Abstract
Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt’s and Zermélo’s paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the irreversible Second Law of Thermodynamics (Loschmidt’s) as well as periodic in the time (Zermélo’s, illustrating Poincaré recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals’ information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here.
Highlights
An understanding of the shrinking phase volume, which leads to an apparent fractal phase-space object is straightforward in the Baker Map example of Figure 2
The two model systems reveal the details of the singular loss of phase volume resulting in these interesting fractal distributions
The maps have shown us that nonequilibrium states are rare, occupying only fractional dimensional portions of phase space
Summary
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