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by means of the doorway variable
ξ(
t
)
, which in turn is perturbed by the “back-reaction
2
term”
. The equation set (
3.40
) consequently models the unperturbed bath,
whereas the set (
3.39
) includes the energy exchange between the two networks.
The solution to the perturbed equations of motion for the doorway variable can be
expressed immediately in terms of Green's function
K
−
w(
t
)
(
)
t
for the unperturbed linear
network as
2
t
0
dt
K
t
)w(
t
),
ξ(
t
)
=
ξ
0
(
t
)
−
(
t
−
(3.41)
where
is the solution to the equations of motion for the unperturbed bath and the
kernel depends only on this unperturbed solution. Inserting this equation into that for
the web variable yields
ξ
0
(
t
)
2
t
0
w(
)
d
t
dt
K
t
)w(
t
),
=
ξ
0
(
)
−
(
−
t
t
(3.42)
dt
resulting in an integro-differential equation for the web variable that in physics is called
the generalized Langevin equation under certain conditions. The unperturbed solution
ξ
0
(
is interpreted as a random force because of the uncertainty of the initial state of
the environment. The integral term is interpreted as a dissipation with memory when the
kernel has the appropriate properties. In the present notation the fluctuation-dissipation
relation has the general form
t
)
2
w
2
eq
K
ξ
0
(
t
)ξ(
0
)
0
=
(
t
),
(3.43)
where the subscript zero denotes an average taken over the distribution of the initial
state of the bath starting at
and the subscript eq denotes an average taken over an
equilibrium distribution of fluctuations of the web variable. The kernel is the response
function of the doorway variable to the perturbation
ξ(
0
)
2
and the relation (
3.43
)
indicates that the kernel is proportional to the autocorrelation function of the doorway
variable
−
w(
t
)
ξ
2
0
K
(
t
)
=
2
w
2
eq
ξ
(
t
).
(3.44)
In this simple model the dissipation is given by the integral in (
3.42
) and the dissipative
nature of the kernel (
3.44
) is a consequence of the infinite number of degrees of freedom
of the heat bath. Poincaré proved that the solution to such a set of linear equations would
periodically return to its initial state if the equations were generated by a Hamiltonian.
Consequently the doorway variable would be periodic, but the large number of degrees
of freedom makes the Poincaré recurrence time extremely long. This practical “irre-
versibility” leads to a characteristic correlation time
τ
c
for the autocorrelation function
of the doorway variable and therefore of the kernel
K
(
t
),
∞
∞
dt
ξ
0
(
t
)ξ
0
ξ
2
0
τ
c
≡
=
dt
ξ
(
t
).
(3.45)
0
0
Assume a time-scale separation such that the web variable changes in time much more
slowly than does the bath and therefore does not change appreciably over the correlation
