Finite Temperature Phase Diagrams by Nested Sampling

The partition function describes the statistical properties of a system in thermodynamic equilibrium and most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. Knowledge of the partition function, thus, gives us access to complete finite temperature phase diagram of any material. However, no one has previously been able to compute the partition function for any atomistic system with realistic models of the interatomic interactions and, indeed, it has generally been believed that such a computation is intractable. However, the introduction of the ‘Nested Sampling’ technique has changed this situation. Nested Sampling is a novel inference algorithm developed by Prof John Skilling [1] that provides a method for searching the whole of configuration space in polynomial computational time – in contrast to methods such a simulated annealing for which there is no such rigorous mathematical bound on the amount of computational time needed. Interestingly, the Nested Sampling algorithm naturally samples configuration space in a way that is closely aligned with the contribution of each region of space to the partition function – high energy regions are sparsely sampled but low energy regions are more densely sampled. In this talk I shall describe how Nested Sampling can be used to calculate partition functions and I will describe a number of applications of the methodology.
[1] AIP Conference Proceedings 735: 395–405 (2004).