I would like working Mathematica (preferred) or C code (with compile
instructions for an AMD Athlon XP under Windows XP with a free (as in
beer or speech) compiler) for determining the value of reduced
collision integrals, especially for l=1 & n=1, as given in "The
Transport Properties for Non-Polar Gases," by Hirschfelder, Bird, and
Spotz in The Journal of Chemical Physics Volume 16, Number 10, p
968-981, October 1948. Note that I do not want a curve fit or table of
values for the integrals. I want a fast, working, coded, program for
evaluating those integrals numerically. Make sure you use the same
effective potential function that they do: Lennard-Jones, plus a term
to account for the relative speed of approach, g, and the impact
parameter, b. Given that Hirschfelder et.al. published that
aforementioned paper in 1948, without the use of computers, I'm sure
there is a ridiculously fast solution that could be coded today. I
have already programmed a solution into Mathematica, but it is too
slow and seems to suffer from a loss of precision.

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Request for Question Clarification by leapinglizard-ga on 08 Nov 2004 20:28 PST

I've ordered a copy of the paper and will set to work once it arrives
later this week. In the meantime, you might like to post your existing
Mathematica solution for comparison. What leads you to believe that it
loses precision? What analytical values do you use as benchmarks?

leapinglizard

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Clarification of Question by cpcender-ga on 09 Nov 2004 06:55 PST

The values I'm using as benchmarks are in the paper itself; they are
simply the results of the integration. I believe them to be correct,
both from what I have read about Hirschfelder's work at U of Wisconsin
and also from his claims of close numerical agreement with other
authors' research on p553 of Molecular Theory of Gases and Liquids
(MTGL).

If you are using Mathematica to numerically evaluate the integrals,
you may see warning messages about setting precision higher to help it
handle the singularities. By increasing the precision, I was able to
get rid of the error messages, but the answers still did not come out
correctly. BTW, the names of the functions in my notebook follow the
convention in MTGL, not the J. Chem. Phys. paper.

Here is the link to my calculations (right click, save as - or, in
Mathematica or Math Reader just file-open and fill in the url):
http://kettering.edu/~chia5013/Collision%20New%202.nb

Here is the link to Math Reader:
http://www.wolfram.com/products/mathreader/

You might find the following pages of MTGL (revised edition, with the
correction list printed at the front) useful: 524-527, 546-547, 553,
557-559. Note the difference between the potential function and the
effective potential function and also the difference in their
notation; that will possibly save you some headaches.

You may wish to check out a copy of MTGL from your local library,
which would possibly have to get the book through InterLibrary Loan
(ILL).
The full reference for MTGL:
MTGL, Hirschfelder, Curtiss and Bird, John Wiley & Sons, Inc., New
York. Chapman & Hall, Limited, London. Copyright 1954. Library of
Congress Catalog Card Number 54-7621.

---

Clarification of Question by cpcender-ga on 09 Nov 2004 06:59 PST

As you may notice in the notebook calculations, there is no
calculation (with high precision) of the collision integral. That is
because my computer just runs the calculations for hours without
returning a result.

Much as I love to crunch transport problems, I have to point out that
"free beer" and "free speech" involve significantly different meanings
of the word free.

But, it's all good!

--mathtalk-ga

While we await leapinglizard-ga's procurement of the paper, it might
be expeditious to sketch out how I read the computation of W^(l)(n;x)
there.

Because the Courier font makes it difficult to distinguish l from 1,
I'm going to take the liberty of replacing l by L in this discussion.

There's a slightly complicated expression involving parameter L that
appears in a couple of places:

               L
       1 + (-1)
  2 - -----------
        1 + L

Now for L even, this is 2L/(1+L); for L odd it is simply 2.  It is a
constant as far as any of the integrals in the definition of
W^(L)(n;x) are concerned, but for the sake of consistency with how the
paper develops their computation, let's call the value U(L).

Then they define in their equation (3):

W^(L)(n;x) = (1/8)U(L)x^(n+2) * 

                INTEGRAL exp(-xK) K^(n+1) S^(L)(K) dK OVER [0,+oo)

in which x = e/kT where k is Boltzmann's constant (and T is
temperature) and e is here an "energy difference".

The evaluation of W^(L)(n;x) from the improper integral is then
complicated mainly by evaluation of the "reduced collision cross
section" S^(L)(K), which is first defined by their paper in equation
(2):

S^(L)(K) = (4/U(L)) * INTEGRAL (1 - cos^L(chi(B,K))*B dB OVER [0,+oo)

Notice that the factor U(L) which appears outside the integral in
W^(L)(n;x) will cancel the division by U(L) that appears in S^(L)(K). 
The purpose in carrying the factor along would appear to be limited to
making the interpretation of S^(L)(K) more "physical", but perhaps
there's something I'm missing.

In any case the evaluation of S^(L)(K) itself depends on an improper
integral, and for how "angle of deflection" chi(B,K) depends on B and
K, we must turn to equation (32) in their paper:

chi(B,K) = pi - 2B * INTEGRAL dy/SQRT(1 - (By)^2 + (4/K)(y^6 - y^12))

where the integral is over [0,y_m] when "y_m is the lowest root for
which the denominator of the integrand vanishes."

In other words chi(B,K) also depends on an improper integral for its
evaluation, because the integrand blows up at the limit of integration
y_m.

Much of the remainder of that Section II.B of the paper is devoted to
rewriting the integral for chi(B,K) into forms that make it apparent
the singularity is really integrable and that "facilitate the
numerical integration".

regards, mathtalk-ga

mathtalk - This is similar to how I interpret the paper.

Also, from at least one of the collision diagrams on the J. Chem.
Phys. paper, one can see that y is the inverse of the radius. Putting
the problem in terms of y (and ym) makes the "chi integral" fit
hyper-elliptic or Alebian form. I have noticed that in MTGL, the
integral is stated in the form of r (and rm), where rm corresponds to
either the r for the minimum of the potential function or the minimum
r achieved in a collision. However, I was not able to get the "r form"
to work. :[

Sorry for the long delay. I've been busy with other projects. I have
reviewed the Hirschfelder paper and concluded that I'm not qualified
to answer your question. It looks like mathtalk is the man for the
job. Again, I apologize to you and to mathtalk for not getting in
touch earlier.

leapinglizard