Edwards Statistical Mechanics of Granular Matter

Granular materials are particulate systems composed of massive macro-scale particles, which are by nature out of equilibrium and dissipative. They exhibit rich rheological properties and are closely related with many challenging engineering problems. While current research primarily relies on macroscopic empirical laws, a microscopically-based theory of granular materials needs a statistical mechanical framework to build the connections between the micro and the macro. In the 1990s, Edwards and collaborators introduced a statistical mechanics framework for jammed granular packings, where volume is postulated to be the quantity equivalent to energy in thermal systems. They subsequently introduced thermodynamic parameters such as compactivity and entropy. Despite extensive research, several issues remain unresolved about the Edwards ensemble, including the influence of friction, the validity of the equal probability hypothesis, and the connection with disordered materials, which warrants further investigation. Using X-ray tomography, we systematically investigated both tapped and sheared granular systems, we validate the ergodicity hypothesis of the Edwards ensemble by analyzing the volume fluctuations under the Edwards canonical ensemble, which is further supported by the experimentally observed equivalence between the Edwards compactivity and the effective temperature based on fluctuation-dissipation relation. Through microscopic structural analysis of sheared granular packings, we provide a statistical mechanical explanation for the critical state of granular solids, elucidating the respective contributions of friction and disorder to granular entropy. Additionally, by adopting the Edwards statistical framework, we can introduce an Edwards-type free energy, which offers a theoretical explanation for the segregation phenomenon observed in granular materials.