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Minor changes in documentation for tabular distributions.

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Paul Romano 2012-07-28 20:17:08 -04:00
parent 7b367dd858
commit e201435763
1 changed files with 10 additions and 10 deletions
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docs/source/methods/physics.rst
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@ -88,15 +88,15 @@ incoming energy grid:
where :math:`E` is the incoming energy of the particle. Then, statistical
interpolation is performed to choose between using the cosines and distribution
functions corresponding to energy :math:`E_i` and :math:`E_{i+1}`. Let
:math:`\ell` be the chosen table where :math:`\ell = i` if :math:`\xi > f` and
:math:`\ell = i + 1` otherwise where :math:`\xi` is a random number. A different
random number is used to sample a scattering cosine bin :math:`j` using the
cumulative distribution function:
:math:`\ell` be the chosen table where :math:`\ell = i` if :math:`\xi_1 > f` and
:math:`\ell = i + 1` otherwise where :math:`\xi_1` is a random number. Another
random number :math:`\xi_2` is used to sample a scattering cosine bin :math:`j`
using the cumulative distribution function:
.. math::
:label: sample-cdf
c_{\ell,j} < \xi < c_{\ell,j+1}
c_{\ell,j} < \xi_2 < c_{\ell,j+1}
The final scattering cosine will depend on whether histogram or linear-linear
interpolation is used. In general, we can write the cumulative distribution
@ -123,7 +123,7 @@ after integration we have that
.. math::
:label: cumulative-dist-histogram
c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi
c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi_2
Solving for the scattering cosine, we obtain the final form for histogram
interpolation:
@ -131,7 +131,7 @@ interpolation:
.. math::
:label: cosine-histogram
\mu = \mu_{\ell,j} + \frac{\xi - c_{\ell,j}}{p_{\ell,j}}
\mu = \mu_{\ell,j} + \frac{\xi_2 - c_{\ell,j}}{p_{\ell,j}}
For linear-linear interpolation, we represent the function :math:`p(\mu')` as a
first-order polynomial in :math:`\mu'`. If we interpolate between successive
@ -159,7 +159,7 @@ Let us now make a change of variables using
:label: introduce-eta
\eta = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}}
(\mu' - \mu_{\ell,j})
(\mu' - \mu_{\ell,j}) + p_{\ell,j}
Equation :eq:`cdf-linlin` then becomes
@ -182,7 +182,7 @@ Integrating equation :eq:`cdf-linlin-eta`, we have
:label: cdf-linlin-integrated
c(\mu) = c_{\ell,j} + \frac{1}{2m} \left ( \left [ m (\mu - \mu_{\ell,j} ) +
p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi
p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi_2
Solving for :math:`\mu`, we have the final form for the scattering cosine using
linear-linear interpolation:
@ -190,7 +190,7 @@ linear-linear interpolation:
.. math::
:label: cosine-linlin
\mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi -
\mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi_2 -
c_{\ell,j} )} - p_{\ell,j} \right )
.. _sample-energy: