The discontinuous behavior demonstrated here may prove useful in investigating the hypothesized phase transition of Kirkwood [87] for a hard sphere gas. The Kirkwood transition hypothetically occurs for a density smaller than the close-packing density, while here the phase transition occurs for two densities of bits smaller than the maximum bit density of one. Because of the hard-sphere potential, the system analyzed there is independent of temperature, while the system described here is de-facto independent of temperature. For the hard-sphere gas it is known that, if a phase transition does occur at a density less than the close-packed density, the dimension must be greater than one. Here the dimension is effectively infinite because only the number of bit locations is considered, not the spatial organization of these locations. Finally, to complete the analogy to the hard sphere gas, let the bits interact with a hard sphere potential: the energy of any configuration of bits where no two bits are at the same location may be considered zero independent of the configuration of the occupied locations, while the energy of any configuration of bits some two of which occupy the same location may be taken as infinite, thus disallowing such configurations at any temperature. That is, the bits interact with a pairwise potential that is zero for bits at different locations, and infinite for bits at the same location. With this interaction potential the bit string system becomes temperature independent when considered as a thermodynamic system. Thus the infinite bit string system is a lattice hard-sphere gas on the infinite simplex. Before taking this too seriously though, note that the hard sphere gas exists in a continuous space, and that all these discontinuities indicate for now is that fractional bits do not occur. At the very least, the analogy to and the exactly solvable nature of the bit system may afford an avenue toward insightful work on the hypothesized Kirkwood phase transition in the hard-sphere gas. For this reason, the plots are given for continuous ranges of q and N values, showing where interesting behavior may be found/utilized in some other investigation. Like the operation sqrt on the integers and the subsequent introduction of the algebraic numbers (like sqrt(2)), perhaps there is a sensible interpretation of these fractional bit objects.
A plot of the correction at the maximal absolute correction (q=1/N) as a function of string length is given in figure 4.1. A plot of the correction as a function of bit density for a string length of N=10 is given in figures 4.2, and again magnified for the second order correction in the region of the lower phase transition in figure 4.3. A plot of the derivative of the correction (one sided) at the lower phase transition point and N=10 is given in figure 4.4.