dp∥/dt=q/c(v⊥×δB)
p⊥=μp
dμ/dt=q/(pc)v(1−μ2)1/2δBcos(Ωt−kx+ψ)
δB(x,t)=δBcos(Ωt−kx+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(1−μ2)(δB)22c2p2cos[(Ω−kvμ)(t−t″)]
.
Integrating over time,
⟨δμ(t′)δμ(t″)⟩t=q2v2(1−μ2)(δB)22c2p2∫dt′∫dt″cos[(Ω−kvμ)(t−t″)]=q2v(1−μ2)(δB)2c2p2μδ(k−Ω/(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,
⟨δμδμ⟩δt=q2(1−μ2)πm2c2γ24πvμ∫dk(δB(k))24πδ(k−Ω/(vμ))
⟨δμδμ⟩δt=π2Ω(1−μ2)kresF(kres)
kres=Ωvμ
dμ=dcos(θ)=sin(θ)dθ
dμdμ=(1−μ2)dθdθ
pitch angle diffusion coefficient:
Dμμ=⟨δθdθ2δt⟩=π4ΩkresF(kres)
How a particle that is diffusing in its pitch angle muse also diffusing in space ?
∂f∂t+vμ∂f∂z=∂∂μ[Dμμ∂f∂μ]
f0=12∫1−1dμf(p,t,μ,z)
∂f0∂t+12v∫1−1dμμ∂f∂z=0
Dμμ(±)=0 ?
∂f0∂t=−∂J∂z(∗)
Note μ=−12∂∂μ(1−μ2)
J=12v∫1−1dμμf=v4∫1−1dμ(1−μ2)∂f∂μ
integration from -1 to μ and multiply by (1−μ2)/Dμμ.
∂f∂t+vμ∂f∂z=∂∂μ[Dμμ∂f∂μ]
get
(1−μ2)∂f∂μ=1−μ2Dμμ∂∂t∫μ−1dμf+1−μ2Dμμ∫μ−1dμvμ∂f∂z
Assuming the distribution function tends to isotropy, so that at the lowest order in the anisotropy one has:
(1−μ2)∂f∂μ=1−μ2Dμμ∂f0∂t(1+μ)+1−μ2Dμμ12v(μ2−1)∂f0∂z
J=12v∫1−1dμμf=v4∫1−1dμ(1−μ2)∂f∂μ=v4∂f0∂t∫1−1dμ1−μ2Dμμ(1+μ)−v28∂f0∂z∫1−1dμ(1−μ2)2Dμμ≡kt∂f0∂t−kz∂f0∂z
From (*), we get ∂f0∂t=−∂J∂z
substituting J to this equation, we get
∂f0∂t=−∂∂z(−kt∂J∂z−kz∂f0∂z)
The first term is negligible,
J=v2∫1−1dμμf0(1+δμ)=13vδf0≪vf0
δ≪1.
diffusion equation
∂f0∂t=∂∂z[kz∂f0∂z]
spatial diffusion coefficient
kz=v28∫1−1dμ(1−μ2)2Dμμ≡13vλ∥
(**)
From (*), we get ∂f0∂t=−∂J∂z
substituting J to this equation, we get
∂f0∂t=−∂∂z(−kt∂J∂z−kz∂f0∂z)
The first term is negligible,
J=v2∫1−1dμμf0(1+δμ)=13vδf0≪vf0
δ≪1.
diffusion equation
∂f0∂t=∂∂z[kz∂f0∂z]
spatial diffusion coefficient
kz=v28∫1−1dμ(1−μ2)2Dμμ≡13vλ∥
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.
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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