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ζ
∗
=
With the choice of surface
0 the condition (
4.116
) with the unperturbed potential
given by (
4.120
) yields the following expression for the manifold
:
1
2
m
1
π
1
2
ξ
2
2
E
−
−
−
ε
[
ξ
−
ξ (
0
)
]≥
0
,
(4.127)
which can be rewritten as the points interior to an ellipse
1
2
m
1
π
1
2
(ξ
−
ε)
2
2
2
+
≤
2
E
+
2
εξ(
0
)
+
ε
(4.128)
centered at
The calculation of the Poincaré map for the unperturbed
energy of the booster is set at
E
(π
c
=
0
,ξ
c
=
ε).
8. With this choice of the energy the booster is in
a very chaotic regime and the intersections of the chaotic trajectories with the plane
ζ
∗
=
=
0
.
0 are randomly distributed over the accessible Poincaré surface of section as
shown in Figure
4.22
. Furthermore, the numerical calculation shows that the modulus
of the greatest Lyapunov exponent is larger than 4. The network is proved to a very good
approximation to be ergodic since the numerical calculation shows that the distribution
of the points on the manifold is a flat function.
We are now in a position to determine the consistency of the calculations of the trans-
port coefficients, these being the diffusion coefficient and the friction coefficient, with
the theoretical predictions. According to the theory presented above, the derivation of
the diffusion coefficient
D
rests on the properties of the Poincaré map of the unperturbed
booster. More precisely,
D
is obtained by multiplying the correlation time
τ
c
defined by
(
3.45
) by the unperturbed second moment of the doorway variable defined by (
4.118
).
Insofar as the friction
γ
is concerned, we use the susceptibility
χ
of the booster, so that
2
in the present calculation
χ
=
1 and consequently, from (
3.47
),
γ
=
.
Thus, under
the physical conditions for which Figure
4.21
was calculated, namely
γ
=
0
.
01 and
The distribution of the intersections of the trajectories of (
4.117
) with the plane
ζ
∗
=
0
in the chaotic case corresponding to the energy value
E
=
0
.
8[
7
]. Reproduced with
permission.
Figure 4.22.
