Diffusive Nested Sampling

We introduce a general Monte Carlo method based on Nested Sampling (NS), for sampling complex probability distributions and estimating the normalising constant. The method uses one or more particles, which explore a mixture of nested probability distributions, each successive distribution occupying ~e^-1 times the enclosed prior mass of the previous distribution. While NS technically requires independent generation of particles, Markov Chain Monte Carlo (MCMC) exploration fits naturally into this technique. We illustrate the new method on a test problem and find that it can achieve four times the accuracy of classic MCMC-based Nested Sampling, for the same computational effort; equivalent to a factor of 16 speedup. An additional benefit is that more samples and a more accurate evidence value can be obtained simply by continuing the run for longer, as in standard MCMC.Comment: Accepted for publication in Statistics and Computing. C++ code available at http://lindor.physics.ucsb.edu/DNes

Surface Phase Diagrams from Nested Sampling

Atomic-scale modeling of surface phase equilibria often focuses on temperatures near zero Kelvin due to the difficulty in computing the free energy of surfaces at finite temperatures. The Bayesian-inference-based nested sampling (NS) algorithm allows modeling surface phase equilibria at arbitrary temperatures by directly and efficiently calculating the partition function, whose relationship with free energy is well known. In this work, we extend NS to calculate surface phase diagrams, including all relevant translational, rotational, and vibrational contributions to the free energy. We apply NS to the surfaces of the Lennard-Jones solid, recording energies through the iterative compression of surface phase space rather than a specific cooling schedule. We construct the partition function from these recorded energies to calculate ensemble averages of thermodynamic properties, such as the constant-volume heat capacity and temperature-dependent order parameters that characterize the surface structure. Key results include determining the nature of phase transitions on flat and stepped surfaces, which typically feature an enthalpy-driven condensation at higher temperatures and an entropy-driven reordering process at lower temperatures, and the presence of critical points on the phase diagrams of most of the flatter facets. Overall, we demonstrate the ability and potential of NS for surface modeling and, ultimately, materials discovery.Comment: 41 pages, 11 figures; figures in Sec. S5 are not included due to size restriction

A general-purpose machine learning Pt interatomic potential for an accurate description of bulk, surfaces and nanoparticles

A Gaussian approximation machine learning interatomic potential for platinum is presented. It has been trained on DFT data computed for bulk, surfaces and nanostructured platinum, in particular nanoparticles. Across the range of tested properties, which include bulk elasticity, surface energetics and nanoparticle stability, this potential shows excellent transferability and agreement with DFT, providing state-of-the-art accuracy at low computational cost. We showcase the possibilities for modeling of Pt systems enabled by this potential with two examples: the pressure-temperature phase diagram of Pt calculated using nested sampling and a study of the spontaneous crystallization of a large Pt nanoparticle based on classical dynamics simulations over several nanoseconds

Neural-Network Force Field Backed Nested Sampling: Study of the Silicon p-T Phase Diagram

Nested sampling is a promising method for calculating phase diagrams of materials, however, the computational cost limits its applicability if ab-initio accuracy is required. In the present work, we report on the efficient use of a neural-network force field in conjunction with the nested-sampling algorithm. We train our force fields on a recently reported database of silicon structures and demonstrate our approach on the low-pressure region of the silicon pressure-temperature phase diagram between 0 and \SI{16}{GPa}. The simulated phase diagram shows a good agreement with experimental results, closely reproducing the melting line. Furthermore, all of the experimentally stable structures within the investigated pressure range are also observed in our simulations. We point out the importance of the choice of exchange-correlation functional for the training data and show how the meta-GGA r2SCAN plays a pivotal role in achieving accurate thermodynamic behaviour using nested-sampling. We furthermore perform a detailed analysis of the exploration of the potential energy surface and highlight the critical role of a diverse training data set

Pressure–temperature phase diagram of lithium, predicted by embedded atom model potentials

In order to study the performance of interatomic potentials and their reliability at higher pressures, the phase diagrams of two different embedded-atom-type potential models (EAMs) and a modified embedded-atom model (MEAM) of lithium are compared. The calculations were performed by using the nested sampling technique in the pressure range 0.01–20 GPa, in order to determine the liquid–vapor critical point, the melting curve, and the different stable solid phases of the compared models. The low-pressure stable structure below the melting line is found to be the body-centered-cubic (bcc) structure in all cases, but the higher pressure phases and the ground-state structures show a great variation, being face-centered cubic (fcc), hexagonal close-packed (hcp), a range of different close-packed stacking variants, and highly symmetric open structures are observed as well. A notable behavior of the EAM of Nichol and Ackland (Phys. Rev. B: Condens. Matter Mater. Phys.2016, 93, 184101) is observed, that the model displays a maximum temperature in the melting line, similarly to experimental results

Nested sampling of materials’ potential energy surfaces : case study of Zirconium

The nested sampling (NS) method was originally proposed by John Skilling to calculate the evidence in Bayesian inference. The method has since been utilised in various research fields, and here we focus on how NS has been adapted to sample the Potential Energy Surface (PES) of atomistic systems, enabling the straightforward estimation of the partition function. Using two interatomic potential models of zirconium, we demonstrate the workflow and advantages of using nested sampling to calculate pressure-temperature phase diagrams. Without any prior knowledge of the stable phases or the phase transitions, we are able to identify the melting line, as well as the transition between the body-centred-cubic and hexagonal-close-packed structures

Insight into liquid polymorphism from the complex phase behavior of a simple model

We systematically explored the phase behavior of the hard-core two-scale ramp model suggested by Jagla [Phys. Rev. E 63, 061501 (2001)] using a combination of the nested sampling and free energy methods. The sampling revealed that the phase diagram of the Jagla potential is significantly richer than previously anticipated, and we identified a family of new crystalline structures, which is stable over vast regions in the phase diagram. We showed that the new melting line is located at considerably higher temperature than the boundary between the low- and high-density liquid phases, which was previously suggested to lie in a thermodynamically stable region. The newly identified crystalline phases show unexpectedly complex structural features, some of which are shared with the high-pressure ice VI phase

Nested sampling for materials: the case of hard spheres

The recently introduced nested sampling algorithm allows the direct and efficient calculation of the partition function of atomistic systems. We demonstrate its applicability to condensed phase systems with periodic boundary conditions by studying the three dimensional hard sphere model. Having obtained the partition function, we show how easy it is to calculate the compressibility and the free energy as functions of the packing fraction and local order, verifying that the transition to crystallinity has a very small barrier, and that the entropic contribution of jammed states to the free energy is negligible for packing fractions above the phase transition. We quantify the previously proposed schematic phase diagram and estimate the extent of the region of jammed states. We find that within our samples, the maximally random jammed configuration is surprisingly disordered