A STOCHASTIC SIMULATION METHOD
FOR FOKKER-PLANCK EQUATIONS
T. YOSHIDA AND S. YANAGITA
Faculty of Science, Ibaraki University, Mito 310, Japan
1.
Introduction
A stochastic simulation method to solve the cosmic-ray transport equation
by It^o's stochastic dierential equations (SDEs) has been developed[1],[2].
This method is based on the equivalence between Fokker-Planck equations
(FPEs) and SDEs. However, simple-minded application of this method is
not enough to achieve a wide dynamic range, as pointed out by Park and
Petrosian[3]. In order to cope with this diculty, we couple this method
with a technique known as particle splitting. Our method is quite adapted
for simulating particle acceleration, propagation, and non-thermal photon
emission processes in astrophysical phenomena[4].
2.
Method of Numerical Calculation
Park and Petrosian[3] have proposed the following time-dependent test
problem
@f
@t
2
@
@
3
= 0 @p
(4p2f ) + @p
2 (p f ) 0 f + (p 0 pinj )(t);
(1)
where f is the distribution function of particles, p is particle momentum,
and pinj is injection momentum. This equation has the impulsive injection
term at t = 0 and the escape term. At t = 0 particles are injected with a
momentum of pinj = 0:1. To treat the escape01term numerically we calculate
the probability of escape P (1t) = 1 0 e at each time step 1t and
determine whether to remove each particle by using a uniform random
number. Equation (1) is equivalent to the following SDEs,
q
dp = 4p2 dt + 2p3 dW;
(2)
t
400
where dW is a Wiener process given by the Gaussian distribution. The
numerical integration of the SDEs is performed by a simple Euler method.
A particle splitting is essential to achieve a wide dynamic range. We
set splitting surfaces p in momentum space. Each time a particle hits the
surface p , the particle is split into w particles with the same momentum
which the particle has attained.
T. YOSHIDA AND S. YANAGITA
i
i
3.
Numerical Results
0
log10f (arbitrary unit)
log10f (arbitrary unit)
In Figure 1(a) we show the numerical results
without particle splitting
to
4
6
make a comparison between the case of 10 particles and that of 10 . As
Park and Petrosian[3] have concluded, the SDE method without particle
splitting is susceptible to Poisson noise when the number of particles is
insucient. Figure 1(b) shows the numerical results with and without
(a)
-2
-4
-6
-8
-2
0
2
log10p
4
0
(b)
-2
-4
-6
-8
-2
0
2
log10p
4
Figure 1.
(a) The solid line shows the spectrum at t = 3:0 without particle splitting,
6
4
where we use 10 particles. The dotted line shows the result of 10 particles without
particle splitting. (b) The solid line shows the spectrum with particle splitting, where we
4
take w = 4 using 10 particles. The dotted line is the same as Figure 1(a).
particle splitting for the same number of test particles of 104. It is clearly
seen that Poisson noise is avoided even for the relatively small number of
test particles when particles are split.
References
1.
2.
3.
4.
A&A, 251, 693.
A&A, 265, L13.
ApJS 103, 255.
Prog. Theor. Phys., 92, 1217.
MacKinnon, A.L. and Craig, I.J.D. (1991),
Achterberg, A. and Kr
ulls, W. M. (1992),
Park, B.T. and Petrosian, V. (1996),
,
Yoshida, T. and Yanagita S. (1994),