Thermodynamic analysis of the jamming transition.— So far we have considered how the angoricity determines the pressure fluctuations in a jammed packing at a fixed φ . The role of the compactivity in the jamming transition can be analyzed in terms of the entropy which is easily calculated in the microcanonical ensemble from the density of states. Figure 4 shows S = ln(Ω( p, φ )) (Ω is the number of states which is the unnormalized version of g (Γ , φ )), which is the non-equilibrium entropy of the system at the given ( p, φ ) in phase space. We analyze the non-equilibrium entropy surface S (ln( φ − φ$_{c}$ ) , ln p ) plotted versus (ln( φ − φ$_{c}$ ) , ln p ) in Fig. 4 and demonstrate that the MD curve 〈 p ( φ ) 〉 dyn passes along the maximum of the entropy surface constrained by the coupling between p and φ , Eq. (3). Thus the points along the entropy surface defined by 〈 p ( φ ) 〉 correspond to the equilibrium entropy; such a curve is superimposed to the entropy surface in Fig. 4. Due to the coupling through the contact force law, the maximization of entropy is not on p or φ alone but on a combination of both. The entropy S reaches a maximum at the point S (ln( 〈 φ 〉 dyn − φ$_{c}$ ) , ln 〈 p 〉 $_{dyn}$) when we move along the direction perpendicular to the jamming curve 〈 p ( φ ) 〉 dyn (see the maximization direction in Fig. 4). This is a direct verification of the second-law of thermodynamics: the dynamical measures maximize the entropy of the system. We can use this result to obtain a relation between angoricity and compactivity. We write ln p = ln p$_{0}$ + a ln( φ − φ$_{c}$ ) where a is the exponent in Eq. (3), such that S (ln( φ − φ$_{c}$ ) , ln p ) is maximized at the MD measures according to the direction of ( − sin θ, cos θ ) (tan θ = a is the slope of the power-law curve in the log − log plot in Fig. 4). Therefore, neither A nor X can play the role of the temperature of the system alone, but a combination of both determined by entropy maximization satisfying the coupling between stress and strain. Since δS = 0 at (ln( 〈 φ 〉 dyn − φ$_{c}$ ) , ln 〈 p 〉 $_{dyn}$) along ( − sin θ, cos θ ) then ( ∂S/∂ ln p ) cos θ − ( ∂S/∂ ln( φ − φ$_{c}$ )) sin θ = 0. We obtain c$_{1}$α + ac$_{2}$β = 0 (where c$_{1}$ = Γ and c$_{2}$ = ( φ − φ$_{c}$ )( NV$_{g}$/φ $^{2}$)) and the relation between A and X (see Fig. 13 and Materials and Methods Section III E): From Eq. (8) we obtain that: X ∝ − ( φ − φ$_{c}$ ) 1+ a − $^{γ}$/φ and near φ$_{c}$ : We notice that the compactivity is negative near the jamming transition. A negative tem- 6 X ∼ − ( φ − φ$_{c}$ ) $^{2}$. (9) X = − aA ( φ − φ$_{c}$ ) /pφ. (8)