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

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Martiniani, Stefano  ORCID logo
Schrenk, K Julian 
Stevenson, Jacob D 
Wales, David J 


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.



cond-mat.stat-mech, cond-mat.stat-mech, cond-mat.dis-nn, cond-mat.soft, physics.comp-ph

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Phys Rev E

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American Physical Society (APS)
Engineering and Physical Sciences Research Council (EP/I000844/1)
Engineering and Physical Sciences Research Council (EP/I001352/1)
Engineering and Physical Sciences Research Council (EP/N035003/1)
European Commission (275544)
We acknowledge useful discussions with Daniel Asenjo, Carl Goodrich, Silke Henkes, and Fabien Paillusson. S.M. acknowledges financial support by the Gates Cambridge Scholarship. K.J.S. acknowledges support by the Swiss National Science Foundation under Grant No. P2EZP2-152188 and No. P300P2-161078. J.D.S. acknowledges support by Marie Curie Grant 275544. D.F. and D.J.W. acknowledge support by EPSRC Programme Grant EP/I001352/1, by EPSRC grant EP/I000844/1 (D.F.) and ERC Advanced Grant RG59508 (D.J.W.)