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extended by Milla & Kudeki (2011). The new procedure allows the calculation of collisional
IS spectra at all magnetic aspect angles including the perpendicular-to-
B
direction (
α
=
0
◦
)as
needed for IS radar applications. In this section, we present the procedure developed by Milla
& Kudeki (2011) to model the effects of Coulomb collisions on the incoherent scatter spectrum.
e
j
k
·
Δ
r
The single-particle ACF
in a collisional plasma including a magnetic field can in
principle be calculated by taking the spatial Fourier transform of the probability distribution
f
(Δ
r
,
τ
)
(Δ
r
,
τ
)
of the particle displacement
Δ
r
appropriate for such plasmas, and
f
in turn
(
r
,
t
)
can be derived from the solution
f
of the Boltzmann kinetic equation with a collision
operator, e.g., the Fokker-Planck kinetic equation of Rosenbluth et al. (1957). Although,
analytical solutions of simplified versions of the Fokker-Planck kinetic equation are available
(e.g., Chandrasekhar, 1943; Dougherty, 1964), determining
f
(Δ
r
,
τ
)
would be a daunting task
when the full Fokker-Planck equation is considered.
We will discuss here an alternative and more practicable approach that involves Monte Carlo
simulations of sample paths
r
(
of particles undergoing Coulomb collisions. A sufficiently
large set of samples of trajectories
r
(
t
)
e
j
k
·
Δ
r
t
)
can then be used to compute
as well as
any other statistical function of
to be ergodic. This
alternate procedure requires the availability of a stochastic equation describing how the
particle velocities
Δ
r
assuming the random process
r
(
t
)
d
r
dt
v
(
t
)
≡
(34)
may evolve under the influence of Coulomb collisions.
v
(
)
Assuming that under Coulomb collisions the velocities
constitute a Markovian random
process — meaning that past values of
v
would be of no help in predicting its future values if
the present value is available — the stochastic evolution equation of
v
(
t
)
will be constrained
by very strict self-consistency conditions discussed by Gillespie (1996a;b) to acquire the form
of a Langevin equation
t
d
v
(
t
)
C
(
v
,
t
=
A
(
v
,
t
)+
)
W
(
t
)
(35)
dt
and matrix
C
(
v
,
t
where vector
A
(
v
,
t
)
)
consist of arbitrary smooth functions of arguments
v
and
t
, and
W
(
t
)
is a random vector having statistically independent Gaussian white noise
components
W
i
(
)=
Δ
t
→
0
N
(
)
t
lim
0, 1/
Δ
t
,
(36)
2
compatible with the requirement that
W
i
(
t
+
τ
)
W
i
(
t
)
=
δ
(
τ
)
. Here and elsewhere
N
(
μ
,
σ
)
2
.
denotes the normal random variable with mean
μ
and variance
σ
A more natural way of expressing the Langevin equation (35) is to cast it in an update form,
namely
C
(
v
,
t
)
Δ
t
1/2
U
(
v
(
t
+
Δ
t
)=
v
(
t
)+
A
(
v
,
t
)
Δ
t
+
t
)
,
(37)
where
is a vector composed of independent
zero-mean Gaussian random variables with unity variance, i.e.,
U
i
(
Δ
t
is an infinitesimal update interval and
U
(
t
)
t
)=
N
(
0, 1
)
.
Note that the Langevin equation describing a Markovian process has the form of Newton's
second law of motion, with the terms on the right representing forces per unit mass exerted on
plasma particles. Considering the Lorentz force on a charged particle in a magnetized plasma
with a constant magnetic field
B
, and not violating the strict format of (35), we can modify the
equation by adding a term
q
v
(
)
×
B
/
m
to its right hand side.
t
