Commit 3bb9e46d authored by Tianbai Xiao's avatar Tianbai Xiao
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Update theory

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......@@ -5,8 +5,8 @@ Theory
The Boltzmann equation
---------
A physical system exhibit different behaviors at characteristic different scales.
We are interested in the transport phenomena of many particle systems.
The particle transport phenomena enjoy rich academic research value and application prospects.
A many-particle system can exhibit different behaviors at characteristic different scales.
Down to the finest scale of a many-particle system, the Newton’s second law depicts particle motions via
.. math::
......@@ -23,6 +23,7 @@ An intuitive numerical solution algorithm is to get the numerous particles on bo
A typical example is the molecular dynamics (MD) method.
This is not going to be efficient since there are more than :math:`2\times 10^{25}` molecules per cubic meter in normal atmosphere,
and things get extremely complicated if the N-body interactions are counted all the time.
Simplifications can be conducted to accelerate the numerical computation.
As an example, the Monte Carlo method employs certain particle models and conduct the interactions in a stochastic manner.
It significantly reduces the computational cost, while the trade-off is the artificial fluctuations.
......@@ -53,15 +54,15 @@ It is often reformulated with polar coordinates
where the particles don't interact with one another but scatter with the background material.
For convenience, we reformulate the particle velocity into polar coordinates :math:`\{r, \phi, \theta \}`
For convenience, we reformulate the particle velocity into polar coordinates :math:`\{r, \phi, \theta \}`.
.. math::
:label: porbz
&\left[\frac{1}{\mathrm{~V}} \frac{\partial}{\partial t}+\Omega \cdot \nabla+\Sigma(r, E, t)\right] \psi(r, \Omega, E, t) \\
&=\int_{0}^{\infty} d E^{\prime} \int_{\mathcal R^2} d \Omega^{\prime} \Sigma_{s}\left(r, \Omega^{\prime} \bullet \Omega, E^{\prime} \rightarrow E\right) \psi\left(r, \Omega^{\prime}, E^{\prime}, t\right)
&\left[\frac{1}{\mathrm{~V}} \frac{\partial}{\partial t}+\Omega \cdot \nabla+\Sigma(r, E, t)\right] \psi(r, \Omega, E, t) \\
&=\int_{0}^{\infty} d E^{\prime} \int_{\mathcal R^2} d \Omega^{\prime} \Sigma_{s}\left(r, \Omega^{\prime} \bullet \Omega, E^{\prime} \rightarrow E\right) \psi\left(r, \Omega^{\prime}, E^{\prime}, t\right)
The particle distribution :math:`\psi(t, r, \Omega, E)` here is often named as angular flux.
The particle distribution :math:`\psi(r, \Omega, E, t)` here is often named as angular flux.
The continuous slowing down approximation
......@@ -199,10 +200,3 @@ which gives
.. math::
D(x) =& -\int_{\infty}^{0} \int_{\mathbb{S}^2} \frac{1}{\rho(x)}\bar \psi(\bar E,x,\Omega)\frac{1}{S(E(\bar E))}\,d\Omega d\bar E\\
=& \int_{0}^{\infty} \frac{1}{\rho(x)S(E(\bar E))}\int_{\mathbb{S}^2} \bar \psi(\bar E,x,\Omega)\,d\Omega d\bar E.
.. math::
&\widehat{\psi}(E,x,\Omega) := \widetilde{\widehat{\psi}}(\widetilde E,x,\Omega) :=\bar{\psi}(\bar{E},x,\Omega),\\
&dE = -S(E) d\bar E(E), \\
&D(x) = -\int_{\infty}^{0} \int_{\mathbb{S}^2} \frac{1}{\rho(x)}\bar \psi(\bar E,x,\Omega)S(E(\bar E))d\Omega d\bar E
= \int_{0}^{\infty} \frac{S(E(\bar E))}{\rho(x)}\int_{\mathbb{S}^2} \bar \psi(\bar E,x,\Omega)\,d\Omega d\bar E.
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