (ξ v′ γ v ) Ge (s′e )
R:
(3)
where the ξ ' s are modal energies normalised with respect to kT ( k = Boltzmann constant,
T = temperature), s is the correlation vector, G is the set of correlation densities,
and subscripts t, r, v, and s denote translational, rotational, vibrational and solid respectively.
Start with the correlation densities:
[
Ge (s′e ) = β (st′ α tγ t , (1 − α t )γ t ) ; (sr′ α rγ r , (1 − α r )γ r ) ; (sv′ α vγ v , (1 − α v )γ v ) ; (s′s α sγ s , (1 − α s )γ s )
]
( 4)
R:
[
]
× β [ (st′ α tγ t , (1 − α t )γ t ) ; (sr′ α rγ r , (1 − α r )γ r ) ; (sv′ α vγ v , (1 − α v )γ v ) ; (s′s α sγ s , (1 − α s )γ s ) ]
Ga (ξ t′ γ t ) ; (ξ s′ γ s ) ; (ξ r′ γ r ) ; (ξ v′ γ v )
(5)
T:
′ ,ξ s′2 ), (ξ r′ ; sr′ ) → (ξ ra
′ ,ξ r′2 ), (ξ v′ ; sv′ ) → (ξ va
′ ,ξ v′ 2 )
(ξt′; st′ ) → (ξ ta′ ,ξ t′2 ), (ξ s′ ; ss′ ) → (ξ sa
R:
[
′ α vγ v ); (ξsa′ α sγ s )
Ga (ξ ta′ α tγ t ); (ξ ra′ α rγ r ); (ξ va
[
]
′ (1 − α r )γ r ), (ξ v2
′ (1 − α v )γ v ), (ξs2
′ (1 − α s )γ s )
× Ga (ξ t′2 (1 − α t )γ t ), (ξ r2
]
( 6)
T:
′ + ξva
′ ; s′I′ ) → ξ Ia
′
(ξra
′
′ + ξta′ ; s′g′ ) → ξ ga
(ξ Ia
′ + ξsa
′ ; sa′′ ) → ξa
(ξ ga
(7)
R:
[
]
′ (1 − α v )γ v ) ; (ξs2
′ (1 − α s )γ s )
Ga (ξ a α tγ t + α rγ r + α vγ v + α sγ s ) Ga (ξ t′2 (1 − α t )γ t ); (ξ r′2 (1 − α r )γ r ) ; (ξ v2
[
(
]
)
× β (s′I′ α rγ r ,α vγ v ) ; s′g′ α rγ r + α vγ v ,α tγ t ; (sa′′ α tγ t + α rγ r + α vγ v ,α sγ s )
We must now decompose the total active energy into post-interaction components. This requires introducing a
number of pre-interaction correlation variates, with the correlation densities:
[
(
]
)
Ge (s′e ) = β (s′I α rγ r ,α vγ v ) ; s′g α rγ r + α vγ v ,α tγ t ; (sa′ α tγ t + α rγ r + α vγ v ,α sγ s )
Note that these densities are implied by their post-interaction versions, which are generated in the previous step.
T:
′′ ,ξsa
′′ ),
(ξa ; sa′ ) → (ξ ga
′′ ; s′g ) → (ξ Ia
′′ ,ξta′′ ),
(ξ ga
′′ ; s′I ) → (ξra
′′ ,ξva
′′ )
(ξ Ia
(8)
R:
[
] [
× β [(s′I′ α rγ r ,α vγ v ) ; (s′g′ α rγ r + α vγ v ,α tγ t ) ; (sa′′ α tγ t + α rγ r + α vγ v ,α sγ s )]
]
′′ α vγ v ); (ξ sa′′ α sγ s ) Ga (ξ t2′ (1 − α t )γ t ) ; (ξ r2
′ (1 − α r )γ r ) ; (ξ v2
′ (1 − α v )γ v ) ; (ξ s2
′ (1 − α s )γ s )
Ga (ξ ta′′ α tγ t ); (ξ ra′′ α rγ r ); (ξ va
Now combine the post-interaction active energies to their corresponding inactive energies.
