Theory of the Jamming Transition at Finite Temperature

A theory for the microscopic structure and the vibrational properties of soft sphere glass at finite temperature is presented. With an effective potential, derived here, the phase diagram and vibrational properties are worked out around the Maxwell critical point at zero temperature $T$ and pressure $p$. Variational arguments and effective medium theory identically predict a non-trivial temperature scale $T^*\sim p^{(2-a)/(1-a)}$ with $a \approx 0.17$ such that low-energy vibrational properties are hard-sphere like for $T \gtrsim T^*$, and zero-temperature soft-sphere like otherwise. However, due to crossovers in the equation of state relating $T$, $p$, and the packing fraction $\phi$, these two regimes lead to four regions where scaling behaviors differ when expressed in terms of $T$ and $\phi$. Scaling predictions are presented for the mean-squared displacement, characteristic frequency, shear modulus, and characteristic elastic length in all regions of the phase diagram.