ilya_prigogine_isabelle_stengers_alvin_tofflerbookfi-org

265 IRREVERSIBILITY-THE ENTROPY BARRIER
that at the end of the nineteenth century Bruns and Poincare
demonstrated that most dynamic systems, starting with the
famous "three body" problem, were not integrable.
On the other hand, the very idea of approaching equilibrium
in terms of the theory of ensembles requires that we go beyond
the idealization of integrable systems. As we saw in Chapter
VII I, according to the theory of ensembles, an isolated system
is in equilibrium when it is represented by a "microcanonical
ensemble," when all points on the surface of given energy
have the same probability. This means that for a system to
evolve to equilibrium, energy must be the only quantity conserved
during its evolution. It must be the only "invariant."
Whatever the initial conditions, the evolution of the system
must allow it to reach all points on the surface of given energy.
But for an integrable system, energy is far from being the only
invariant. In fact, there are as many invariants as degrees of
freedom, since each generalized momentum remains constant.
Therefore we have to expect that such a system is "imprisoned"
in a very small "fraction" of the constant-energy (see
Figure 32) surface formed by the intersection of all these invar­
-iant surfaces.
p
Figure 32. Temporal evolution of a cell in phase space p, q. The "volume"
of the cell and its form are maintained in time; moreover, most of the phase
space is inaccessible to the system.
q