Molecular Dynamics, Fractal Phase-Space Distributions, the Cantor Set, and Puzzles Involving Information Dimensions for Two Compressible Baker Maps

Nonequilibrium molecular dynamics simulations generate ''fractal'' [ fractional-dimensional ] phase-space distributions. Their steady-state ''Information Dimensions'' are defined by $D_I(\epsilon) = \sum P(\epsilon) \ln P(\epsilon)/\ln (\epsilon)$ . The bin probabilities $\{ \ P(\epsilon) \ \}$ are computed using hypercube bins of sidelength $\epsilon$. Similarly, with one-dimensional bins, the simplest fractal distribution is the zero-measure 0.630930-dimensional Cantor dust -- that singular set of $2^{\aleph_0}$ points composed solely of ternary 0's and 2's. The Cantor dust can be constructed iteratively, by first removing the middle third from the unit interval, next removing the middle thirds of the two remaining segments, then the middle thirds of the remaining four segments, and so on. Because the resulting ordered ternary sets of 0's and 2's have the same cardinality as corresponding binary sets of 0's and 1's the ''02 dust'' and the binary ''01 continuum'' have the ''same size'' with identical cardinalities, $2^{\aleph_0} \equiv \aleph_1$. With two-dimensional bins we consider two similar linear Baker Mappings, N2$(q,p)$ and N3$(q,p)$. Their ''Information Dimensions'' $D_I$, are identical, as estimated from Cantor-like sequential mappings. But the estimates generated by pointwise iteration differ. Further, with N2 both these different dimensions disagree with the Kaplan-Yorke dimension $D_{KY}$ based on Lyapunov exponents and thought by most authorities to agree with $D_I$. The surprising differences among the three $\{ \ D_I \ \}$ estimates for N2 merit more investigation.