Integral form of the Boltzmann equation

The Boltzmann equation can be put in an integral form. This can be made by mathematical manipulation of equation 3.22. Here we give a physical derivation of the integral form based on physical arguments, in the simplifying case where the macroscopic cross-sections are time-independent, the scattering cross-section is isotropic, the medium homogeneous and the system is stationary in time. Then the flux at position $\overrightarrow{r}$ is created by neutrons created elsewhere, be they scattered neutrons or fission neutrons with the right velocity. The probability that a neutron with velocity v at position $\overrightarrow{r^{\shortmid}}$ reaches position $\overrightarrow{r}$ is $\frac{e^{-\Sigma_{T}(v)\left\vert \overrightarrow {r} -\overrightarrow{r^{\shor... ...\left\vert \overrightarrow {r} -\overrightarrow{r^{\shortmid}}\right\vert ^{2}}$ The number of neutrons scattered and created at position $\overrightarrow{r^{\shortmid}}$is $\int\varphi(\overrightarrow{r^{\shortmid}} ,v^{\shortmid})\left( \Sigma _{s}(\o... ...\Sigma_{f}(\overrightarrow{r^{\shortmid}} ,v^{^{\prime}})\right) dv^{\shortmid}$. Thus, the flux at position $\overrightarrow{r}$reads:

 

$$\varphi(\overrightarrow{r} ,v) \iiint d^{3}r\left( \frac{e^{-\Sigma _{T}(v)\left\vert \overright... ...ert \overrightarrow{r} -\overrightarrow{r^{\shortmid}}\right\vert ^{2}} \right.$$ = (4.20)

$$\int\varphi(\overrightarrow{r^{\shortmid}} ,v^{\shortmid})\left( ... ..._{f}(\overrightarrow{r^{\shortmid}} ,v^{^{\prime}})\right) dv^{\shortmid}\Bigg)$$