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the surface displacement is finite the central limit theorem (CLT) predicts that
F
has
normal statistics. The physics literature is replete with this kind of phenomenology to
describe the statistical behavior of observations when the exact dynamics of the web
was not in hand. We return to this point repeatedly in our discussions.
Describing the statistical properties of physical webs phenomenologically was not
very satisfying to theorists in statistical physics, who looked around for a more sys-
tematic way to understand physical uncertainty. The most satisfying explanation of
statistical ambiguity, of course, should spring from a mechanical model and that
meant the equations of Hamilton or Newton. Or at least that was the thinking in
the statistical-physics community. So let us sketch the kind of argument that was
constructed.
We construct a network of linear equations following the discussion of Bianucci
et al
.[
8
] and write
(
t
)
w(
)
d
t
=
ξ
(
)
,
t
dt
d
ξ(
t
)
2
=−
w(
t
)
+
v(
t
),
dt
d
v(
t
)
2
(3.39)
=−
1
ξ(
t
)
+
π(
t
),
dt
d
π(
t
)
2
=−
2
v(
t
)
+···
dt
.
.
.
It should be stressed that this linear model is completely general and constitutes an infi-
nite hierarchy of equations, where for the present discussion
is the web variable
and all the other variables are associated with the soon-to-be-ignored degrees of freedom
in the dynamics and constitute the environment or heat bath. Why we use the nomen-
clature of a heat bath for the environment will become clear shortly. Subsequently the
web variable is identified with the velocity of a Brownian particle, as it so often is in
physical models.
The dynamics of an isolated linear environment are described by the subset of
equations
w(
t
)
d
ξ (
t
)
=
v(
t
),
dt
d
v(
t
)
2
=−
1
ξ(
t
)
+
π(
t
),
dt
(3.40)
d
π(
t
)
2
=−
2
v(
t
)
+···
dt
.
.
.
2
2
where the coupling coefficients
2
, ..., canallbedifferentandin(
3.40
)wehave
detached the environment from the web variable of interest. On comparing the sets of
equations (
3.39
) and (
3.40
) it is clear that the heat bath perturbs the web variable
1
,
w(
t
)
