Information Technology Reference
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We return to this discussion after having introduced the formal notion of LRT and
justified some of the phenomenological terms used in the discussion above. For the
moment we emphasize that if a variable
is weakly coupled to a nonlinear network, of
any dimension, provided that this web is chaotic and ergodic, the resulting deterministic
motion of the variable
w
w
conforms to that of a standard fluctuation-dissipation pro-
cess found in most texts on statistical physics. The irregularities of the deterministic
statistics are washed out by the large-time-scale separation between the variables in
the web of interest and those in the chaotic booster. In other words, we find that the
influence exerted by the chaotic and ergodic booster on the variable
is indistin-
guishable from that produced by a standard heat bath with infinitely many degrees of
freedom.
Thus, we might expect complex social webs to exhibit Brownian motion even when
the traditional heat-bath argument is not appropriate. This could well apply to the DDM
model discussed earlier (Section 3.3.2). The uncertainty in the information on which
decisions are based would not be due to an infinite number of variables in the environ-
ment, but would be a consequence of the nonlinear interactions among the finite number
of variables describing society.
For the present approach to the standard fluctuation-dissipation process to apply to
the variable of interest, the booster has to satisfy the basic condition of responding
linearly to a constant external perturbation. If we adopt a mathematical mapping as a
booster using the Poincaré surface of section from the last section, then we need to
understand the problem of the linear response of a mapping to an external perturbation,
which we take up subsequently. For the present motivational purposes we merely indi-
cate some computer calculations that support the predictions of the theoretical approach
discussed here, using a two-dimensional booster, in which the description of a chaotic
web with a well-defined phase space is given by a microcanonical distribution. As a
result of an external perturbation the phase-space manifold on which the dynamics
unfolds is modified and the distribution is accordingly deformed, thereby making it
possible to derive an analytic expression for the corresponding stationary web response.
The details of this argument are given later.
We derive the standard fluctuation-dissipation-relation process from the following
three-dimensional mapping:
w
w
n
+
1
=
w
n
+
ξ
n
+
1
,
ξ
n
+
1
=
ξ
n
+
f
(ξ
n
,π
n
,
−
γw
n
),
(4.104)
π
n
+
1
=
π
n
+
g
(ξ
n
,π
n
,
−
γw
n
).
The values
w
n
are to be regarded as being the values taken at discrete times by a contin-
uous variable
, the same symbol as that used in (
4.100
). This choice is made because,
as we shall see, (
4.104
) leads to dynamics for
w
w
n
that is virtually indistinguishable from
the conventional picture of (
3.52
). Thus, we refer to
as the velocity of the Brownian
particle. The functions
f
and
g
of this mapping are obtained as follows. A microscopic
network with two degrees of freedom is assumed to have an unperturbed Hamiltonian,
for example,
w
