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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