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where
u
.x/ is the
drift function
, i.e. a function related to the derivative of the external
potential V . In that case Eq. (
7.53
) can be rewritten as
@
@t
D H
(7.54)
h
2
2 @
2
@x
u
.x/
i
, while the probability density function .x;t/ is
H D
@
where
@x
2
found to be
.x;t/ D e
H
.x/
(7.55)
It has to be noted that the Fokker-Planck equation is equivalent to Langevin's
equation and can be also used for the calculation of the mean position of the diffused
particle, as well as for the calculation of its variance [
66
].
Now, the solution .x/ is a wave-function for which holds .x/ Dj .x/j
2
with
.x/ D
P
iD1
c
k
k
.x/, where
k
.x/ are the associated eigenfunctions [
34
,
127
]. It
can be assumed that
0
.x/ Dj
0
.x/j, i.e. the p.d.f. includes only the basic mode,
while higher order modes are truncated, and the drift function
u
.x/ of Eq. (
7.53
)is
takentobe[
56
]
1
2
2
1
0
.x/
@
0
.x/
@x
u
.x/ D
(7.56)
Thus it is considered that the initial probability density function is .x/ D
0
.x/,
which is independent of time, thus from Eq. (
7.54
) one has H
0
D 0, which means
that the p.d.f. remains independent of time and the examined diffusion process is a
stationary one, i.e. .x;t/ D
0
.x/ 8t. A form of the probability density function
for the stationary diffusion is that of shifted, partially overlapping Gaussians, which
is depicted in Fig.
7.1
b. In place of Gaussian p.d.f., symmetric triangular possibility
distributions have been also proposed [
163
]. The equation that describes the shifted
Gaussians is (Fig.
7.1
b)
1
2
C
2
e
1
2
C
2
e
!
2
.xa/
2
!
2
.xCa/
2
0
.x/ D
C
(7.57)
7.6.3
Study of the QHO Model Through
the Ornstein-Uhlenbeck Diffusion
The Ornstein-Uhlenbeck diffusion is a model of the Brownian motion [
17
]. The
particle tries to return to position x.0/ under the influence of a linear force vector,
i.e. there is a spring force applied to the particle as a result of the potential V.x/.The
corresponding phenomenon in quantum mechanics is that of the QHO [
34
,
66
]. In
the QHO the motion of the particle is affected by the parabolic (harmonic) potential
