Turning intractable counting into sampling: Computing the configurational entropy of three-dimensional jammed packings

We present a numerical calculation of the total number of disordered jammed configurations $\mathrm{\Omega }$ of $N$ repulsive, three-dimensional spheres in a fixed volume $V$. To make these calculations tractable, we increase the computational efficiency of the approach of Xu et al. [Phys. Rev. Lett. 106, 245502 (2011)] and Asenjo et al. [Phys. Rev. Lett. 112, 098002 (2014)] and we extend the method to allow computation of the configurational entropy as a function of pressure. The approach that we use computes the configurational entropy by sampling the absolute volume of basins of attraction of the stable packings in the potential energy landscape. We find a surprisingly strong correlation between the pressure of a configuration and the volume of its basin of attraction in the potential energy landscape. This relation is well described by a power law. Our methodology to compute the number of minima in the potential energy landscape should be applicable to a wide range of other enumeration problems in statistical physics, string theory, cosmology, and machine learning that aim to find the distribution of the extrema of a scalar cost function that depends on many degrees of freedom.

Figure 1

Entropy as a function of system size $N$ for two- (Ref. [34]) and three-dimensional (this work) jammed sphere packings. Dashed curves are lines of best fit of the form $S=aN$.

Figure 2

Hard sphere fluid at ${\varphi }_{\text{HS}}=0.5$, left, and HS-WCA jammed packing at ${\varphi }_{\text{SS}}=0.7$, right, for a system of 44 polydisperse hard spheres with mean radius $\langle r_h\rangle =1$ and standard deviation ${\sigma }_{\text{HS}}=0.05$. We prepare the polydisperse HS fluid configurations at fixed packing fraction ${\varphi }_{\mathrm{HS}}=0.5$ by a Monte Carlo simulation. Particles are then inflated by the same factor, proportional to their radius (spheres are coloured according to their radius), to obtain an overcompressed soft spheres jammed packing at ${\varphi }_{\mathrm{SS}}=0.7$ by an infinitely fast quench (energy minimization).

Figure 3

Top left: Entropy as a function of the system size $N$ computed, in order, according to the Gibbs configurational entropy and the Edwards configurational entropy using a nonparametric fit by kernel density estimation (KDE), a parametric fit to a generalized Gaussian c.d.f. using a nonlinear least-squares method, and a fit to the corresponding p.d.f. using maximum likelihood (ML). Comparison of generalized Gaussian best-fit parameters for 2D (see Ref. [34]) and 3D sphere packings: scale parameter $\sigma$ (bottom left) and mean log-volume $\mu$ (top right) scale linearly with system size $N$; distributions are more peaked for 2D packings. In 2D we observe much stronger dependence of the shape parameter $\zeta$ (bottom right) as a function of system size than in 3D.

Figure 4

Top: Dimensionless free energy versus pressure of mechanically stable states at fixed volume for several system sizes. Best-fit lines are in black. In the bottom left and right plots we show slope and intercept for each of the best fit lines as a function of system size. Both slope and intercept scale linearly with system size.

Figure 5

Empirical cumulative distribution functions of the pressures for several system sizes. Dashed lines in the corresponding color are curves of best fit to a generalized log-normal distribution. The curves are mostly indistinguishable. Inset: Best-fit parameters for the generalized log-normal distribution as a function of system size. The mean $\mu$ and scale parameter $\sigma$ scale linearly with $1/N$, while the shape parameter $\zeta$ is approximately insensitive with respect to system size.

Figure 6

Top: Generalized Edwards entropy at fixed volume fraction for various system sizes. The curves show a well-defined maximum for all sizes, while their shape depends on the specific parameters of the generalized log-normal that best fits the underlying distribution of pressures. Bottom left: Comparison between the Edwards entropy and the maximum value attained by each curve: $max\left[S_B(\mathcal{P},V)\right]$ scales linearly with size and its value is progressively closer to the marginal (total) Edwards entropy $S_B\left(V\right)$, consistent with the fact that $S_B(\mathcal{P},V)$ is a negative exponential function, and the area under the curve is dominated by the mode for increasing system size. $S_B\left(V\right)$ should constitute an upper limit to $max\left[S_B(\mathcal{P},V)\right]\le S_B\left(V\right)$ and the two should be equivalent only in the thermodynamic limit. Bottom right: Ensemble average of the pressure computed as a function of inverse angoricity $\alpha$ and system size. The curves, in the same color as the top figure, do not diverge and the arrows indicates their value at $\alpha =0$.

Figure 7

Average-squared displacement ${〈|\mathbf{r}-{\mathbf{r}}_0{|}^2〉}_k$ as a function of the spring constant $k$ (symbols). The dashed line shows the expression in Eq. (B8). The data are measured for a packing of $N=32$ spheres, with ${\varphi }_{\text{HS}}=0.5$ and ${\varphi }_{\text{SS}}=0.7$ via Hamiltonian PT. Inset: Corresponding integrand for the thermodynamic integration, resulting from the change of variables in Eq. (B13).