Information Technology Reference
In-Depth Information
of one another. The velocity variable
can be considered as a function of the other
variables and of the energy
E
in the following way:
v
2
m
2
E
1
2
m
1
π
v
=
−
2
−
U
(ξ,ζ,)
.
(4.113)
In the three-dimensional phase space
ξ,ζ,π
the trajectories of the network lie inside a
domain
(
E
)
defined by the condition
2
v
≥
0
(4.114)
imposed by energy conservation. This domain is enclosed by the surface
S
(
E
)
defined
by the region of the phase space where
v
=
0, namely
1
2
m
1
π
2
E
−
−
U
(ξ,ζ,)
=
0
.
(4.115)
(
)
The surface
S
is shown schematically in Figure
4.20
.
In the present situation the heat bath for our Brownian particle with velocity
E
w
is
a booster obtained as a Poincaré map of the generating network (
4.104
). Consider the
Poincaré map corresponding to the intersection of the trajectories inside the domain
(
ζ
=
ζ
∗
, where
ζ
∗
denotes a generic fixed value of
E
)
with the plane defined by
the variable
ζ
. The points of the Poincaré map lie within a manifold
given by the
ζ
=
ζ
∗
. The manifold
intersection of the domain
(
E
)
with the plane
is formally
defined by the condition
1
2
m
1
π
2
(ξ,ζ
∗
,)
≥
E
−
−
U
0
.
(4.116)
The manifold
0. The Poincaré map is
area-preserving so that the invariant measure is the flat (or Lebesgue) measure
d
is illustrated in Figure
4.21
for the case
=
σ
=
ζ
=
ζ
∗
; that is, they are the
independent variables describing the dynamics of the booster.
Owing to the chaotic and ergodic dynamics of the generating network given by
(
4.100
), the dynamics of this Poincaré map is mixing, in which case
d
ξ
d
π
, where
ξ
and
π
are the coordinates on the plane
μ
is the unique
Figure 4.20.
A schematic illustration of the domain
(
E
)
enclosed by the surface
S
(
E
)
[
7
]. Reproduced with
permission.
