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The variable
; that is, the web is coupled to
the bath dynamics by means of this variable. The web variable is eventually identified
with the “free” velocity of the Brownian particle, at which time the doorway variable
is identified with the random force. The symbol
w
is driven by the “doorway variable”
ξ
denotes the set of additional phase-
space variables necessary to complete the description of the bath dynamics and the
“booster” refers to the dynamical network described by the equations
d
dt
=
R
ξ
(ξ,,),
(4.101)
d
π
j
dt
=
j
(ξ,,)
;
=
,
,...,
j
1
2
where, for the moment, the function
representing the coupling between the web and
the bath is a constant. If
=
0 the booster is isolated and unperturbed; however, if
=
0 then the booster is perturbed. In (
4.100
) the perturbation
=−
γw
is the back-
reaction of the booster to the web of interest.
A necessary condition for this prototype model is that it has no memory of initial
conditions. This lack of memory implies that its dynamics are chaotic. Of course, no
memory does not mean that the booster has an infinite number of degrees of freedom.
A putative candidate could be a finite set of nonlinear oscillators in the chaotic regime.
The response of the oscillators to an external perturbation is “naturally” different from
zero. Broadly speaking, we expect that a chaotic network, if perturbed by an external
force, rearranges itself to accommodate the perturbation.
Let us suppose that the value of the coupling between the web and bath is weak; the
“macroscopic” time scale of the evolution of the variable
γτ
c
]
−
1
,
and we expect that a time-scale separation holds for the dynamics of the web of interest
and for the variables of the booster. From (
4.100
), the velocity
w
is then of order [
w(
t
)
is the integral over
time of the pseudo-stochastic variable
ξ(
t
)
and, on applying the central limit theorem,
we immediately conclude that
has Gaussian statistics, with a width that grows lin-
early in time. The properties of the web variable should not be too surprising: it is well
known that chaotic processes can result in long-time diffusional processes, and success-
ful methods to predict the corresponding diffusion coefficients have been developed
[
14
]. However, although this research essentially solves the problem of a determinis-
tic approach to Brownian motion, it refers only to the unperturbed case, that is,
w(
t
)
0,
which corresponds to the case of no coupling, or pure diffusion. The coefficient of
diffusion obtained in the unperturbed case reads
γ
=
2
0
τ
c
,
D
=
ξ
(4.102)
where the characteristic time scale
τ
c
is defined as the integral over the autocorrelation
function
∞
0
ξ
(
τ
c
≡
t
)
dt
.
(4.103)
Note that we have defined the microscopic time scale as the time-integral over the
doorway variable's unperturbed autocorrelation function, which is assumed to be
stationary.
