Nested sampling of isobaric phase space for the direct evaluation of the isothermal-isobaric partition function of atomic systems

Efficient approaches to solutions of partition function for condensed matters

The key problem of statistical physics standing over one hundred years is how to exactly calculate the partition function (or free energy), which severely hinders the theory to be applied to predict the thermodynamic properties of condensed matters. Very recently, we developed a direct integral approach (DIA) to the solutions and achieved ultrahigh computational efficiency and precision. In the present work, the background and the limitations of DIA were examined in details, and another method with the same efficiency was established to overcome the shortage of DIA for condensed system with lower density. The two methods were demonstrated with empirical potentials for solid and liquid cooper, solid argon and C₆₀ molecules by comparing the derived internal energy or pressure with the results of vast molecular dynamics simulations, showing that the precision is about ten times higher than previous methods in a temperature range up to melting point. The ultrahigh efficiency enables the two methods to be performed with ab initio calculations and the experimental equation of state of solid copper up to ∼600 GPa was well reproduced, for the first time, from the partition function via density functional theory implemented.

Comparison of Two Efficient Methods for Calculating Partition Functions

In the long-time pursuit of the solution to calculating the partition function (or free energy) of condensed matter, Monte-Carlo-based nested sampling should be the state-of-the-art method, and very recently, we established a direct integral approach that works at least four orders faster. In present work, the above two methods were applied to solid argon at temperatures up to 300 K. The derived internal energy and pressure were compared with the molecular dynamics simulation as well as experimental measurements, showing that the calculation precision of our approach is about 10 times higher than that of the nested sampling method.

Nested Transition Path Sampling

We introduce a novel transition path (TPS) sampling scheme employing nested sampling. Analogous to how nested sampling explores the entire configurational phase space for atomistic systems, nested TPS samples the entire available trajectory space in one simulation. Thermodynamic and path observables can be constructed a posteriori for all temperatures simultaneously. We exploit this to compute the rate of rare processes at arbitrarily low temperature through the coupling to easily accessible rates at high temperature. We illustrate the method on several model systems.