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## I. RESULTS 

In general, if the density of states g (Γ , φ ) in the space of jammed configurations (defined as the probability of finding a jammed state at a given (Γ , φ ) at A = ∞ ) is known, then calculations of macroscopic observables, like pressure p and average coordination number Z as a function of φ , can be performed by the canonical ensemble average [9, 10] at a given volume: ∞ 

and 

where the canonical partition function is = ∫ ∞ 0 g (Γ , φ ) e − α $^{Γ}$dΓ and the density of states is normalized as ∫ ∞ 0 g (Γ , φ )dΓ = 1. The inverse angoricity is defined as α ≡ 1 /A = ∂S/∂ Γ. At the jamming transition the system reaches isostatic equilibrium, such that the stresses 

<img src='content_image/343069.jpg'>

<img src='content_image/343068.jpg'>

are exactly balanced in the resulting configuration, and there exists a unique solution to the interparticle force equations satisfying mechanical equilibrium. It is well known that observables present power-law scaling [4, 5, 7]: 

<img src='content_image/343067.jpg'>

where a = 3 / 2 and b = 1 / 2 for Hertzian spheres and Z$_{c}$ = 6 is the coordination number at the isostatic point (J-point) [4]. The average 〈· · · 〉 dyn indicates that these quantities are obtained by averaging over packings generated dynamically in either simulations or experiments as opposed to the ensemble average over configurations 〈· · · 〉 ens of Eqs. (1)–(2). Comparing the ensemble calculations, Eq. (1)–(2), with the direct dynamical measurements, Eq. (3), provides a basic test of the ergodic hypothesis for the statistical ensemble. 

Our approach is the following: We first perform an exhaustive enumeration of configura- tions to calculate g (Γ , φ ) and obtain 〈 p ( α, φ ) 〉 ens as a function of α for a given φ using Eq. (1). Then, we obtain the angoricity by comparing the pressure in the ensemble average with the one obtained following the dynamical evolution with Molecular Dynamics (MD) simula- tions. By setting 〈 p ( α, φ ) 〉 ens = 〈 p 〉 $_{dyn}$, we obtain the angoricity as a function of φ . By virtue of obtaining α ( φ ), all the other observables can be calculated in the ensemble formulation. The ultimate test of ergodicity is realized by comparing the remaining ensemble observables with the corresponding direct dynamical measures. 

