Nonequilibrium Time Reversibility with Maps and Walks

Time-reversible simulations of nonequilibrium systems exemplify both Loschmidt's and Zermelo's paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the {\it irreversible} Second Law of Thermodynamics (Loschmidt's) as well as {\it periodic} in the time (Zermelo's, illustrating Poincare recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying one of the simplest such model systems, the time-reversible, but nevertheless dissipative and periodic, piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler one-dimensional random walk confined to the unit interval.