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

Abstract

We report a numerical calculation of the total number of disordered jammed configurations $\Omega$ 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.

18 pages, 7 figures

References in corpus (23)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Flux Compactification
• Determination of phase diagrams via computer simulation: Methodology and applications to water, electrolytes and proteins
• Stationary state volume fluctuations in a granular medium
• A deductive statistical mechanics approach for granular matter
• Entropy and Temperature of a Static Granular Assembly
• Topological and Dynamical Complexity of Random Neural Networks
• The Statistical Physics of Athermal Materials
• Understanding the Frequency Distribution of Mechanically Stable Disk Packings
• Cosmology From Random Multifield Potentials
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• The role of structure and entropy in determining differences in dynamics for glass formers with different interaction potentials
• Exploring a string-like landscape
• On the number of connected components of random algebraic hypersurfaces
• Multi-field Inflation with a Random Potential
• On the Unlikeliness of Multi-Field Inflation: Bounded Random Potentials and our Vacuum
• Random Polynomials and the Friendly Landscape
• Communication: Newton homotopies for sampling stationary points of potential energy landscapes
• On the Explicit Height Distribution and Expected Number of Local Maxima of Isotropic Gaussian Random Fields
• Devising a protocol-related statistical mechanics framework for granular materials
• An Inversion-Relaxation Approach for Sampling Stationary Points of Spin Model Hamiltonians
• Statistics of Stationary Points of Random Finite Polynomial Potentials
• Certified counting of roots of random univariate polynomials

Cited by in corpus (26)

• Temperature in and out of equilibrium: a review of concepts, tools and attempts
• Perspective: Energy Landscapes for Machine Learning
• Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming
• Does the configurational entropy of polydisperse particles exist?
• Predicting the phase diagram of titanium dioxide with random search and pattern recognition
• Geometry of energy landscapes and the optimizability of deep neural networks
• Equation of state of polydisperse hard-disk mixtures in the high-density regime
• Structural analysis of high-dimensional basins of attraction
• Monte Carlo sampling for stochastic weight functions
• Another resolution of the configurational entropy paradox as applied to hard spheres
• An exploratory study of the glassy landscape near jamming
• Dynamical computation of the density of states and Bayes factors using nonequilibrium importance sampling
• Information density, structure and entropy in equilibrium and non-equilibrium systems
• Complete mathematical theory of the jamming transition: A perspective
• Tuning and jamming reduced to their minima
• Absence of hyperuniformity in amorphous hard-sphere packings of nonvanishing complexity
• Attraction Controls the Inversion of Order by Disorder
• A local view on the role of friction and shape
• Exploring canyons in glassy energy landscapes using metadynamics
• Quantitative assessment of increasing complexity
• When you can't count, sample! Computable entropies beyond equilibrium from basin volumes
• Numerical method for computing the free energy of glasses
• Affine and topological structural entropies in granular statistical mechanics: explicit calculations and equation of state
• Geometric and Topological Entropies of Sphere Packing
• Basins with tentacles
• Rényi complexity in mean-field disordered systems