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term yields the Mittag-Leffler function as found in the homogeneous case. The inverse
Laplace transform of the second term is the convolution of the random force and a
stationary kernel.
The explicit inverse of (
5.71
) yields the formal solution [
31
]
t
−
τ))
α
)ξ(τ)
E
α,α
(
−
(λ(
t
)
α
)
+
Q
(
t
)
=
Q
(
0
)
E
α
(
−
(λ
t
d
τ
,
(5.72)
1
−
α
(
t
−
τ)
0
where we see that the fluctuations from the random force are smoothed or filtered by the
generalized MLF, as well as by the inverse power law. The kernel in this convolution is
given by the series
∞
z
k
(β
+
E
α,β
(
z
)
≡
α)
;
α,β >
0
,
(5.73)
k
k
=
0
which is the generalized MLF and for
1 this series reduces to the standard MLF.
Consequently both the homogeneous and the inhomogeneous terms in the solution to the
fractional Langevin equation can be expressed in terms of these MLF series. The aver-
age of the solution (
5.72
) over an ensemble of realizations of a zero-centered random
force is given by
β
=
)
α
),
Q
(
t
)
ξ
=
Q
(
0
)
E
α
(
−
(λ
t
(5.74)
which we see coincides with the deterministic solution to the fractional diffusion
equation (
5.46
).
In the case
1, the MLF reduces to the exponential, so that the solution to the
fractional Langevin equation becomes that for an Ornstein-Uhlenbeck (OU) process,
α
=
t
e
−
λ
t
e
−
λ(
t
−
τ)
ξ(τ),
Q
(
t
)
=
Q
(
0
)
+
d
τ
(5.75)
0
as it should. Equation (
5.75
) is simple to interpret since the dissipation has the typical
form of a filter and it smoothes the random force in a familiar way.
The properties of the solutions to such stochastic equations can be determined only
after the statistics of the random force have been specified. The usual assumption made
is that the random force is zero-centered, delta-correlated in time with Gaussian statis-
tics. From these assumed statistics and the fact that the equation of motion is linear,
even if fractional, the statistics of the response must also be Gaussian, and consequently
its properties are completely determined by its first two moments. It is interesting to
note that the variance for the dissipation-free (
λ
=
0
)
fractional Langevin equation can
be calculated to yield
Q
)
ξ
2
t
2
α
−
1
(
t
)
−
Q
(
t
,
(5.76)
ξ
∝
where the proportionality constant is given in terms of the strength of the random force.
This time dependence of the variance agrees with that obtained for anomalous diffusion
if we make the identification
2
H
=
2
α
−
1
,
(5.77)
