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Monday, April 20, 2020

From pitch angle diffusion to spatial diffusion coefficient

dp/dt=qv/c×(B0+δB)

dp/dt=q/c(v×δB)

p=μp

dμ/dt=q/(pc)v(1μ2)1/2δBcos(Ωtkx+ψ)

δB(x,t)=δBcos(Ωtkx+ψ)


Averaging over random phase of the waves ψ,
2π0cos2ψdψ=π

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=μvt,
δμ(t)δμ(t)ψ=q2v2(1μ2)(δB)22c2p2cos[(Ωkvμ)(tt)]
.

Integrating over time,
δμ(t)δμ(t)t=q2v2(1μ2)(δB)22c2p2dtdtcos[(Ωkvμ)(tt)]=q2v(1μ2)(δB)2c2p2μδ(kΩ/(vμ))δt
.



In general one does not have a single wave but rather a power spectrum p(k)=B2k/(4π).
Integrating over all of them,
δμδμδt=q2(1μ2)πm2c2γ24πvμdk(δB(k))24πδ(kΩ/(vμ))

δμδμδt=π2Ω(1μ2)kresF(kres)

kres=Ωvμ

dμ=dcos(θ)=sin(θ)dθ
dμdμ=(1μ2)dθdθ

pitch angle diffusion coefficient:
Dμμ=δθdθ2δt=π4ΩkresF(kres)


How a particle that is diffusing in its pitch angle muse also diffusing in space ?
ft+vμfz=μ[Dμμfμ]


f0=1211dμf(p,t,μ,z)
f0t+12v11dμμfz=0

Dμμ(±)=0 ?

f0t=Jz()


Note μ=12μ(1μ2)
J=12v11dμμf=v411dμ(1μ2)fμ

integration from -1 to μ and multiply by (1μ2)/Dμμ.
ft+vμfz=μ[Dμμfμ]

get
(1μ2)fμ=1μ2Dμμtμ1dμf+1μ2Dμμμ1dμvμfz

Assuming the distribution function tends to isotropy, so that at the lowest order in the anisotropy one has:
(1μ2)fμ=1μ2Dμμf0t(1+μ)+1μ2Dμμ12v(μ21)f0z


J=12v11dμμf=v411dμ(1μ2)fμ=v4f0t11dμ1μ2Dμμ(1+μ)v28f0z11dμ(1μ2)2Dμμktf0tkzf0z
(**)

From (*), we get f0t=Jz
substituting J to this equation, we get
f0t=z(ktJzkzf0z)


The first term is negligible,
J=v211dμμf0(1+δμ)=13vδf0vf0

δ1.

diffusion equation
f0t=z[kzf0z]


spatial diffusion coefficient
kz=v2811dμ(1μ2)2Dμμ13vλ



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






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