Emerging quasi-0D states at vanishing total entropy of the 1D hard sphere system: a coarse-grained similarity to the car parking problem

A coarse-grained system of one-dimensional (1D) hard spheres (HSs) is created using the Delaunay tessellation, which enables one to define the quasi-0D state. It is found from comparing the quasi-0D and 1D free energy densities that a frozen state due to the emergence of quasi-0D HSs is thermodynamically more favorable than fluidity with a large-scale heterogeneity above crossover volume fraction of ${\varphi }_c=e\mathrm{/}(1+e)=0.731\cdots$, at which the total entropy of the 1D state vanishes. The Delaunay-based lattice mapping further provides a similarity between the dense HS system above ${\varphi }_c$ and the jamming limit in the car parking problem.