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 $\phi_c=e/(1+e)=0.731\cdots$, 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 $\phi_c$ and the jamming limit in the car parking problem. 
Hiroshi Frusawa
cond-mat.dis-nn cond-mat.mes-hall cond-mat.soft cond-mat.stat-mech