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

Abstract

We present a numerical calculation of the total number of disordered jammed configurations Ω 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)10.1103/PhysRevLett.106.245502] and Asenjo et al. [Phys. Rev. Lett. 112, 098002 (2014)10.1103/PhysRevLett.112.098002] 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.

Affine and topogical structural entropies in granular statistical mechanics: Explicit calculations and equation of state

“…Consequently, equation (16) substantiates the generality of previous results obtained in the high angoricity limit. Keeping only these two terms, we obtain…”

Absence of Hyperuniformity in Amorphous Hard-Sphere Packings of Nonvanishing Complexity

“…2 is plotted the right-hand side of Eq. (5). Notice that it only goes to zero, indicating the presence of perfect hyperuniformity, at two exceptional packing fractions, φ min and φ max ; the corresponding densities are ρ min = 2/(σ + δ) and ρ max = 1/δ.…”

“…We next describe a situation where one can calculate exactly both S(k → 0) and also Σ(ρ) for a set of jammed states and show that they are indeed related by Eq. (5). These are the jammed states in the system of disks in a narrow channel depicted in Fig.…”

“…Let us call the complexity Σ N,V , the logarithm of the number of jammed states of N spheres in a volume V . Numerical methods have been recently devised for determining Σ N,V [5]. When it is extensive, it is expected to be of the form Σ N,V = N Σ(ρ) as N → ∞ at fixed density ρ ≡ N/V .…”

Another resolution of the configurational entropy paradox as applied to hard spheres

“…This "colloidal" indistinguishability term is needed to prevent the Gibbs paradox and define entropy in a reasonable way (so that the entropy is extensive). [8][9][10] For hard spheres, U N ( r) = 0 if there are no intersections between particles and U N ( r) = ∞ otherwise.…”