2.1. Relaxation functions, variances and MSD
We use the Laplace transform method to analyze the FGLE. Thus,
relations (2.1) and (4.17) yield
ˆ
V
(
s
) =
v
0
s
μ
−
1
s
μ
+ ˆ
γ
(
s
)
+
1
s
μ
+ ˆ
γ
(
s
)
ˆ
F
(
s
)
,
(2.2)
ˆ
X
(
s
) =
x
0
1
s
+
v
0
s
μ
−
ν
−
1
s
μ
+ ˆ
γ
(
s
)
+
s
−
ν
s
μ
+ ˆ
γ
(
s
)
ˆ
F
(
s
)
,
(2.3)
where ˆ
V
(
s
) =
L
[
v
(
t
)](
s
), ˆ
X
(
s
) =
L
[
x
(
t
)](
s
), ˆ
γ
(
s
) =
L
[
γ
(
t
)](
s
), ˆ
F
(
s
) =
L
[
ξ
(
t
)](
s
). Here we introduce the following functions
ˆ
g
(
s
) =
1
s
μ
+ ˆ
γ
(
s
)
,
(2.4)
ˆ
G
(
s
) =
s
−
ν
s
μ
+ ˆ
γ
(
s
)
,
(2.5)
VELOCITY AND DISPLACEMENT CORRELATION . . .
431
ˆ
I
(
s
) =
s
−
2
ν
s
μ
+ ˆ
γ
(
s
)
.
(2.6)
By applying inverse Laplace transform to relations (2.2) and (2.3), for the
displacement
x
(
t
) and velocity
v
(
t
) =
C
D
ν
0+
x
(
t
) it follows:
v
(
t
) =
v
(
t
)
+
t
0
g
(
t
−
t
)
ξ
(
t
)d
t
,
(2.7)
x
(
t
) =
x
(
t
)
+
t
0
G
(
t
−
t
)
ξ
(
t
)d
t
,
(2.8)
with
G
(0) = 0, where
v
(
t
)
=
v
0
·
C
D
μ
+
ν
−
1
0+
G
(
t
)
,
(2.9)
x
(
t
)
=
x
0
+
v
0
·
C
D
μ
+
ν
−
1
0+
I
(
t
)
,
(2.10)
and
g
(
t
) =
L
−
1
[ˆ
g
(
s
)] (
t
),
G
(
t
) =
L
−
1
ˆ
G
(
s
)
(
t
),
I
(
t
) =
L
−
1
ˆ
I
(
s
)
(
t
) are
known as relaxation functions. From relations (2.4), (2.5), (2.6) and (4.17)
it follows that
C
D
ν
0+
G
(
t
) =
g
(
t
) and
C
D
ν
0+
I
(
t
) =
G
(
t
).
From relations (2.7), (2.8) and (1.3), in case of an internal noise, follow
the following general expressions of variances
σ
xx
=
x
2
(
t
)
−
x
(
t
)
2
= 2
t
0
d
t
1
G
(
t
1
)
t
1
0
d
t
2
G
(
t
2
)
C
(
t
1
−
t
2
)
= 2
k
B
T
1
Γ(
ν
)
t
0
d
ξG
(
ξ
)
ξ
ν
−
1
−
t
0
d
ξG
(
ξ
)
C
D
μ
0+
G
(
ξ
)
,
(2.11)
σ
xv
=
(
v
(
t
)
−
v
(
t
)
) (
x
(
t
)
−
x
(
t
)
)
=
t
0
d
t
1
g
(
t
1
)
t
0
d
t
2
G
(
t
2
)
C
(
t
1
−
t
2
)
=
k
B
T
1
Γ(
ν
)
t
0
d
ξg
(
ξ
)
ξ
ν
−
1
−
t
0
d
ξg
(
ξ
)
C
D
μ
0+
G
(
ξ
)
−
t
0
d
ξG
(
ξ
)
RL
D
μ
0+
g
(
ξ
)
,
(2.12)
σ
vv
=
v
2
(
t
)
−
v
(
t
)
2
= 2
t
0
d
t
1
g
(
t
1
)
t
1
0
d
t
2
g
(
t
2
)
C
(
t
1
−
t
2
)
=
−
2
k
B
T
t
0
d
ξg
(
ξ
)
RL
D
μ
0+
g
(
ξ
)
,
(2.13)
432
T. Sandev, R. Metzler, ˇ
Z. Tomovski
where it is used the symmetry of the correlation function
C
(
t
1
−
t
2
). For
μ
= 1 and 0
< ν <
1 it is obtained
σ
xx
=
k
B
T
2
Γ(
ν
)
t
0
d
ξG
(
ξ
)
ξ
ν
−
1
−
G
2
(
t
)
,
(2.14)
σ
xv
=
k
B
T
1
Γ(
ν
)
t
0
d
ξg
(
ξ
)
ξ
ν
−
1
−
g
(
t
)
G
(
t
)
,
(2.15)
σ
vv
=
k
B
T
1
−
g
2
(
t
)
.
