1. Stochastic Differential Equation Approach to Cosmic Ray Propagation and Acceleration
International Workshop
New Perspectives on Cosmic Rays in the Heliosphere
14:00,March 25, 2010
Ming Zhang
Florida Institute of Technology

2. Brownian Motion Experiment (1824)
100 Particle simulation

3. Properties of Brownian Motion
Particle density
distribution:
Average distance:
Diffusion:

4. Einstein Theory of Brownian Motion
for steady state H O 2
Pollen
Langevin Equation:
(poorly defined)
Stochastic differential equation:

5. Fokker-Planck Equation
For a stochastic process:
(Ito)
The probability density to find the process at a given
time and location follows a Fokker-Planck equation:
(Time forward)
(Time backward)
diffusion coefficient is:

6. Solve second-order partial differential equations
Advantages: (1) Straight-forward, (2) get solution for full space
Disadvantages: (1) Limited to source or initial value problems,
(2) Low statistics in high dimensions.

7. Advantages: (1) Efficient if no need for global solution
(2) No statistics problem in high dimension
Disadvantage: Global solution too computational intense

8. Numerical solution to stochastic differential equation
Stochastic differential equation is similar to ordinary differential equation
Euler integration scheme works: error cancellation due to stochastics
Range-Kutta scheme to speed up calculation.
Simulation in high-dimensions needs little computer resource
Flexible geometry

9. Cosmic ray transport is most likely a diffusion process in phase space
Cosmic ray transport is most likely a diffusion process in phase space. Stochastic differential equation can be applied to the following studies of cosmic rays:
Solar modulation of cosmic rays
Cosmic ray propagation through interstellar medium with nuclear interaction network
Diffusive shock acceleration
Solar (cosmic ray) energetic particle transport

10. 1. Solar Modulation of Cosmic Ray

11. Cosmic ray transport mechanisms in the heliosphere
Drift Vd
Solar wind
Convection Vsw
Adiabatic
deceleration
ISM
Inward diffusion

12. Stochastic method to solve modulation spectra
Backward trajectory of particles
(pe2)
Interstellar
Spectrum fism(p)
(pe1)
all starting at (x,p,t)
Modulated spectrum

13. Ulysses observation of a north-south asymmetry of the heliosphere
(Simpson, Zhang, Bame, 1996)
-80
-60
-40
-20 20 40 60 80
Ulysses Latitude
1.00
0.95
0.90
0.85
0.80
0.75
Ulysses/IMP-8 >90 MeV p
E>90 MeV GCR

14. Modeling the asymmetry
Heliosphere
Solar wind
Bnorth
Bsouth

15. Distribution of particle (1 GeV/c) intensity on
a sphere at 1 AU (3000 particle calculation)

16. Latitudinal distribution of particles when entering the heliosphere (all particles arrive at north pole 1 AU)

17. Model of the Heliosphere and GMIR propagation
HCS tilt 45o
Polarity: negative
SW speed 400km/s
TS compression 3.3
GMIR coverage
+- 45o Lat
GMIR shock speed:
600 km/s in SW
400 km/s in sheath
GMIR shock compression
3 in SW
1.5 in sheath

21. Cosmic ray modulation in 3-d MHD model heliosphere (Ball et al. 2005)
MHD model heliosphere provided by Linde et al. (1998)

22. Ball et al. 2005

23. From Florinski and Pogorelov (2009)
MHD heliosphere including interaction with neutral ISM

24. Add Second-order Fermi Acceleration
Enhanced Fermi acceleration in the heliosheath:
Compression of plasma increases
Strong scattering reduces

27. 2. Cosmic Ray Propagation in Interstellar Medium
Elemental Abundance

28. (p+p —> p0 —> g and Inverse Compton)

29. Diffusion Model for cosmic ray propagation
in the interstellar medium with halo
At least 97 isotopes need to be considered. For each nuclear isotope:
Matrix representation (with an assumption that all species have the same diffusion coefficient with a function of energy per nucleon)
Nuclear reaction matrix

30. Stochastic solution to diffusion equation with source problem
Integration is carried along stochastic trajectories described by stochastic differential equation

31. Results with input of measured interstellar gas distribution
Elemental contribution to 12C and 10B observed at the solar system
Spectrum of B/C ratio at the
solar system

32. 3. Diffusive Shock Acceleration
Nearly isotropic distribution
Time-forward stochastic simulation:

34. Transport equation for particles with large anisotropy
Backward Stochastic differential equation solver

35. From McKibben et al. (2003)

37. SEP Propagation Model includes:
Pitch angle diffusion
focusing,
Streaming
Convection
Perpendicular diffusion
Adiabatic cooling
(pitch-angle dependent)

40. Stochastic differential equation vs
Stochastic differential equation vs. Fokker-Planck partial differential equation
Monte-Carlo simulation with SDE can be used to solve Fokker-Planck equation.
Track stochastic trajectories to look into the details of particle transport.
Problems in high dimensions do not necessarily increase computation demand.
Programming is extremely simple and less numerical instability.
Statistics in some problems may require special consideration.
Global solutions to Fokker-Planck equation need too much computation resources.
Cannot handle non-linear problems.