It was long known that dissipative time-irreversible equations of motion (like
dx/dt = -x) lead to phase-space objects of reduced dimensionality.
The realization that strange attractors (reduced dimensionality combined
with Lyapunov instability) result from time-reversible equations of motion
became clear from phase-space distributions for the Galton Board problem,
as discussed in a well-known, but seldom-seen, paper by Bill Moran,
William Hoover, and Stronzo Bestiale
[Journal of Statistical Physics 48, 709 (1987)],
which led to many more
investigations. See for instance our work in Chaos 2, 599 (1992) and
[Physical Review A 40, 5319 (1989)]
After 15 years of exploration the fractal
objects have been closely connected to the older ways of describing and
understanding nonequilibrium systems. This work has shown very clearly
that attractor states (which obey the Second Law of Thermodynamics), though
Lyapunov unstable, are much more stable than the reversed-velocity repellor
states (violating the Law), which are completely unobservable. See
the discussions in my 2001 World Scientific book on Time Reversibility,
Computer Simulation, and Chaos. Particularly clear demonstrations of the
loss of phase-space dimensionality in strange attractors follow from simulations
of heat flow in tethered harmonic lattices; for these see particularly the two
papers on "Dimensionality Loss" listed in the Recent Publications on the
Main
Page. Despite considerable progress
in devising computational recipes for nonequilibrium simulations, it appears
that theoretical approaches have very little to contribute in this
area. It is also clear that the description of "real experiments"
with classical forcelaw models is seldom worthwhile, though much has been
learned by studying the correspondence between the microscopic and
macroscopic descriptions of material behavior.
Time-Reversible Molecular Motion and Macroscopic Irreversibility (with
Harald Posch and Brad Holian) [Berichte Bunsengesellschaft 94, 250 (1990)]

Christoph Dellago
and I rediscovered, and then extended, some extremely interesting results
from the efforts of Beck, Grebogi, Kruskal, Ott, Yorke, et alii. By using five-digit
finite-precision arithmetic we had stumbled upon an 11,951-collision periodic orbit
for the Galton Board problem. (See the illustration on page 134 of my Time
Reversibility, Computer Simulation, and Chaos book.) The lengths of such
periodic orbits (of which there are only a few) can be estimated quite nicely
in terms of the fractal correlation dimension of the underlying strange
attractor.
[Physical Review E 62, 6275 (2000)].

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