Stochastic Differential Equation Approach to Cosmic Ray Propagation and Acceleration

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. 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 ISM Adiabatic deceleration 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. E>90 MeV GCR 1.00 0.95 0.90 Ulysses/IMP-8 >90 MeV p 0.85 0.80 0.75 -80 -60 -40 -20 0 20 40 60 80 Ulysses Latitude Ulysses observation of a north-south asymmetry of the heliosphere (Simpson, Zhang, Bame, 1996)

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. 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.