T:
(ξta′′ + ξ t′2 ; st′′) → ξ t′′
′′ + ξ s′2 ; ss′′) → ξ s′′
(ξ sa
′′ + ξ r′2 ; sr′′) → ξ r′′
(ξ ra
′′ + ξ v′ 2 ; sv′′ ) → ξ v′′
(ξ va
( 9)
R:
[
]
× β [(st′′ α tγ t , (1 − α t )γ t ) ; (sr′′ α rγ r , (1 − α r )γ r ) ; (sv′′ α vγ v , (1 − α v )γ v ) ; (ss′′ α sγ s , (1 − α s )γ s )]
× β [(s′I′ α rγ r ,α vγ v ) ; (s′g′ α rγ r + α vγ v ,α tγ t ) ; (sa′′ α tγ t + α rγ r + α vγ v ,α sγ s )]
Ga (ξ t′′ γ t ); (ξ r′′ γ r ); (ξ v′′ γ v ); (ξ s′′ γ s )
The modelling is now complete because we have obtained the post-interaction energies. We know this because
all terms in this equation are post-interaction terms. In fact, the left hand side of the reciprocity equation has
transformed into the right hand side. This means that the series of transformations satisfy reciprocity. It also
means that the overall transformation will satisfy reciprocity.
O:
ξ t′′ = ~st′ξ t′ + ~sg′ sa′ ξ a , ξ r′′ = ~sr′ξ r′ + s′I s′g sa′ ξ a , ξ v′′ = ~sv′ξ v′ + ~sI′ s′g sa′ ξ a , ξ s′′ = ~ss′ξ s′ + ~sa′ ξ a
where
ξ a = st′ξ t′ + sr′ξ r′ + sv′ξ v′ + ss′ξ s′ = st′′ξ t′′ + sr′′ξ r′′ + sv′′ξ v′′ + ss′′ξ s′′
(10a)
The correlation densities are:
Ge (s′e ) = β (st′ α tγ t , (1 − α t )γ t )β (sr′ α rγ r , (1 − α r )γ r )β (sv′ α vγ v , (1 − α v )γ v )β (ss′ α sγ s , (1 − α s )γ s )
(
(10b)
)
× β (s′I α rγ r ,α vγ v )β s′g α rγ r + α vγ v ,α tγ t β (sa′ α tγ t + α rγ r + α vγ v ,α sγ s )
These are the densities utilised in the modelling process. We can also combine the elementary transformations to
obtain the inverse process.
I:
ξ t′ = ~st′′ξt′′ + ~sg′′ sa′′ξ a , ξ r′ = ~sr′′ξ r′′ + s′I′s′g′ sa′′ξ a , ξ v′ = ~sv′′ξ v′′ + ~sI′′s′g′ sa′′ξ a , ξ s′ = ~ss′′ξ s′′ + ~sa′′ξ a
where
ξ a = st′ξt′ + sr′ξ r′ + sv′ξ v′ + ss′ξ s′ = st′′ξ t′′ + sr′′ξ r′′ + sv′′ξ v′′ + ss′′ξ s′′
The correlation densities are:
Ge (s′e′ ) = β (st′′α tγ t , (1 − α t )γ t )β (sr′′ α rγ r , (1 − α r )γ r )β (sv′′ α vγ v , (1 − α v )γ v )β (ss′′ α sγ s , (1 − α s )γ s )
(
)
× β (s′I′ α rγ r ,α vγ v )β s′g′ α rγ r + α vγ v ,α tγ t β (sa′′ α tγ t + α rγ r + α vγ v ,α sγ s )
The inverse transformation has the same form as the forward transformation, thus confirming that the model
satisfies symmetry.
Deriving the Overall Transformation
The overall model transformation is derived by judicious expansion of appropriate elementary transformations.
Basically we scan the elementary transformations for the required components from which the overall equation
can be derived. For instance to construct the first of the overall equations of (10a) we scan for required components and find that we need parts of equations (9), (8) and (6), respectively. Expand these required parts as follows:
ξ t′′ = ξta′′ + ξ t′2 ,
′′ , ξ ga
′′ = sa′ ξ a ,
ξ ta′′ = (1 − s′g )ξ ga
ξt′2 = (1 − st′ )ξ t′
Combining these parts gives the required overall equation for the post-collision translational energy. We can
repeat this procedure to determine the overall equations for the post-collision rotational, vibrational and solid
energies.
