The paper "Is Random Close Packing of Spheres Well Defined?" by S. Torquato, T. M. Truskett, and P. G. Debenedetti addresses the fundamental issue of defining random close packing (RCP) of spheres. The authors argue that the current understanding of RCP is not mathematically precise and is influenced by the specific protocols used to achieve random packings. They support this claim through molecular dynamics simulations of hard spheres using the Lubachevsky-Stillinger compression algorithm. The paper highlights that the RCP state is ill-defined and dependent on the system's characteristics, such as the pouring rate and vibration amplitude in experiments, or the choice of algorithms in simulations. The authors propose a new concept called the "maximally random jammed (MRJ) state," which is defined as the state that minimizes an order parameter among all statistically homogeneous and isotropic jammed structures. To support their arguments, the authors use molecular dynamics simulations to show that the MRJ packing fraction for 500 identical hard spheres is approximately 0.64, close to the commonly reported RCP value of 0.636. They also introduce measures of translational and bond-orientational order to quantify the randomness in jammed structures. The results demonstrate that the notion of RCP as the highest possible density is ill-defined, as packings with arbitrarily small increases in volume fraction can be achieved at the expense of small increases in order. The paper concludes by emphasizing the need for new and efficient protocols to generate jammed states and for further research into better order parameters to study randomness in dense sphere packings.The paper "Is Random Close Packing of Spheres Well Defined?" by S. Torquato, T. M. Truskett, and P. G. Debenedetti addresses the fundamental issue of defining random close packing (RCP) of spheres. The authors argue that the current understanding of RCP is not mathematically precise and is influenced by the specific protocols used to achieve random packings. They support this claim through molecular dynamics simulations of hard spheres using the Lubachevsky-Stillinger compression algorithm. The paper highlights that the RCP state is ill-defined and dependent on the system's characteristics, such as the pouring rate and vibration amplitude in experiments, or the choice of algorithms in simulations. The authors propose a new concept called the "maximally random jammed (MRJ) state," which is defined as the state that minimizes an order parameter among all statistically homogeneous and isotropic jammed structures. To support their arguments, the authors use molecular dynamics simulations to show that the MRJ packing fraction for 500 identical hard spheres is approximately 0.64, close to the commonly reported RCP value of 0.636. They also introduce measures of translational and bond-orientational order to quantify the randomness in jammed structures. The results demonstrate that the notion of RCP as the highest possible density is ill-defined, as packings with arbitrarily small increases in volume fraction can be achieved at the expense of small increases in order. The paper concludes by emphasizing the need for new and efficient protocols to generate jammed states and for further research into better order parameters to study randomness in dense sphere packings.