(2.16)
The case
ν
= 1 and 0
< μ <
1 yields the following results [15, 12]
σ
xx
= 2
k
B
T
I
(
t
)
−
t
0
d
ξG
(
ξ
)
C
D
μ
0+
G
(
ξ
)
,
(2.17)
σ
xv
=
1
2
d
σ
xx
d
t
=
k
B
T G
(
t
)
1
−
C
D
μ
0+
G
(
t
)
,
(2.18)
σ
vv
=
−
2
k
B
T
t
0
d
ξg
(
ξ
)
RL
D
μ
0+
g
(
ξ
)
.
(2.19)
Furthermore, for
μ
=
ν
= 1 we obtained the well known expressions [56, 53]
σ
xx
=
k
B
T
2
I
(
t
)
−
G
2
(
t
)
,
(2.20)
σ
xv
=
k
B
T G
(
t
) [1
−
g
(
t
)]
,
(2.21)
σ
vv
=
k
B
T
1
−
g
2
(
t
)
.
(2.22)
From relation (2.11) for the MSD we obtain
x
2
(
t
)
=
x
2
0
+ 2
x
0
v
0
C
D
μ
+
ν
−
1
0+
I
(
t
) +
v
2
0
C
D
μ
+
ν
−
1
0+
I
(
t
)
2
+2
k
B
T
t
0
d
ξG
(
ξ
)
ξ
ν
−
1
Γ(
ν
)
−
2
k
B
T
t
0
d
ξG
(
ξ
)
C
D
μ
0+
G
(
ξ
)
,
(2.23)
from where it follows
D
(
t
)
=
1
2
d
d
t
x
2
(
t
)
=
x
0
v
0
C
D
μ
+
ν
0+
I
(
t
) +
v
2
0
C
D
μ
+
ν
−
1
0+
I
(
t
)
C
D
μ
+
ν
0+
I
(
t
)
+
k
B
T G
(
t
)
t
ν
−
1
Γ(
ν
)
−
k
B
T G
(
t
)
C
D
μ
0+
G
(
t
)
.
(2.24)
For
μ
= 1 and 0
< ν <
1 it follows
D
(
t
) =
x
0
v
0
G
(
t
) +
v
2
0
−
k
B
T
G
(
t
)
G
(
t
) +
k
B
T G
(
t
)
t
ν
−
1
Γ(
ν
)
.
(2.25)
VELOCITY AND DISPLACEMENT CORRELATION . . .
433
The case
ν
= 1 and 0
< μ <
1 yields the time-dependent diffusion coefficient
D
(
t
)
=
x
0
v
0
C
D
μ
0+
G
(
t
) +
v
2
0
C
D
μ
0+
I
(
t
)
C
D
μ
0+
G
(
t
) +
k
B
T G
(
t
)
−
k
B
T G
(
t
)
C
D
μ
0+
G
(
t
)
,
(2.26)
and for
μ
=
ν
= 1 we obtain D
(
t
) =
x
0
v
0
g
(
t
) +
v
2
0
−
k
B
T
G
(
t
)
g
(
t
) +
k
B
T G
(
t
)
,
(2.27)
which in case of thermal initial conditions (
x
0
= 0,
v
2
0
=
k
B
T
) turns to the
well known results D
(
t
) =
k
B
T G
(
t
)
,
(2.28)
and
x
2
(
t
)
= 2
k
B
T I
(
t
)
.
(2.29)
Dostları ilə paylaş: |