APPLICATION TO RECIPROCITY PROOFS
Reciprocity proofs have been formulated for binary collisions (Ref. 2) by demonstrating that the overall model
transformation will convert one side of the reciprocity equation to the other side. Physically speaking, this
means that the model transformation will convert the rate of forward processes into the (equal) rate of inverse
processes and vice versa. That presentation is now reformulated using skeleton notation to condense and simplify the exposition. Rotationally excited and vibrationally excited binary collisions with restricted energy exchange are used to illustrate the procedures.
Rotationally Excited Binary Collisions with Restricted Exchange
R:
[
][
]
′ γ r ); (ξ r2
′ γ r ) β (st′ α tγ t , (1 − α t )γ t ) ; (s′I α I γ I , (1 − α I )γ I ) ; (s1′ γ r , γ r ) ; (sa′ α tγ t ,α I γ I )
Ga (ξ t′ γ t ); (ξ r1
Lump the rotational energies together and then decompose all energies into active and inactive parts.
T:
′ ,ξ I′ 2 )]
T [(ξ r′1,ξ r′2 ) ↔ (ξ I′ ; s1′′); (ξ t′; st′ ) ↔ (ξ ta′ ,ξ t′2 ); (ξ I′ ; s′I ) ↔ (ξ Ia
R:
[
] [
× β [(s1′′ γ r , γ r ) ; (s1′ γ r , γ r ) ; (sa′ α tγ t ,α I γ I )]
]
′ α I γ I ) Ga (ξ t′2 (1 − α t )γ t ) ; (ξ I′ 2 (1 − α I )γ I )
Ga (ξ ta′ α tγ t ); (ξ Ia
Now recombine the active parts into the total active energy.
′ ) ↔ (ξ a ; sa′′ )]
T [(ξ ta′ ,ξ Ia
T:
R:
[
]
× β [(s1′′ γ r , γ r ) ; (s1′ γ r , γ r ) ; (sa′′ α tγ t ,α I γ I ) ; (sa′ α tγ t ,α I γ I )]
Ga (ξ a α tγ t + α I γ I ) Ga (ξ t′2 (1 − α t )γ t ) ; (ξ I′ 2 (1 − α I )γ I )
Decompose the total active energy into post-collision components.
′′ )]
T [(ξ a ; sa′ ) ↔ (ξ ta′′ ,ξ Ia
T:
R:
[
] [
× β [(s1′′ γ r , γ r ) ; (s1′ γ r , γ r ) ; (sa′′ α tγ t ,α I γ I )]
]
′′ α I γ I ) Ga (ξ t′2 (1 − α t )γ t ) ; (ξ I′ 2 (1 − α I )γ I )
Ga (ξ ta′′ α tγ t ); (ξ Ia
Combine the post-collision active and inactive components, and then decompose the post-collision internal energy.
′′ ,ξ I′ 2 ) ↔ (ξ I′′; s′I′ ) ; (ξ I′′; s1′ ) ↔ (ξ r′′1 ,ξ r′′2 )]
T [(ξ ta′′ ,ξ t′2 ) ↔ (ξ t′′; st′′) ; (ξ Ia
T:
R:
[
][
]
′′ γ r ); (ξ r2
′′ γ r ) β (st′′α tγ t , (1 − α t )γ t ) ; (s′I′ α I γ I , (1 − α I )γ I ) ; (s1′′ γ r , γ r ) ; (sa′′ α tγ t ,α I γ I )
Ga (ξ t′′γ t ); (ξ r1
Thus the left hand side has transformed into the right hand side, thus confirming that the sequence of transformations satisfies reciprocity.
O:
ξ t′′ = ~st′ξ t′ + sa′ ξ a , ξ r′′1 = s1′ξ I′′, ξ r′′2 = ~s1′ξ I′′,
where ξ a = st′ξ t′ + s′I ξ I′ , ξ I′′ = ~
sa′ ξ a + ~
sI′ξ I′ , ξ I′ = ξ r′1 + ξ r′2
This is the model originally proposed for rotational excitations. We can say that it satisfies reciprocity. The
transformations can also be combined to give the inverse process, in which case it is found that the inverse
transformation has the same form as the forward transformation thus confirming that the model satisfies symmetry.
