A simple derivation for parker transport equation

The continuity equation in differential form
$$\begin{equation*} \frac{\partial f}{\partial t} =-\nabla \cdot \overrightarrow{S.} \end{equation*}$$
In a given volume, the rate of change of the number of particles,$\displaystyle \frac{\partial f}{\partial t}$, is given by the numer of particles that across the surface enclosing the volume, assuming no source of particle is present.
$\displaystyle \vec{S}$ is the streaming crrent density of particles across the surface.

$$\begin{gather*} \vec{S} \ =\ \vec{V} f\ -K\cdot \nabla f\ \\ \nabla _{p} \cdot \overrightarrow{S_{p}} =\frac{1}{p^{2}}\frac{\partial }{\partial p}\left( p^{2} \langle \dot{p} \rangle f\right) \ ,\ \overrightarrow{S_{p}} =\langle \dot{p} \rangle f\\ \end{gather*}$$
$$\begin{equation*} \frac{\partial f}{\partial t} +\nabla \cdot \left(\vec{V} f-K\cdot \nabla f\right) +\left(\frac{1}{p^{2}}\frac{\partial }{\partial p}\left( p^{2} \langle \dot{p} \rangle f\right)\right) =0 \end{equation*}$$
note. The streaming is included in $\vec{S}$.
adiabatic cooling resulting the average change in momentum is $\displaystyle \langle \dot{p} \rangle =-\frac{p}{3} \nabla \cdot \vec{V}$.
$$\begin{gather*} \frac{\partial f}{\partial t} +f\nabla \cdot \vec{V} +\vec{V} \cdot \nabla f-\frac{( \nabla \cdot \overrightarrow{V)}}{3p^{2}}\left( 3p^{2} f+p^{3}\frac{\partial f}{\partial p}\right)\\ =\frac{\partial f}{\partial t} +f\nabla \cdot \vec{V} +\vec{V} \cdot \nabla f-( \nabla \cdot \overrightarrow{V)}\left( f+\frac{p}{3}\frac{\partial f}{\partial p}\right)\\ =\frac{\partial f}{\partial t} +\vec{V} \cdot \nabla f-\left( \nabla \cdot \vec{V}\right)\frac{p}{3}\frac{\partial f}{\partial p} \end{gather*}$$

$$\begin{array}{l} note.\ \frac{dT}{dt} =\frac{dp}{dt}\frac{dT}{dp} =\frac{-\nabla \cdot \vec{V}}{3}\left(p*\frac{p}{T+m}\right) =\frac{-( \nabla \cdot \overrightarrow{V)}}{3} T\left(\frac{T+2m}{T+m}\right) .\\ \frac{dE}{dt} =\frac{dp}{dt}\frac{dE}{dp} =\frac{-\left( \nabla \cdot \vec{V}\right)}{3}(pv) \end{array}$$

When differential energy density, $U_p=4\pi p^2f$, is convected with velocity $V$, the flux observed is not $VU_p$, but rather $S_p=CVU_p$, where $C=\frac{-1}{3}\frac{\partial lnf}{\partial lnp} = \frac{-1}{3}\frac{p}{f}\frac{\partial f}{\partial p}$ is called the Compton-Getting coefficien.
For $f=Ap^{-\gamma}$, C is $\frac{\gamma}{3}$. if $\gamma$ is 4.7, then the C is $4.7/3\sim 1.57$.

The adiabatic rate of change of momentum $\dot p = −(1/3)p\nabla\cdot V$ referred to above, is actually the rate of change in the non-inertial frame. The rate in the stationary frame is $\dot p=−(1/3)pV \cdot (\nabla f/f)$.


Reference
Radiation Environments and Damage in Silicon Semiconductors Silicon chapter4
Cosmic-Ray Modulation Equations