Intrinsic density of states and the origin of Gamma volume distributions in jammed granular matter.

Athermal systems such as granular materials exhibit reproducible volume fluctuations despite the absence of thermal equilibration, yet a quantitative statistical-mechanical description remains incomplete. Within the Edwards ensemble, volume replaces energy as the conserved quantity and compactivity provides Boltzmann-like weighting, but the form of the intrinsic configurational density of states and the origin of experimentally observed Gamma-like volume distributions have remained unresolved. Here, we show analytically and experimentally that the intrinsic density of states of jammed granular packings follows a Gamma distribution when expressed in terms of free volume. This shows that the widely observed Gamma form originates from the underlying structural statistics of mechanically stable configurations, rather than from the measured distribution itself. Incorporating this density of states into the Edwards ensemble yields a factorized description in which structure and thermodynamic weighting are explicitly separable. The framework provides closed-form predictions for the partition function, mean volume, fluctuations, compactivity, and entropy without fitting probability distributions. Large-scale x-ray tomography experiments confirm these predictions and demonstrate thermodynamic consistency with independent fluctuation- and histogram-based measurements. These results resolve the interpretation of Gamma volume distributions and establish a quantitative statistical-mechanical foundation for jammed granular matter.