Vibrationally Excited Binary Collisions with Loosely Restricted Exchange
R:
[
]
(st′ α tγ t , (1 − α t )γ t ) ; (s′R α Rγ R , (1 − α R )γ R ) ; (sV′ αV γ V , (1 − αV )γ V ) ;
×β

(s1′ γ r , γ r )β (s2′ γ v , γ v ) ; (s′I α Rγ R ,αV γ V ) ; (sa′ α Rγ R + αV γ V ,α tγ t ) 
′ γ r ); (ξ r2
′ γ r ); (ξ v1
′ γ v ); (ξ v2
′ γv)
Ga (ξ t′ γ t ); (ξ r1
where
γ R = 2γ r
γ V = 2γ v
T:
′ ,ξ R′ 2 ) ; (ξV′ ; sV′ ) ↔ (ξVa
′ ,ξV′ 2 )]
T [(ξ r′1,ξ r′2 ) ↔ (ξ R′ ; s1′′) ; (ξ v′1,ξ v′ 2 ) ↔ (ξV′ ; s2′′ ) ; (ξ t′; st′ ) ↔ (ξ ta′ ,ξ t′2 ) ; (ξ R′ ; sR′ ) ↔ (ξ Ra
R:
[
] [
× β [(s1′′ γ r , γ r ) ; (s2′′ γ v , γ v ) ; (s1′ γ r , γ r ) ; (s2′ γ v , γ v ) ; (s′I α Rγ R ,αV γ V ) ; (sa′ α Rγ R + αV γ V ,α tγ t )]
]
′ α Rγ R ); (ξ Va
′ αV γ V ) Ga (ξ t′2 (1 − α t )γ t ) ; (ξ R2
′ (1 − α R )γ R ) ; (ξ V2
′ (1 − αV )γ V )
Ga (ξ ta′ α tγ t ); (ξ Ra
′ ,ξVa
′ ) ↔ (ξ Ia
′ ; s′I′ ) ; (ξ Ia
′ ,ξ ta′ ) ↔ (ξ a ; sa′′ )]
T [(ξ Ra
T:
R:
[
]
′ (1 − α R )γ R ) ; (ξ V2
′ (1 − αV )γ V )
Ga (ξ a α tγ t + α Rγ R + αV γ V ) Ga (ξ t′2 (1 − α t )γ t ) ; (ξ R2
(s1′′ γ r , γ r ) ; (s2′′ γ v , γ v ) ; (s1′ γ r , γ r ) ; (s2′ γ v , γ v ) ; (s′I′ α Rγ R ,αV γ V ) ;

×β

 (sa′′ α Rγ R + αV γ V ,α tγ t ) ; (s′I α Rγ R ,αV γ V ) ; (sa′ α Rγ R + αV γ V ,α tγ t )
′′ ,ξ ta′′ ) ; (ξ Ia
′′ ; s′I ) ↔ (ξ Ra
′′ ,ξVa
′′ )]
T [(ξ a ; sa′ ) ↔ (ξ Ia
T:
R:
[
] [
× β [(s1′′ γ r , γ r ) ; (s2′′ γ v , γ v ) ; (s1′ γ r , γ r ) ; (s2′ γ v , γ v ) ; (s′I′ α Rγ R ,αV γ V ) ; (sa′′ α Rγ R + αV γ V ,α tγ t )]
]
′′ α Rγ R ); (ξ Va
′′ αV γ V ) Ga (ξ t′2 (1 − α t )γ t ) ; (ξ R2
′ (1 − α R )γ R ) ; (ξ V2
′ (1 − αV )γ V )
Ga (ξ ta′′ α tγ t ); (ξ Ra
T:
′′ ,ξ R′ 2 ) ↔ (ξ R′′ ; s′R′ ) ; (ξVa
′′ ,ξV′ 2 ) ↔ (ξV′′ ; sV′′ ) ; (ξ R′′ ; s1′ ) ↔ (ξ r′′1 ,ξ r′′2 ) ; (ξV′′ ; s2′ ) ↔ (ξ v′′1,ξ v′′2 )]
T [(ξta′′ ,ξ t′2 ) ↔ (ξt′′; st′′) ; (ξ Ra
R:
[
]
(st′′ α tγ t , (1 − α t )γ t ) ; (sR′′ α Rγ R , (1 − α R )γ R ) ; (sV′′ αV γ V , (1 − αV )γ V ) ;
×β

(s1′′ γ r , γ r )β (s2′′ γ v , γ v ) ; (s′I′ α Rγ R ,αV γ V ) ; (sa′′ α Rγ R + αV γ V ,α tγ t ) 
′′ γ r ); (ξ r2
′′ γ r ); (ξ v1
′′ γ v ); (ξ v2
′′ γ v )
Ga (ξ t′′ γ t ); (ξ r1
The left hand side has transformed into the right hand side, thus confirming that the sequence of transformations
satisfies reciprocity.
