Cosmic rays propagation equation

the particles number through energy boundary in $dt$ is $dN$

(E+dE)-------->|(E)-------->(E-dE)

boundary

\[ dN = \int_E^{E+dE} dN/dE(E) dE = dN/dE(E_0)dE \approx dN(E)/dEdE  = \frac{dN(E)}{dE}\frac{dE}{dt} dt\]

the number of particles that pass through the energy boundary per unit of time,
\[ J_E = \frac{dN}{dt} = \frac{dN}{dE}\frac{dE}{dt} \]

----------------->[(E+dE)------------------](E)--------------->

the change of particle number due to energy variation within energy interval $[E,E+dE]$ per unit time if $dE/dt<0$.

\begin{equation}
\begin{split}
 \frac{dN}{dt} & =  dN/dE(E+dE)d(E+dE)/dt  - dN/dE(E)dE/dt \\
& = [dN/dE(E)+\frac{d^2N}{dE^2}(E) dE] \cdot [dE/dt + \frac{d^2E}{dEdt}(E)dE ] - dN/dE(E)dE/dt \\
& =  \frac{d}{dE}(\frac{dN}{dE}(E)\frac{dE}{dt}) \cdot dE
\end{split}
\end{equation}

thus,  \[ \frac{dN}{dEdt} = \frac{d}{dE}(\frac{dN}{dE}(E)\frac{dE}{dt}) = \frac{dJ_E}{dE} \]



the change of particle number due to motion



random walk and diffusion in one dimension

<--------o-------->

1/2 = p   p =1/2

random walk, the probability of motion to left or right is same.

        ------------x-dx---------><|(x)---------|(x+dx)><--------x+2dx---------------

N(x)           N(x+dx)

the change of particle number in the interval $[x,x+dx]$ and time interval $dt$ .
\[ \int_{x-dx}^x dN/dx dx \equiv N(x-dx) \]
\[  dN = 1/2 N(x-dx) - 1/2N(x) +  1/2N(x+dx)-1/2N(x+dx)  = 1/2(dx)^2  \frac{d^2N}{dx^2} \]
\[dN/dt = D \frac{d^2N}{dx^2} \]

diffusion coefficient
\[D =1/2 \frac{(dx)^2}{dt} \]

-------------
the number of particles that pass through the left position boundary
\[ \int_{x-dx}^x  1/2 dN/dx(x) dx - \int_x^{x+dx} 1/2 dN/dx(x) dE \approx  1/2 \frac{dN}{dx}(x-dx)dx - 1/2 \frac{dN}{dx}(x)dx = \qquad -1/2(dx)^2 \frac{d}{x}(dN/dx) \]
\[ \frac{dN}{dt} = -1/2D \frac{d}{dx}(\frac{dN}{dx}) \]
-------------

The number of particles which cross area dA due to diffusion.

                    /|
                  /  |
                / d |
                | A|
  _______|    |_________________________>
________|___________________________>
                x

\begin{equation}
\begin{split}
 dN = J dAdt = &1/2 [ \int_{x-dx}^{x} dN/dV(x) dx dA - \int_x^{x+dx} dN/dV(x)dxdA]  \\
&  \approx  1/2dN/dV(x-dx/2)dxdA - dN/dV(x+dx/2)dxdA \\
& = -1/2 \frac{d}{dx}(\frac{dN}{dV})(dx)^2dA
\end{split}
\end{equation}
\[ J_x = \frac{dN}{dAdt} = -\frac{(dx)^2}{2dt}\frac{d}{dx}(\frac{dN}{dV}) = -D\frac{dn}{dx} \]

The change of particle number in the volume dxdA
\[ dN = J(x)dAdt - J(x+dx)dAdt =  - dJ/dxdAdtdx \]
\[\frac{dn}{dt} = \frac{d}{dx}(D\frac{dn}{dx}) \]

The number of particles which cross area dA due to convection.
\[dN = n(x)v(x)dtdA - n(x+dx)v(x+dx)dtdA = \frac{d}{dx}(nv(x))dxdAdt \]
\[J_x = nv_x \]
\[\frac{dn}{dt} = -\frac{d}{dx}(nv_x) \]

propagation equation

              E |
                 |     
                 |         ⬜dE
                 |          dx
                 |--------------------> x

change of particle number due to diffusion
\[ \frac{dn}{dt} = \sum_i \frac{d}{x_i}(D\frac{dn}{dx_i}) \]

-----------
$n = \int dn/dp dp$
\[\frac{dn}{dt} = \frac{d}{dt}(\int dn/dp dp) = \int \frac{d^2n}{dtdp}dp \]
\[ \frac{d}{dx}(D\frac{d}{dx}(\int dn/dp dp)) \]
invalid.
-----------
for the change of particle number due to motion,
substitute $\frac{d}{dt} \rightarrow \frac{d}{dtdE}$ and $n \rightarrow \frac{dn}{dE}$.

diffusion
\[ \frac{dn}{dEdt} = \sum_i \frac{d}{x_i}(D\frac{dn}{dEdx_i}) \]
convection
\[\frac{dn}{dEdt} = -\frac{d}{dx}(dn/dEv_x) \]
change of particle number due to energy variation
define $dE/dt>0$
\[ \frac{dn}{dEdt} = - \frac{d}{dE}(\frac{dn}{dE}(E)\frac{dE}{dt}) \]

$u = dn/dE$
\[ \frac{\partial u}{\partial t} = \nabla \cdot(D\nabla u)  -\nabla \cdot (u\vec{v}) - \frac{d}{dE}(u\frac{dE}{dt}) \]


REFERENCE:
split equation more than one line
https://tex.stackexchange.com/questions/3782/how-can-i-split-an-equation-over-two-or-more-lines
fick's law
https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion