Structural signature of jamming transition

In thermal amorphous systems, the first peak of the pair correlation function $g(r)$ reaches the maximum height $g_1^ {\rm max}$ at a crossover volume fraction $\phi_v$ when the volume fraction $\phi$ is varied. In the $T=0$ limit, $\phi_v$ approaches $\phi_c$, the critical volume fraction of the $T=0$ jamming transition, accompanied by a diverging $g_1^{\rm max} $. The occurrence of $g_1^{\rm max}$ at $T>0$ thus reminisces the $T=0$ jamming transition. By measuring typical quantities such as the pressure, bulk modulus, shear modulus, and characteristic frequency of the boson peak, which all show power law scalings with $\phi-\phi_c$ in marginally jammed solids at $T=0$, we observe that $\phi=\phi_v$ separates the thermal amorphous systems into two regimes with distinct material properties: these quantities show similar power law scalings with $\phi-\phi_c$ to marginally jammed solids when $\phi>\phi_v$, which break down when $\phi<\phi_v$. Therefore, the occurrence of $g_1^{\rm max}$ signifies the jamming transition at $T>0$. Because the scalings are manipulated by $\phi_c$, the $T=0$ jamming transition should be the only critical point that controls the jamming transition and properties of jammed solids at $T>0$.