Emergent interparticle interactions in thermal amorphous solids

Amorphous media at finite temperatures, be them liquids, colloids, or glasses, are made of interacting particles that move chaotically due to thermal energy, continuously colliding and scattering off each other. When the average configuration in these systems relaxes only at long times, one can introduce effective interactions that keep the mean positions in mechanical equilibrium. We introduce a framework to determine the effective force laws that define an effective Hessian that can be employed to discuss stability properties and the density of states of the amorphous system. We exemplify the approach with a thermal glass of hard spheres; these experience zero forces when not in contact and infinite forces when they touch. Close to jamming we recapture the effective interactions that at temperature $T$ depend on the gap $h$ between spheres as $T/h$ [C. Brito and M. Wyart, Europhys. Lett. 76, 149 (2006)]. For hard spheres at lower densities or for systems whose binary bare interactions are longer ranged (at any density), the emergent force laws include ternary, quaternary, and generally higher-order many-body terms, leading to a temperature-dependent effective Hessian.

Figure 1

Shown on top is a comparison of the computed effective forces for $\epsilon ={10}^{-4}$. The blue line shows the results obtained from the present algorithm, i.e., $T/h_{ij}$. The red dots show the results of estimating the forces from the direct momentum transfer method. For small values of $h_{ij}$ the agreement is excellent; it deteriorates at higher values. The black dashed line is the expected result $C/h_{ij}$, with $C>T$ to allow comparison. The bottom panel shows the same data plotted after excluding any particle pair that collided less than 100 times during the measurement period and the corresponding measured forces.

Figure 2

Comparison of the measured forces (red dots) and the best fit to binary forces (dashed black line), for $\epsilon ={10}^{-2}$. The measured forces are no longer a graph of $h_{ij}$ since they reflect the existence of multiple-body interactions. The dashed line is a graph by construction, but obviously it does not represent the data well (see also Fig. 3).

Figure 3

Net forces on each particle, rescaled by $T/\langle h_{ij}\rangle$. We have reordered the indices of the particles according to the increasing magnitude of the net force. Black circles show the net forces computed from the best binary approximation. Blue squares represent the net forces computed with binary and some ternary contributions. The addition of even a limited number of ternary interactions improves the approximation.