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

TL;DR

This paper introduces a coarse-grained model of 1D hard spheres using Delaunay tessellation, revealing a thermodynamically favored frozen state with quasi-0D particles above a critical density, and draws parallels to the car parking problem.

Contribution

It develops a Delaunay-based coarse-grained approach to identify quasi-0D states and links the dense hard sphere system to the car parking problem at high densities.

Findings

Frozen quasi-0D states are thermodynamically favored above $oldsymbol{\phi_c}$.

Total entropy of the 1D system vanishes at $oldsymbol{\phi_c}$.

Similarity between dense HS system and the car parking problem at high density.

Abstract

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.