content_image/341735.jpg content_image/341734.jpg content_image/341731.jpg content_image/341733.jpg content_image/341730.jpg content_image/341729.jpg
that the entropy S (ln( 〈 φ 〉 dyn − φ$_{c}$ ) , ln 〈 p 〉 $_{dyn}$) is maximum along the direction of ( − sin θ ,cos θ ) and the slope of the entropy along this direction ( − sin θ ,cos θ ) is 0 (see Fig. 13), that is, 

<img src='content_image/341735.jpg'>

Thus we verify the second law of thermodynamics for jammed systems: δS = 0 at (ln( 〈 φ 〉 dyn − φ$_{c}$ ) , ln 〈 p 〉 $_{dyn}$). 

By the definition of angoricity A = ∂ Γ /∂S and compactivity X = ∂V /∂S , we have: 

<img src='content_image/341734.jpg'>

∂S ∂ ln( φ − φ$_{c}$ ) = ( φ − φ$_{c}$ ) ∂S ∂φ = ( φ − φ$_{c}$ ) ∂V ∂φ 1 X = − ( φ − φ$_{c}$ ) NV$_{g}$ φ 2 1 X = − c$_{2}$ X , (23) where φ = NV$_{g}$/V , c$_{1}$ = Γ and c$_{2}$ = ( φ − φ$_{c}$ )( NV$_{g}$/φ $^{2}$). By Eq. (22) and Eq. (23), we can simplify Eq. (21): 

The relation between X and A can be obtained then: 

<img src='content_image/341731.jpg'>

<img src='content_image/341733.jpg'>

Since we obtain the angoricity A ∝ ( φ − φ$_{c}$ ) γ with γ = 2 . 5 in the main text and the pressure p ∝ ( φ − φ$_{c}$ ) a with a = 1 . 5 (actually we get 1.65 for the small system size used in the main text but the difference can be neglected to simplify). The compactivity X ∝ − ( φ − φ$_{c}$ ) $^{2}$/φ . We can therefore define the “jamming temperature” T$_{J}$ as a function of the slope along the direction (cos θ ,sin θ ): 

<img src='content_image/341730.jpg'>

That is: 

<img src='content_image/341729.jpg'>

Furthermore, the “jamming energy” E$_{J}$ , corresponding to the “jamming temperature” T$_{J}$ 

