Stefano Martiniani

Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming

A decades-old proposal that all distinct packings are equally probable in granular media has gone unproven due to the sheer number of packings involved. Numerical simulation now demonstrates that it holds — precisely at the jamming threshold. In the late 1980s, Sam Edwards proposed a possible statistical-mechanical framework to describe the properties of disordered granular materials1. A key assumption underlying the theory was that all jammed packings are equally likely. In the intervening years it has never been possible to test this bold hypothesis directly. Here we present simulations that provide direct evidence that at the unjamming point, all packings of soft repulsive particles are equally likely, even though generically, jammed packings are not. Typically, jammed granular systems are observed precisely at the unjamming point since grains are not very compressible. Our results therefore support Edwards’ original conjecture. We also present evidence that at unjamming the configurational entropy of the system is maximal.

Superposition Enhanced Nested Sampling

The theoretical analysis of many problems in physics, astronomy and applied mathematics requires an efficient numerical exploration of multimodal parameter spaces that exhibit broken ergodicity. Monte Carlo methods are widely used to deal with these classes of problems, but such simulations suffer from a ubiquitous sampling problem: the probability of sampling a particular state is proportional to its entropic weight. Devising an algorithm capable of sampling efficiently the full phase space is a long-standing problem. Here we report a new hybrid method for the exploration of multimodal parameter spaces exhibiting broken ergodicity. Superposition enhanced nested sampling (SENS) combines the strengths of global optimization with the unbiased/athermal sampling of nested sampling, greatly enhancing its efficiency with no additional parameters. We report extensive tests of this new approach for atomic clusters that are known to have energy landscapes for which conventional sampling schemes suffer from broken ergodicity. We also introduce a novel parallelization algorithm for nested sampling.

Structural analysis of high-dimensional basins of attraction

We propose an efficient Monte Carlo method for the computation of the volumes of high-dimensional bodies with arbitrary shape. We start with a region of known volume within the interior of the manifold and then use the multistate Bennett acceptance-ratio method to compute the dimensionless free-energy difference between a series of equilibrium simulations performed within this object. The method produces results that are in excellent agreement with thermodynamic integration, as well as a direct estimate of the associated statistical uncertainties. The histogram method also allows us to directly obtain an estimate of the interior radial probability density profile, thus yielding useful insight into the structural properties of such a high-dimensional body. We illustrate the method by analyzing the effect of structural disorder on the basins of attraction of mechanically stable packings of soft repulsive spheres.

Exploiting the potential energy landscape to sample free energy

We review a number of recently developed strategies for enhanced sampling of complex systems based on knowledge of the potential energy landscape. We describe four approaches, replica exchange, Kirkwood sampling, superposition‐enhanced nested sampling, and basin sampling, and show how each of them can exploit information for low‐lying potential energy minima obtained using basin‐hopping global optimization. Characterizing these minima is generally much faster than equilibrium thermodynamic sampling, because large steps in configuration space between local minima can be used without concern for maintaining detailed balance. WIREs Comput Mol Sci 2015, 5:273–289. doi: 10.1002/wcms.1217

Monte Carlo sampling for stochastic weight functions

Significance Markov chain Monte Carlo is the method of choice for sampling high-dimensional (parameter) spaces. The method requires knowledge of the weight function (or likelihood function) determining the probability that a state is observed. However, in many numerical applications the weight function itself is fluctuating. Here, we present an approach capable of tackling this class of problems by rigorously sampling states proportionally to the average value of their fluctuating likelihood. We demonstrate that the method is capable of computing the volume of a basin of attraction defined by stochastic dynamics as well as being an efficient method to identify a transition state along a known reaction coordinate. We briefly discuss how the method might be extended to experimental settings. Conventional Monte Carlo simulations are stochastic in the sense that the acceptance of a trial move is decided by comparing a computed acceptance probability with a random number, uniformly distributed between 0 and 1. Here, we consider the case that the weight determining the acceptance probability itself is fluctuating. This situation is common in many numerical studies. We show that it is possible to construct a rigorous Monte Carlo algorithm that visits points in state space with a probability proportional to their average weight. The same approach may have applications for certain classes of high-throughput experiments and the analysis of noisy datasets.