Commit 3018795e authored by Tianbai Xiao's avatar Tianbai Xiao
Browse files

Fix physics.rst

parent 7d1e77c0
Pipeline #138012 passed with stage
in 19 minutes and 2 seconds
......@@ -119,10 +119,11 @@ Then, we substitute
.. math::
\widehat{\psi}(E,x,\Omega):= S(E)\rho(x)\psi(E,x,\Omega)
into \eqref{eq:CSD3}, which yields
into :eq:`CSD3`, which yields
.. math::
:label: CSD4
-S(E)\partial_E\widehat{\psi}(E,x,\Omega)+\Omega\cdot\nabla_x \frac{\widehat{\psi}(E,x,\Omega)}{\rho}+\Sigma_t(E)\widehat{\psi}(E,x,\Omega) = \int_{\mathbb{S}^2}\Sigma_s(E,\Omega\cdot\Omega')\widehat{\psi}(E,x,\Omega')d\Omega'.
Now, to get rid of the stopping power in front of the energy derivative, we make use of the transformation
......@@ -140,7 +141,7 @@ Now let us change to
In this case, the energy derivative becomes
.. math::
\partial_{\widetilde{E}}\widetilde{\widehat{\psi}}(\widetilde E,x,\Omega) = \partial_{E}\widetilde{\widehat{\psi}}( E,x,\Omega)\partial_{\widetlde E }E(\widetilde E(\widetilde E) = \partial_{ E}\widetilde{\widehat{\psi}}(\widetilde E,x,\Omega){S(E(\widetilde E))}.
\partial_{\widetilde{E}}\widetilde{\widehat{\psi}}(\widetilde E,x,\Omega) = \partial_{E}\widetilde{\widehat{\psi}}( E,x,\Omega)\partial_{\widetilde E }E(\widetilde E(\widetilde E) = \partial_{ E}\widetilde{\widehat{\psi}}(\widetilde E,x,\Omega){S(E(\widetilde E))}.
And by rearranging the terms, we finally get
......@@ -161,7 +162,7 @@ Here, we define :math:`\widetilde\Sigma_{t}(\widetilde E):=\Sigma_t(E(\widetilde
.. math::
\bar{E}(\widetilde{E}) = \widetilde{E}_{\text{max}}-\widetilde{E}.
Then, with :math:`\bar{\psi}(\bar{E},x,\Omega):=\widetilde{\widehat{\psi}}(\widetilde{E}(\bar{E}),x,\Omega)`, :math:`\bar\Sigma_{t}(\bar E):=\widetilde{\Sigma}_t(\widetilde{E}(\bar{E}))` as well as :math:`\bar\Sigma_{s}(\bar E,\Omega\cdot\Omega'):=\widetilde{\Sigma}_s(\widetilde{E}(\bar{E}),\Omega\cdot\Omega')` equation \eqref{eq:CSD4} becomes
Then, with :math:`\bar{\psi}(\bar{E},x,\Omega):=\widetilde{\widehat{\psi}}(\widetilde{E}(\bar{E}),x,\Omega)`, :math:`\bar\Sigma_{t}(\bar E):=\widetilde{\Sigma}_t(\widetilde{E}(\bar{E}))` as well as :math:`\bar\Sigma_{s}(\bar E,\Omega\cdot\Omega'):=\widetilde{\Sigma}_s(\widetilde{E}(\bar{E}),\Omega\cdot\Omega')` equation :eq:`CSD4` becomes
.. math::
:label: CSD6
......
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