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, 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: ∂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: That is: Furthermore, the “jamming energy” E$_{J}$ , corresponding to the “jamming temperature” T$_{J}$ 13 T$_{J}$ = A sin θ c$_{1}$ = − X cos θ c$_{2}$ = sin θ Γ A = a √$_{1 +}$ a 2 A Γ ∼ ( φ − φ$_{c}$ ) γ − a ∼ ( φ − φ$_{c}$ ) . (27) 1 T$_{J}$ = c$_{1}$ A sin θ − c$_{2}$ X cos θ = cos θ ( a c$_{1}$ A − c$_{2}$ X ) = c$_{1}$ A sin θ = − c$_{2}$ X cos θ . (26) X = − a c$_{2}$ c$_{1}$ A = − a φ − φ$_{c}$ pφ A. (25) 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 ): c$_{1}$ A + a c$_{2}$ X = 0 . (24) ∂S ∂ ln p = p ∂S ∂p = Γ ∂S ∂ Γ = Γ A = c$_{1}$ A , (22) ∂S ∂ ln( φ − φ$_{c}$ ) sin θ = ∂S ∂ ln p cos θ. (21)