Nested sampling, statistical physics and the Potts model

Abstract

We present a systematic study of the nested sampling algorithm based on the example of the Potts model. This model, which exhibits a first order phase transition for $q>4$, exemplifies a generic numerical challenge in statistical physics: The evaluation of the partition function and thermodynamic observables, which involve high dimensional sums of sharply structured multi-modal density functions. It poses a major challenge to most standard numerical techniques, such as Markov Chain Monte Carlo. In this paper we will demonstrate that nested sampling is particularly suited for such problems and it has a couple of advantages. For calculating the partition function of the Potts model with $N$ sites: a) one run stops after $O(N)$ moves, so it takes $O(N^{2})$ operations for the run, b) only a single run is required to compute the partition function along with the assignment of confidence intervals, c) the confidence intervals of the logarithmic partition function decrease with $1/\sqrt{N}$ and d) a single run allows to compute quantities for all temperatures while the autocorrelation time is very small, irrespective of temperature. Thermodynamic expectation values of observables, which are completely determined by the bond configuration in the representation of Fortuin and Kasteleyn, like the Helmholtz free energy, the internal energy as well as the entropy and heat capacity, can be calculated in the same single run needed for the partition function along with their confidence intervals. In contrast, thermodynamic expectation values of magnetic properties like the magnetization and the magnetic susceptibility require sampling the additional spin degree of freedom. Results and performance are studied in detail and compared with those obtained with multi-canonical sampling. Eventually the implications of the findings on a parallel implementation of nested sampling are outlined.