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
$$D =1/2 \frac{(dx)^2}{dt}$$

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