Application of nested sampling in statistical physics: the Potts model

We present a systematic benchmark study of the nested sampling algorithm on the basis of the Potts model. This model exhibits a first order phase transition for q>4 at the critical temperature. The numerical evaluation of the partition function and thermodynamic observables, which involves high dimensional sums of sharply structured multi-modal density functions, represents a major challenge to most standard numerical techniques, such as Markov Chain Monte Carlo. Nested sampling, on the other hand, is particularly suited for such problems. In this paper we will employ both, nested sampling and thermodynamic integration to evaluate the partition function of the Potts model. In both cases individual moves are based on Swendsen-Wang updates. A autocorrelation time analysis of both algorithms shows that the severe slowing down of thermodynamic integration around the critical temperature does not occur in nested sampling. In addition we show, how thermodynamic variables can be computed with high accuracy from the results of a single nested sampling run, without numerical derivatives. Results for the internal energy are compared to known results obtained by means of a multi-canonical simulation. Eventually an approach for a parallel implementation of nested sampling is presented and analysed in detail