Bayesian statistical modelling of microcanonical melting times at the superheated regime

TL;DR

This paper develops a Bayesian statistical model to analyze the distribution of melting times in superheated crystals, revealing a gamma distribution shape and accounting for simulation time truncation, enhancing understanding and prediction of melting behavior.

Contribution

It introduces a Bayesian approach to model melting time distributions, showing they are gamma-distributed, and accounts for simulation time limitations, improving analysis accuracy.

Findings

Melting times follow a gamma distribution, not exponential.

Probability of very short melting times is low.

Model reduces uncertainty in melting temperature estimates.

Abstract

Homogeneous melting of superheated crystals at constant energy is a dynamical process, believed to be triggered by the accumulation of thermal vacancies and their self-diffusion. From microcanonical simulations we know that if an ideal crystal is prepared at a given kinetic energy, it takes a random time $t_w$ until the melting mechanism is actually triggered. In this work we have studied in detail the statistics of $t_w$ for melting at different energies by performing a large number of Z-method simulations and applying state-of-the-art methods of Bayesian statistical inference. By focusing on the short-time tail of the distribution function, we show that $t_w$ is actually gamma-distributed rather than exponential (as asserted in previous work), with decreasing probability near $t_w \sim 0$. We also explicitly incorporate in our model the unavoidable truncation of the distribution function due to the limited total time span of a Z method simulation. The probabilistic model presented in this work can provide some insight into the dynamical nature of the homogeneous melting process, as well as giving a well-defined practical procedure to incorporate melting times from simulation into the Z-method in order to reduce the uncertainty in the melting temperature.