Boltzmann equation

The Boltzmann equation expresses the variation with time of the number of neutrons present in an elementary volume V of surface S. We can write this as:
 $$\begin{align}\frac{d}{dt}\iiint{ n(}\overrightarrow{r} { ,v,t)d}^{3}r & =\mbox{(... ...created-absorbed)} \\ & \mbox{+(inscaterred-outscattered))neutrons} \end{align}$$
We explicit each of the terms of the r.h.s of 3.20:

$$\mbox{entering-exiting}=\underset{S}{-\iint}\overrightarrow{J... ...iint} div(\overrightarrow{J} (\overrightarrow {r} ,v,t))d^{3}r$$

expresses the total current entering the volume if the normal is directed outwards

$$\mbox{created}=\underset{V}{\iiint}\left[ S(\overrightarrow{r... ...htarrow{r} ,v^{^{\prime}})dv^{^{\prime}}\right) \right] d^{3}r$$

$S(\overrightarrow{r} ,v,t)$ is the external neutron source, the second term of the r.h.s., the fission source, $\psi_{f}(v)$ is the velocity spectrum of the fission neutrons, and the macroscopic cross-sections are assumed to be time-independent, i indexes fissioning nuclei

$$\mbox{inscattered}=\underset{V}{\iiint}\underset{j}{\sum}\lef... ...ow{r} ,v^{^{\prime}}\rightarrow v)dv^{^{\prime}}\right) d^{3}r$$

where j indexes all species of scattering nuclei.

$$\mbox{outscattered+absorbed}=\underset{V}{\iiint}\varphi(\ove... ...underset{j}{\sum}\Sigma_{T}^{(j)}(\overrightarrow{r} ,v)d^{3}r$$

where $\Sigma_{T}=\Sigma_{s}+\Sigma_{a}$.

In the above expressions we have, for the sake of simplicity, neglected the $\overrightarrow {\Omega}$ dependance of the cross-sections and integrated over $\overrightarrow {\Omega}$. The Boltzmann equation is obtained:

 $$\begin{align}\frac{\partial\varphi(\overrightarrow{r} ,v,t)}{v\partial t} & =-di... ...)\underset{j}{\sum}\Sigma_{T}^{(j)} (\overrightarrow{r} ,v)\nonumber \end{align}$$
where we made use of $\varphi(\overrightarrow{r} ,v,t)=vn(\overrightarrow {r} ,v,t)$.