Information Technology Reference
In-Depth Information
1
4
0.5
2
0
1
3
-0.5
6
5
-1
-1
-0.5 0 0.5
Displacement
1
Figure 4.17.
The initial point lies on the intersection of a torus with the surface of section but the mapping
does not eventually trace out a continuous curve for the cross section of the torus. Instead one
obtains a random display of points.
A. The notion of an
irrational
number is non-physical. It is impossible to prove that a
physical quantity is rational or irrational.
B. The notion of commensurability is non-physical, when the quantities to be compared
are known only to within some experimental errors.
C. It is impossible to study the properties of a single (mathematical) trajectory. The
physicist knows only
bundles of trajectories
, corresponding to slightly different ini-
tial conditions
any information obtained from an experiment must be paid for
by a corresponding increase of entropy in the measuring device: infinite accuracy
could cost an infinite amount of entropy increase and require infinite energy! This is
absolutely unthinkable.
...
4.4.2
Chaos and linear response theory
Let us introduce a class of nonlinear dynamical microscopic webs that reproduce Brow-
nian motion at the macroscopic level. It is impossible to study all of the possible kinds of
nonlinear networks which belong to this class, so we study a prototype of these nonlin-
ear models. An important feature of our prototype network is the presence of a variable
of interest
coupled to a thermal bath (in what follows this thermal bath is called the
booster). The dynamical bath is here characterized by very few degrees of freedom that
are nonlinearly coupled to one another and are therefore described by chaotic dynamics,
and have no memory of the bath's initial conditions. The prototype model we study can
be cast in the form [
7
,
8
]
d
w
dt
=
ξ,
d
dt
=
R
ξ
(ξ,,
−
γw),
(4.100)
d
π
j
dt
=
j
(ξ,,
−
γw)
;
j
=
1
,
2
,...
