428
T. Sandev, R. Metzler, ˇ
Z. Tomovski
function
ξ
(
t
)
ξ
(
t
)
=
C
(
t
−
t
)
.
(1.2)
The correlation function
C
(
t
) may be dependent on the frictional mem-
ory kernel via the second fluctuation-dissipation theorem [27, 30] in the
following way:
C
(
t
) =
k
B
T γ
(
t
)
,
(1.3)
where
k
B
is the Boltzmann constant and
T
is the absolute temperature
of the environment. This is a case of internal noise, when the fluctuation
and dissipation come from same source, and the system will reach the
equilibrium state. From the other side, the fluctuation and dissipation may
come from different sources, so the fluctuation-dissipation theorem (1.2)
does not hold, and the system will not reach a unique equilibrium state
[56, 53, 47]. Note that in case of internal white Gaussian noise
ξ
(
t
), the
GLE (1.1) would correspond to the classical Langevin equation [30]. The
FBM and GLE motion are ergodic (time and ensemble averages are same
[3, 9, 24]).
In this paper the anomalous diffusion is investigated by analyzing FGLE
with a three parameter Mittag-Leffler frictional memory kernel. It is a
generalization of the GLE (1.1) in which the integer order derivatives are
substituted by fractional order derivatives, for example, of the Caputo form
[8]:
C
D
γ
0+
f
(
t
) =
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1
Γ(
m
−
γ
)
t
0
f
(
m
)
(
τ
)
(
t
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