From pitch angle diffusion to spatial diffusion coefficient

\[ dp/dt = qv/c \times (B_0 + \delta B) \]
\[dp_{\parallel}/dt  = q/c (v_{\perp} \times \delta B) \]
\[p_{\perp} = \mu p \]
\[ d\mu/dt = q/(pc)v(1-\mu^2)^{1/2} \delta B \cos(\Omega t -kx +\psi) \]
\[ \delta B(x,t) = \delta B \cos(\Omega t -kx +\psi) \]

Averaging over random phase of the waves $\psi$,
\[ \int_0^{2\pi} \cos^2\psi d\psi = \pi\]
\[\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b) \]
in the frame in which the wave is at rest, we can write $x=\mu v t$,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{\psi} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \cos[(\Omega -kv\mu)(t-t'')] \].

Integrating over time,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{t} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2}  \int dt' \int dt'' \cos[(\Omega -kv\mu)(t-t'')]  \\
= \frac{q^2 v(1-\mu^2)(\delta B)^2}{c^2 p^2 \mu} \delta(k-\Omega/(v\mu)) \delta t \].



In general one does not have a single wave but rather a power spectrum $p(k) = B_k^2/(4\pi)$.
Integrating over all of them,
\[ \frac{\langle \delta \mu \delta \mu \rangle}{\delta t} = \frac{q^2(1-\mu^2)\pi}{m^2c^2\gamma^2}\frac{4\pi}{v\mu} \int dk \frac{(\delta B(k))^2}{4\pi} \delta (k-\Omega/(v\mu) ) \]
\[ \langle \frac{\delta \mu \delta \mu \rangle}{\delta t} = \frac{\pi}{2} \Omega (1-\mu^2) k_{res} F(k_{res}) \]
$k_{res} = \frac{\Omega}{v\mu}$

$d\mu = d\cos(\theta) = \sin(\theta) d\theta$
$d\mu d\mu = (1-\mu^2) d\theta d\theta $

pitch angle diffusion coefficient:
\[D_{\mu \mu} = \langle \frac{\delta \theta d\theta}{2\delta t} \rangle = \frac{\pi}{4} \Omega k_{res} F(k_{res}) \]

How a particle that is diffusing in its pitch angle muse also diffusing in space ?
\[ \frac{\partial f}{\partial t} + v\mu \frac{\partial f}{\partial z} = \frac{\partial}{\partial \mu} [D_{\mu \mu} \frac{\partial f}{\partial \mu}] \]

$f_0 = \frac{1}{2} \int_{-1}^{1} d\mu f(p,t,\mu,z)$
\[ \frac{\partial f_0}{\partial t} + \frac{1}{2} v \int_{-1}^{1} d\mu \mu \frac{\partial f}{\partial z} = 0\]
$D_{\mu\mu}(\pm) = 0$ ?

\[ \frac{\partial f_0}{\partial t} = - \frac{\partial J}{\partial z} (*) \]

Note $\mu = -\frac{1}{2} \frac{\partial}{\partial \mu} (1-\mu^2)$
\[ J = \frac{1}{2} v \int_{-1}^{1} d\mu \mu f = \frac{v}{4} \int_{-1}^{1} d\mu (1-\mu^2) \frac{\partial f}{\partial \mu} \]
integration from -1 to $\mu$ and multiply by $(1-\mu^2)/D_{\mu\mu}$.
\[ \frac{\partial f}{\partial t} + v\mu \frac{\partial f}{\partial z} = \frac{\partial}{\partial \mu} [D_{\mu \mu} \frac{\partial f}{\partial \mu}] \]
get
\[ (1-\mu^2) \frac{\partial f}{\partial \mu} = \frac{1-\mu^2}{D_{\mu\mu}} \frac{\partial }{\partial t}\int_{-1}^{\mu} d\mu f + \frac{1-\mu^2}{D_{\mu\mu}} \int_{-1}^{\mu} d\mu v \mu  \frac{\partial f}{\partial z} \]
Assuming the distribution function tends to isotropy, so that at the lowest order in the anisotropy one has:
\[ (1-\mu^2) \frac{\partial f}{\partial \mu} = \frac{1-\mu^2}{D_{\mu\mu}} \frac{\partial f_0}{\partial t} (1+\mu) + \frac{1-\mu^2}{D_{\mu\mu}} \frac{1}{2}v (\mu^2 -1)  \frac{\partial f_0}{\partial z} \]

\[ J = \frac{1}{2} v \int_{-1}^{1} d\mu \mu f = \frac{v}{4} \int_{-1}^{1} d\mu (1-\mu^2) \frac{\partial f}{\partial \mu} \\
=\frac{v}{4} \frac{\partial f_0}{\partial t} \int_{-1}^{1} d\mu \frac{1-\mu^2}{D_{\mu\mu}}(1+\mu) - \frac{v^2}{8} \frac{\partial f_0}{\partial z} \int_{-1}^{1} d\mu \frac{(1-\mu^2)^2}{D_{\mu\mu}}  \\
\equiv k_t \frac{\partial f_0}{\partial t} - k_z \frac{\partial f_0}{\partial z} \] (**)

From (*), we get $\frac{\partial f_0}{\partial t} = - \frac{\partial J}{\partial z}$
substituting $J$ to this equation, we get
\[ \frac{\partial f_0}{\partial t} = - \frac{\partial}{\partial z}(-k_t \frac{\partial J }{\partial z}-k_z\frac{\partial f_0}{\partial z}) \]

The first term is negligible,
\[ J = \frac{v}{2} \int_{-1}^{1} d\mu \mu f_0(1+ \delta \mu)  = \frac{1}{3} v \delta f_0 \ll vf_0 \]
$\delta \ll 1$.

diffusion equation
\[ \frac{\partial f_0}{\partial t} = \frac{\partial }{\partial z} [ k_z \frac{\partial f_0}{\partial z}] \]

spatial diffusion coefficient
\[ k_z = \frac{v^2}{8} \int_{-1}^{1} d\mu \frac{(1-\mu^2)^2}{D_{\mu\mu}} \equiv \frac{1}{3} v \lambda_{\parallel} \]

The basic concept is that a charged particle in a magnetic field with a small turbulence component performs a random diffusive motion parallel to the ordered magnetic field.
There is a small diffusive motion perpendicular to the ordered field. The two become comparable when the turbulence component is large.

Reference:
https://fermi.gsfc.nasa.gov/science/mtgs/summerschool/2012/week1/CR1_Blasi.pdf
https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999JA900356