In the last decade, significant progress has been made in the many-body Hamiltonian formulation of liquid dynamics theory. Earlier analysis of experimental data for a wide variety of elemental liquids had provided the reliable but qualitative description of the atomic motion: vibrations in a representative $3N$-dimensional potential energy valley, plus transits, in which atoms cross the intersections between these valleys. Recent comparisons of first-principles theory with experiment for several elemental liquids at melt revealed a highly accurate and versatile theory. In the present work, we report on an extensive quench study of the entire condensed-matter structure-energy distribution for a metallic Na MD system. With these results, all that was learned from experimental data is confirmed, refined in detail, and made more accurate. We show the entire structure-energy distribution, composed of widespread symmetrics and higher lying randoms, we show the increasing dominance of the randoms as $N$ increases, until the symmetrics vanish completely, and we show the random distribution continue its spectacular narrowing as $N$ continues to increase. This behavior certifies our early assignment of the random distribution to the liquid phase and our prediction of macroscopic uniformity of the random structures. Procedures are discussed to identify and calibrate a single random structure to represent the liquid, and the role of the structure energy in liquid thermodynamics is described. A comparison with other liquid dynamics theories is observed in the Introduction, and the relation to the equation-of-state program is noted in the Conclusion.

• Received 19 June 2018
• Revised 23 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.054201