Green's Functions of Fokker-Planck Equations
Park, B. T. & Petrosian, V. 1995, ApJ, 446, 699
Fokker-Planck Equations of Stochastic
Acceleration: Green's Functions and Boundary Conditions
Postscript files, 2 figures.
Abstract
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 analytical 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:
$\tau_D$, the timescale for energy diffusion, $\tau_A$, the timescale
for advection, $\tau_T$, the timescale for escape, and $\tau_B$, 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.
erratum
The following are the typographical errors which were detected after
the paper was published in the Astrophysical Journal (ApJ, 446, 699).
- In Table 4, 4/5th column, 2nd row from the bottom. The "delta"
symbol in the exponent should be replaced with a "delta bar", both
defined by equation (67).
- In Equation (68), in the expression for "delta plus/minus".
The "a plus/minus 1" should be replaced by "a + 1".
- In Equation (68), in the expression for "lamba sub 0".
The "alpha" should be replaced by an "a".
The version given here corrects these errors.
Last updated 95/5/18
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