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by the network dynamics. The relative amount of time the web trajectory spends in cell
i
is
t
i
and a metric of cell
p
i
is proportional to this time,
t
i
t
T
,
p
i
=
(2.74)
where
t
T
is the total duration of the trajectory. If the probability of the trajectory being in
that cell does not change in time, as the length of the trajectory becomes infinitely long,
the probability is metrically invariant. This invariance property is realized for ergodic
webs that may, but need not, reach equilibrium in a finite time.
Physical theories describe the dynamics of microscopic and macroscopic webs that
conserve probability in the sense that once a probabilistic point of view is adopted the
total probability remains unchanged in time. Consequently, if the local probability in
phase space increases, it must decrease somewhere else in phase space. Recall that the
phase space has all the dynamical variables describing the complex web as independent
axes. In classical mechanics, if one selects a restricted volume in the phase space and
uses each point in this volume as an initial condition for the equations of motion, then
the evolution of the volume may change its geometry, but not the magnitude of the
volume itself. Equilibrium, however, means that the volume itself does not change. You
may recall that this was the basis of Boltzmann's definition of entropy (
1.27
). So how
is this invariance property compatible with ergodicity? How is all of the available phase
space explored as the web evolves?
To achieve equilibrium the volume of initial states develops twisted whorls and long
slender tendrils that can become dense in the available phase space. Like the swirls of
cream in your morning coffee, these tendrils of probability interlace the phase space,
making the coffee uniformly tan. This property is called mixing and refers to ther-
modynamically closed networks. Not all dynamical webs exhibit mixing, but mixing
is required in order for the dynamics of a closed web to reach equilibrium, starting
from a non-equilibrium initial condition. The notion of a closed system or network
was convenient in physics in order to retain a Hamiltonian description of an isolated
mechanical system such as a harmonic oscillator or more elaborate nonlinear dynam-
ical webs. Such a dynamic web could be opened up and allowed to interact with the
environment. If the interaction satisfies certain energy-balance conditions the compos-
ite system is considered to be thermodynamically closed. However, this is not the case
in general. A web that is open to the environment may receive energy and not nec-
essarily give energy back, thereby becoming unstable. These concepts of open and
closed, linear and nonlinear, dynamical webs are subtle and are discussed in subsequent
chapters.
Let us denote the random dynamic variable by
Q
(
t
)
and its statistics by the probability
density function
p
,
so that
q
is the corresponding phase-space variable. Conse-
quently, the average of the random variable is determined by integrating over the range
of the variate
R
,
(
q
)
=
Q
qp
(
q
)
dq
,
(2.75)
R
