1. P.N.Lebedev Physical Institute
Diffusion of charged particles in a stochastic force-free magnetic field
Ya. Istomin, A. Kiselev
P.N.Lebedev Physical Institute
V.Ginzburg conference, 2017

2. Random magnetic field, <B>=0.
Pair correlation function <B(r)B(r+R)>=Φ(R), correlation length L.
B0^2=<B^2>, ω=eB0/mcγ, rL=v/ω, b=B/B0
Diffusion coefficient D=<rv>,
D=vrL<r curl b>/2

3. Force-free magnetic field
curl B=α B
α≈const, B(r)=B0(sinαz, cosαz, 0)
Parameter a=rLα, l=2π/α

9. D=vrL F(a), a=rLα

10. Spectrum of random magnetic field
B0 (r)=BLS(r/L)^β
β=1/3 for Kolmogorov turbulence
Averaging over spectrum: A=2πv/Lω(B=BLS)

12. For A<1 (rLS<L/2π) D=0.1vLA,
For A>1 D=0.1vLA^2
Cosmic rays in the Galaxy
10^-6 Gauss<B<10^-4 Gauss
10^12 cm<L<100 pc
Averaging over BLS and L:
BLS(L)=Bm(L/Lm)^β,

13. dN=f(L)dL, f(L)=σLm^(-1)(L/Lm)^(σ-1)
Averaging gives
D=vLm(2πrm/Lm)^δ, δ=(1-σ)/(1+β)
For Kolmogorov spectrum β=1/3 δ<0.75.
For almost flat distribution over scales,
σ=1/15, δ=0.7

14. Conclusions
1. For force-free magnetic field we
traced the continuous transition from cyclotron rotation to trapped, then to untrapped and free motion of particles.
2. The diffusion coefficient is proportional to Larmor radius rL for rL<L, and rL^2 for rL>L.

15. 3. Averaging over large scale magnetic field BLS and scales L gives the dependance D \propto rL^δ, δ=(1-σ)/(1+β),
For β=1/3 and σ=1/15 δ=0.7
4. More probable spectrum of Galactic
cosmic rays in sources is universal, E^-2