Geoscience Reference
In-Depth Information
Another relevant fact is that a special type of Markov process characterized by a linear
A
(
v
,
t
D
1/2
I
, independent of
v
and
t
, is known as
Brownian motion process (e.g., Chandrasekhar, 1942; Uhlenbeck & Ornstein, 1930), which is
often invoked in simplified models of collisional plasmas (e.g., Dougherty, 1964; Holod et al.,
2005; Woodman, 1967) including our earlier result (24) with
C
=
)=
−
β
v
and a constant matrix
ν
=
β
. In these models, friction
and diffusion coefficients,
β
and
D
, are constrained to be related by
2
KT
m
β
D
=
(38)
for a plasma in thermal equilibrium.
In return for having restricted
v
(
)
to the space of Markovian processes, we have gained a
stochastic evolution equation (35) with a plausible Newtonian interpretation and with the
potential of taking us beyond Brownian motion based collision models.
t
Furthermore, the
evolution of probability density
f
(
v
,
t
)
of a random variable
v
(
t
)
is known to be governed,
when
v
(
is Markovian, by the Fokker-Planck kinetic equation having a “friction vector” and
“diffusion tensor”
t
)
Δ
v
Δ
c
=
A
(
v
,
t
)
,
(39)
t
and
Δ
v
Δ
v
T
Δ
t
C
(
v
,
t
C
T
(
v
,
t
c
=
)
)
,
(40)
respectively, specified in terms of the input functions of the Langevin equation. This
intimate link between the Langevin and Fokker-Planck equations — in describing Markovian
processes from two different but mutually compatible perspectives — was first pointed out
by Chandrasekhar (1943) and discussed in detail by Gillespie (1996b).
Since the Fokker-Planck friction vector and diffusion tensor for equilibrium plasmas with
Coulomb interactions have already been worked out by Rosenbluth et al. (1957) as
Δ
v
Δ
c
=
−
β
(
v
)
v
(41)
t
and
Δ
v
Δ
v
T
Δ
D
vv
T
v
2
D
⊥
(
v
)
D
⊥
(
v
)
I
c
=
+
(
v
)
−
,
(42)
t
2
2
in terms of scalar functions
β
(
v
)
,
D
(
v
)
,
D
⊥
(
v
)
, it follows that the Langevin update equation,
magnetized version of (37), can be written as
q
m
v
(
v
(
+ Δ
)=
v
(
)+
)
×
B
Δ
t
t
t
t
t
D
D
)
Δ
t
2
(
− β
(
)Δ
t
v
(
)+
(
)Δ
+
⊥
(
+
v
t
v
tU
1
v
v
U
2
v
U
3
v
)
,
(43)
⊥
⊥
1
2
where
denote an orthogonal set of unit vectors parallel and
perpendicular to the particle trajectory and
U
i
(
v
(
t
)
,
v
(
t
)
,
and
v
(
t
)
⊥
1
⊥
2
t
)=
N
(
0, 1
)
are independent random
numbers.
For weakly magnetized plasmas of interest here,
where Debye lengths are
smaller than the mean gyro radii, the “friction coefficient”
β
(
v
)
and velocity-space diffusion
