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1
2
m
1
π
1
2
m
2
v
2
2
=
+
+
(ξ,ζ,),
H
U
(4.105)
where the two particles have masses
m
1
and
m
2
. The Hamiltonian represents an oscil-
lator with displacement
ξ
π
and velocity
interacting via a nonlinear interaction with
ζ
v
another oscillator, with displacement
and velocity
. The equations of motion are
given by
dt
=
π,
d
d
dt
=−
m
1
∂
(ξ,ζ,)
∂ξ
1
U
,
(4.106)
d
dt
=
v,
d
dt
=−
m
2
∂
1
U
(ξ,ζ,)
∂ζ
and the masses are coupled to one another by means of the interaction potential
2
2
2
+
ξζ
3
3
+
ξ
(ξ,ζ,)
≡
ξ
2
+
ζ
−
ζξ
2
4
4
U
ζ
+
,
(4.107)
where
=
0 gives the unperturbed potential. The first oscillator is perturbed by a con-
stant field
, acting as the contribution to the potential
−
γw
. Notice that the intersection
of the solution trajectories with the plane
0 results in a harmonic potential, thereby
simplifying the subsequent calculation. We make the choice
m
1
=
ζ
=
1 and
m
2
=
0
.
54 and
refer to (
4.106
) as the microscopic booster for our numerical example.
The connection between the continuous equation (
4.106
) and the mapping (
4.104
)
is obtained by following the trajectory determined by the potential (
4.107
)fromthe
pair
(ξ
n
,π
n
)
on the
ζ
=
0 plane to the next intersection of that plane at
(ξ
n
+
1
,π
n
+
1
)
.
Throughout this motion the value of the perturbation field
is kept fixed at
=−
γw
n
.
The subsequent crossing determines the new pair
(ξ
n
+
1
,π
n
+
1
)
and, through the first
equation of (
4.106
), the new velocity
are
obtained from the Poincaré surface of section of the continuous system (
4.106
)at
ζ
=
w
n
+
1
. In conclusion, the values
(ξ
n
,π
n
)
undergoing an abrupt change at any and every crossing of this surface.
The interval of time between the two consecutive crossings is set equal to unity for
convenience.
We draw your attention to the fact that the time evolution of the variable
0, with
,as
driven by this three-dimensional map, looks impressively similar to a Brownian-motion
process. With the numerical calculation of the mapping we evaluate the equilibrium
autocorrelation function of the velocity,
w
)
=
w(
0
)w(
t
)
eq
w
2
eq
(
t
(4.108)
and show that, as in Brownian motion, the autocorrelation function is an exponential
with damping parameter
. Then let us consider the equilibrium autocorrelation function
of the second-order velocity variable
λ
