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velocity autocorrelation function cannot generate an exponential decay. The search for
a representation that may make the exponential decay compatible with the Hamiltonian
treatment has been the subject of many studies (see [
65
] and references therein). The
discussion of this problem is closely related to the derivation of irreversible processes
from a picture of either quantum or classical physics that is invariant under time reversal.
We cannot rule out the possibility that this transition is realized by the presence of
renewal events that are invisible in conditions very far from the exponential relaxation.
These events become important to eliminate the sign of the original reversibility when
the exponential form becomes the predominant contribution to relaxation as investigated
by Vitali and Grigolini [
65
]. Let us here adopt the view that the exponential decay may
be a good approximation to the relaxation process. In this sense, the linear response
function
t
)
=
t
)
]
χ(
t
,
exp
[−
γ(
t
−
(7.117)
is acceptable, in spite of its exponential character.
7.3.2
Applying the formalism
To put the formalism of the last subsection into perspective, recall the simple example
of LRT given in Section 3.2.2 where we considered the linear Langevin equation for the
velocity with linear dissipation, a white-noise random force and a driving force
E
.
Recall that the exact solution for the velocity with zero initial value and the perturbation
switched on at
t
(
t
)
=
0is
t
e
−
λ(
t
−
t
)
E
t
)
dt
V
(
t
)
=
(
(7.118)
0
and the exponential is the autocorrelation function
V
t
)
eq
V
2
eq
.
(
t
)
V
(
e
−
λ(
t
−
t
)
=
(7.119)
If we again assume equipartition of energy
V
2
eq
=
k
B
T
abs
(7.120)
(
7.118
) becomes
t
V
(
t
)
=
β
ds
χ
VQ
(
t
−
s
)
E
(
s
).
(7.121)
0
We note that this result fits perfectly well the prediction of (
7.114
) and (
7.115
). In fact,
the Hamiltonian interacting with the external perturbation in this case reads
H
1
(
t
)
=−
qE
(
t
),
(7.122)
t
)
and, of course, since
dQ
/
dt
=
V
, the linear response function
χ
VQ
(
t
−
coincides
with the velocity-displacement cross-covariance function.
