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The event-based theory, in the ordinary ergodic and stationary case, yields results that
apparently coincide with the predictions of conventional LRT.
7.3.1
The Hamiltonian approach to LRT
In this section we discuss LRT in the context of Hamiltonian webs following the concise
form proposed by Zwanzig [
73
]. In the presence of a perturbation the classical Liouville
equation can be written in terms of two Liouville operators,
∂
∂
t
ρ(
t
)
=
L
ρ(
t
)
≡[−
L
0
+
L
1
E
(
t
)
]
ρ(
t
),
(7.101)
where
denotes the trajectory density in phase space and its average value yields a
probability density. The quantity
L
0
is the unperturbed Liouville operator and
L
0
ρ
ρ(
t
)
is
the Poisson bracket of
H
0
, the unperturbed Hamiltonian, and the density
will be as
shown explicitly below. The perturbation enters through the second Liouville operator
L
1
, which is the Poisson bracket of
ρ
M
, the web variable used to couple the web to the
external perturbation. The intensity of the time-dependent perturbation is proportional to
the function
E
−
(
t
)
. In other words, the total Hamiltonian
H
of the web is time-dependent
and expressed by
H
(
t
)
=
H
0
−
ME
(
t
).
(7.102)
Let us separate the phase-space distribution into two pieces,
ρ(
t
)
=
ρ
0
(
t
)
+
ρ
1
(
t
)
,
and split the Liouville equation into the lowest-order piece
∂
∂
t
ρ
0
(
)
=−
L
0
ρ
0
(
)
t
t
(7.103)
and the first-order piece
∂
∂
t
ρ
1
(
t
)
=−
L
0
ρ
1
(
t
)
+
L
1
ρ
0
(
t
)
E
(
t
).
(7.104)
The first-order solution to the Liouville equation (
7.101
) is given by solving the
inhomogeneous equation (
7.104
) to obtain
t
dt
exp
t
)
t
)ρ
0
(
t
).
ρ
1
(
t
)
=
[−
(
t
−
L
0
]
L
1
E
(
(7.105)
0
The Liouville operator
L
1
acts on the unperturbed distribution
ρ
0
(
t
)
as follows:
∂
M
∂
p
∂ρ
0
(
t
)
−
∂
M
∂
q
∂ρ
0
(
t
)
L
1
ρ
0
(
t
)
=−
.
(7.106)
∂
q
∂
p
The vectors
p
and
q
, the canonical variables for the web degrees of freedom, are
multidimensional and the scalar product is assumed in (
7.106
). Thus, in practice this
picture may apply to an arbitrarily large number of particles.
Now we assume that prior to the application of the perturbation the network is in
equilibrium and is characterized by the canonical distribution
exp
(
−
β
H
0
)
ρ
eq
(
,
)
=
,
q
p
(7.107)
Z
