Fokker-Planck Equations of Stochastic Acceleration: Green's Functions and Boundary Conditions

Details Fokker-Planck Equations of Stochastic Acceleration: Green's Functions and Boundary Conditions Fokker-Planck Equations of Stochastic Acceleration: Green's Functions and Boundary Conditions

Jun 1995

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995apj...446..699p&link_type=abstract

Astrophysical Journal v.446, p.699

Astronomy and Astrophysics

Astronomy

42

Acceleration Of Particles, Methods: Analytical, Plasmas, Scattering, Waves

Scientific paper

Stochastic wave-particle interaction is an important mechanism for accelerating particles to suprathermal energies in many astrophysical situations. Any sufficiently turbulent magnetized plasma will accelerate particles through resonant interactions with chaotic plasma waves. The effect of random multiple scatterings of particles can be described as a diffusion in energy, pitch angle, and physical space through the Fokker-Planck equation. This equation can be reduced to a simple form, function of energy and time, if the acceleration region is homogeneous and the scattering mean free path is much smaller than both the energy change mean free path and the size of the acceleration region. In spite of its simplicity, analytic solutions can be found only for limited and simple cases; numerical methods must be used for more general cases. In this paper, we discuss the analytic solutions. Previous analytic solutions of the Fokker-Planck equation suffered from ambiguous or incorrect treatment of the boundary conditions. We show that the equations under the above approximations are singular (defined fully in this paper), and therefore cannot be solved using the usual boundary conditions. We obtain the proper singular boundary conditions using the spectral theory of second-order differential equations, which is an extension of the familiar Sturm-Liouville eigenfunction expansion theory. By solving for both the steady state Green's function and the time-dependent Green's function for three specific cases, we examine the dependence of the resulting particle distribution on the coefficients of the Fokker-Planck equation. We discover that the steady state solution does not always exist so we determine the conditions for which this is possible and give physical interpretations of these situations. In general, the steady state solution has a power-law or an exponential form, depending on the energy dependences of the following characteristic timescales: τD, the timescale for energy diffusion; τA the timescale for advection; τT the timescale for escape; and τB, the timescale for secondary advective processes not directly related to the stochastic acceleration process. Implications of the singular nature of the Fokker-Planck equation to numerical analysis are also discussed.

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