perature is a general property of systems with bounded energy like spins [27]: the system attains the larger volume (or magnetization in spins) at φ$_{c}$ when X → 0 − and not X → + ∞ [The bounds φ$_{c}$ ≤ φ ≤ 1 imply that the jamming point at X → 0 − is “hotter” than X → + ∞ . At the same time A → 0 + since the pressure vanishes]. We conclude that, A and X alone cannot play the role of temperature. Instead, there is an actual “jamming temperature” T$_{J}$ that determines the direction ( − sin θ, cos θ ) in the log − log plot of Fig. 4 along the jamming equation of state (see Fig. 13). By maximizing the entropy along this direction we obtain T$_{J}$ as a function of A and X (see Materials and Methods Section III E): T$_{J}$ = sin θ A Γ = a √$_{1 +}$$_{a}$ 2 A Γ ∼ ( φ − φ$_{c}$ ) γ − $^{a}$. (10) By the definition of “heat” capacity, we obtain two jamming capacities as the response to changes in A and X : C$_{Γ}$ ≡ ∂ Γ /∂A ∼ ( φ − φ$_{c}$ ) − 1 ∼ A − 2 / $^{5}$, and C$_{V}$ ≡ ∂V/∂X ∼ ( φ − φ$_{c}$ ) − 1 ∼ | X | − 1 / $^{2}$. (11) From Eq. (11), the jamming capacities diverge at the jamming transition as A → 0 + and X → 0 $^{−}$. However, this result does not imply that the transition is critical since from Einstein fluctuation theory applied to pressure and volume [27] we obtain (we consider k$_{B}$ = 1 for simplicity): 〈 (ΔΓ) $^{2}$〉 = A $^{2}$C$_{Γ}$ ∼ A 1 . $^{6}$, and 〈 (Δ V ) $^{2}$〉 = X $^{2}$C$_{V}$ ∼ | X | 1 . $^{5}$. (12) Thus, the pressure and volume fluctuations near the jamming transition do not diverge, but instead vanish as A → 0 + and X → 0 $^{−}$. From a thermodynamical point of view, the transition is not of second order due to the lack of critical fluctuations. As a consequence, no diverging static correlation length can be found at the jamming point during isotropic compression. However, other correlation lengths of dynamic origin may still exist in the response of the jammed system to perturbations, such as those imposed by a shear strain or in vibrating modes [7, 28]. Such a dynamic correlation length would not appear in a purely thermodynamic static treatment as developed here. We note though that responses to shear can be treated in the present formalism by allowing the inverse angoricity to be tensorial [10]. The intensive jamming temperature Eq. (10) gives use to a jamming effective energy E$_{J}$ as the extensive variable satisfying T$_{J}$ = ∂E$_{J}$/∂S and a full jamming capacity 7