Nested sampling, statistical physics and the Potts model

摘要

We present a systematic study of the nested sampling algorithm based on theexample of the Potts model. This model, which exhibits a first order phasetransition for $q>4$, exemplifies a generic numerical challenge in statisticalphysics: The evaluation of the partition function and thermodynamicobservables, which involve high dimensional sums of sharply structuredmulti-modal density functions. It poses a major challenge to most standardnumerical techniques, such as Markov Chain Monte Carlo. In this paper we willdemonstrate that nested sampling is particularly suited for such problems andit has a couple of advantages. For calculating the partition function of thePotts 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 computethe 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 alltemperatures while the autocorrelation time is very small, irrespective oftemperature. Thermodynamic expectation values of observables, which arecompletely determined by the bond configuration in the representation ofFortuin and Kasteleyn, like the Helmholtz free energy, the internal energy aswell as the entropy and heat capacity, can be calculated in the same single runneeded for the partition function along with their confidence intervals. Incontrast, thermodynamic expectation values of magnetic properties like themagnetization and the magnetic susceptibility require sampling the additionalspin degree of freedom. Results and performance are studied in detail andcompared with those obtained with multi-canonical sampling. Eventually theimplications of the findings on a parallel implementation of nested samplingare outlined.