O:
ξ t′′ = ~st′ξt′ + ~sa′ ξ a , ξ r′′1 = s1′ξ R′′ , ξ r′′2 = ~s1′ξ R′′ , ξ v′′1 = s2′ ξV′′ , ξ v′′2 = ~s2′ξV′′
where :
ξ R′′ = ~sR′ ξ R′ + s′I sa′ ξ a , ξ V′′ = ~sV′ ξV′ + ~sI′ sa′ ξ a , ξa = st′ξ t′ + sR′ ξ R′ + sV′ ξV′ , ξ R′ = ξ r′1 + ξ r′2 , ξV′ = ξ v′1 + ξ v′ 2
This is the model originally proposed for vibrational excitations. We can say that it satisfies reciprocity. The
transformations can also be combined to give the inverse process. We find that the inverse transformation has
the same form as the forward transformation, thus confirming that the model satisfies symmetry.
CONCLUSIONS
This paper has explained the underlying principles behind skeleton notation and has applied the notation to a
number of specific problems drawn from reciprocity modelling and reciprocity proofs. The examples illustrated
the use of skeleton notation with either bracket notation or transformation notation. The examples show that:
1) Skeleton notation condenses the presentation of reciprocity modelling.
2) Skeleton notation simplifies the presentation of reciprocity modelling.
3) Skeleton notation encourages the mind to focus on, and track changes in, the critical variables appearing in
reciprocity modelling.
Although skeleton notation has been illustrated using only the explicit method of reciprocity modelling the notation can also be used to condense other reciprocity modelling techniques, such as the fast-explicit, expert, and
fast-expert methods. However, the visual method embodies an optimal level of compaction so that skeleton notation cannot be used to condense its presentation any further.
Skeleton notation is especially relevant when reciprocity modelling is applied to highly complex phenomena such as many body interactions. For such phenomena skeleton notation expedites and simplifies the
process of constructing reciprocity models and reciprocity proofs. In fact, skeleton notation will be used to formulate reciprocity proofs for many body collisions and many body gas surface interactions.
REFERENCES
1
Agbormbai A A (2002), “Reciprocity Modelling Techniques for Rarefied Hypersonic Flows”, AIAA 40th Aerospace Sciences Meeting and Exhibit, Paper AIAA-2002-0223.
2
Agbormbai A A, “Reciprocity Proofs for Binary Inelastic Collisions in Rarefied Gas Dynamics”, AIAA 8th Joint Thermophysics and Heat Transfer Conference, Paper AIAA-2002-2764.
3
Agbormbai A A (2001), “Dynamical and Statistical Modelling of Many Body Collisions I”, Rarefied Gas Dynamics, 22nd
Symposium, p439.
4
Agbormbai A A (2001), “Dynamical and Statistical Modelling of Many Body Collisions II”, Rarefied Gas Dynamics, 22nd
Symposium, p452.
5
Agbormbai A A (2001), “Reciprocity Theory of Vibrationally Excited Many Body Collisions”, Rarefied Gas Dynamics,
22nd Symposium, CD-ROM.
6
Agbormbai A A (2001), “Reciprocity Modelling of Vibrationally Excited Four Body Collisions”, Rarefied Gas Dynamics,
22nd Symposium, CD-ROM.
7
Agbormbai A A (2001), “Reciprocity Theory of Many Body Monatomic Gas Surface Interactions”, AIAA 35th Thermophysics Conference, Paper No 2001-2973.
8
Agbormbai A A (2001), “Reciprocity Theory of Vibrationally Excited Many Body Gas Surface Interactions”, AIAA 35th
Thermophysics Conference, Paper No 2001-2964.