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where
and
Z
is the classical partition function. Assuming that the web
is originally equilibrated allows us to express the LRT in terms of an equilibrium auto-
correlation function, but at the same time establishes a condition beyond which we plan
to work. Assuming that
β
=
1
/(
k
B
T
abs
)
ρ
0
=
ρ
eq
makes it possible to turn (
7.106
)into
L
1
ρ
eq
=
βρ
eq
∂
M
∂
p
∂
H
0
∂
q
−
∂
M
∂
q
∂
H
0
∂
.
(7.108)
p
The observable time-evolution is driven by
L
+
, the operator adjoint to
L
.Inthe
Hamiltonian case
L
+
=−
L
, thus
L
0
=−
L
0
.
(7.109)
The time-evolution of
M
as an effect of the zeroth Liouville operator is given by
d
dt
M
M
L
0
M
=
=
=−
L
0
M
=
∂
M
∂
p
∂
H
0
∂
q
−
∂
M
∂
q
∂
H
0
∂
p
.
(7.110)
Thus, comparing (
7.110
) with (
7.108
) yields
L
1
ρ
eq
=
β
M
ρ
eq
(7.111)
and by inserting (
7.111
)into(
7.105
) we obtain the first-order phase-space distribution
t
dt
exp
t
)
L
0
]
ME
t
)ρ
eq
ρ
1
(
t
)
=
β
[−
(
t
−
(
(7.112)
0
in terms of the initial equilibrium phase-space distribution.
Consider the web variable
A
and assume that with the web in equilibrium its mean
value vanishes:
A
eq
=
0
.
(7.113)
To establish the web response to perturbation we multiply (
7.112
) by the web variable
A
and integrate over the phase-space variables to obtain for the average response
t
dt
E
t
)χ
AM
(
t
),
A
(
t
)
=
β
(
t
−
(7.114)
0
t
)
=
χ
AM
(
t
)
where the stationary linear response function
χ
AM
(
t
,
t
−
can be formally
expressed as
d
q
d
p
A
t
)
=
t
)
L
0
]
M
χ
AM
(
t
−
(
q
,
p
)
exp
[−
(
t
−
(
q
,
p
)ρ
eq
(
q
,
p
).
(7.115)
We emphasize that the Hamiltonian approach to establishing the form of LRT is
not confined to the exponential case. Quite the contrary, the exponential case is an
idealization that is not compatible with the Hamiltonian approach. We have seen that
t
)
=
A
t
)
,
)
M
χ
AM
(
−
(
(
t
t
(7.116)
namely the linear response function is the cross-covariance function between the web
variables
A
and
M
. Years ago Lee [
40
] proved that the Hamiltonian approach to the
