Random Very Loose Packings

We measure the number $\Omega (\varphi )$ of mechanically stable states of volume fraction $\varphi$ of a granular assembly under gravity. The granular entropy $S(\varphi )=\log \Omega (\varphi )$ vanishes both at high density, at $\varphi \simeq {\varphi }_{\mathrm{rcp}}$, and a low density, at $\varphi \simeq {\varphi }_{\mathrm{rvlp}}$, where ${\varphi }_{\mathrm{rvlp}}$ is a new lower bound we call random very loose pack. ${\varphi }_{\mathrm{rlp}}$ is the volume fraction where the entropy is maximal. These findings allow for a clear explanation of compaction experiments and provide the first first-principle definition of the random loose volume fraction. In the context of the statistical mechanics approach to static granular materials, states with $\varphi <{\varphi }_{\mathrm{rlp}}$ are characterized by a negative temperature.

Figure 1

(a) Number of mechanically stable configurations of $N=20$ disks on a lattice and on the continuous for different values of the interparticle friction coefficient. (b) The volume fraction where ${\Omega }_{\mu }(\varphi )$ is maximal approaches the estimated value of ${\varphi }_{\mathrm{rlp}}$ as the friction coefficient increases.

Figure 2

Schematic landscape of the energy as a function of the configurational coordinates. The red (light gray) disk represents a random configuration, used as the starting point of energy minimization procedures to find mechanically stable states (energy minima). The blue (dark gray) disks represent the “almost stable” initial configurations we use as starting points of our minimization procedure (see text).

Figure 3

(a) The main panel [inset] shows the probability $P({\varphi }_i\to {\varphi }_f)$ [$P({\varphi }_f>\varphi |{\varphi }_i)$] that a sedimentation procedure, which starts from an unstable state configuration of volume fraction ${\varphi }_i$, ends in a stable state of volume fraction ${\varphi }_f$ (greater than $\varphi$). (b) Evolution of the volume fraction in a sequence of taps. (c) The volume fraction reached at stationary in a tapping dynamics as a function of the intensity of the tap $\delta \varphi$. (d) Equation of state in the statistical mechanics approach to granular media.