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In-Depth Information
t
dt
1
t
0
2
t
0
Q
2
2
(
t
)
ξ
=
dt
2
ξ(
t
1
)ξ(
t
2
)
ξ
+
Q
(
0
)
+
dt
1
ξ(
t
1
)
Q
(
0
)
ξ
.
(3.59)
0
If the fluctuations and the initial displacement of the particle are statistically indepen-
dent the last term in (
3.59
) vanishes. If the fluctuations are delta-correlated in time with
strength
D
,
ξ(
t
)
t
),
)ξ(
ξ
=
δ(
−
t
2
D
t
(3.60)
we obtain from (
3.59
) the variance of the dynamical variable
Q
t
dt
1
t
0
2
2
ξ
−
)
(
t
)
Q
(
t
ξ
=
dt
2
2
D
δ(
t
2
−
t
1
)
=
2
Dt
.
(3.61)
0
Consequently, the variance of the displacement of an ensemble of Brownian particles
increases linearly in time. For an ergodic process, namely one for which the ensemble
average is equivalent to a long-time average, we can also interpret (
3.61
) as the variance
of the displacement of the trajectory of a single Brownian particle from its starting
position increasing linearly in time.
The assumption of independence between the initial state of the dynamical variable
holds when there exists a large time-scale separa-
tion between the dynamics of the
fast variable
Q
(
0
)
and the random force
ξ(
t
)
and the
slow variable Q
. However,
this hypothesis must be used with caution in applications, which often refer to situa-
tions in which the random force can have a slow long-time decay, thereby violating the
assumption of time-scale separation. For example, the autocorrelation function
ξ
)
=
ξ(
0
)ξ(
t
)
ξ
ξ
(
t
ξ
2
(3.62)
ξ
can have a number of different forms. We have described the case of no memory, for
which the autocorrelation function is given by the Dirac delta function
ξ
(
t
)
∝
δ(
t
).
(3.63)
A memory in which the autocorrelation function is exponential,
e
−
γ
t
ξ
(
t
)
=
,
(3.64)
and 1
is the decay time of the memory is often used. But in general the autocorrelation
function can have either a positive or a negative long-time tail. In molecular dynamics,
processes characterized by a long-time regime with an inverse power-law correlation
function
/γ
1
t
β
ξ
(
t
)
≈±
(3.65)
have been called slow-decay processes ever since the pioneering work of Adler and
Wainwright [
1
]. Note that such an inverse power-law autocorrelation function implies
an exceptionally long memory and this is what we refer to as long-time memory. The
literature on this topic is quite extensive, but does not directly relate to the purpose
of our present discussion, so we just mention Dekeyser and Lee [
15
], who provided the
first microscopic derivation of an inverse power-law autocorrelation function in physics.
