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An efficient algorithm for numerical computations of continuous densities of states

K. Langfeld, Biagio Lucini Orcid Logo, R. Pellegrini, A. Rago

The European Physical Journal C, Volume: 76, Issue: 6

Swansea University Author: Biagio Lucini Orcid Logo

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Abstract

In Wang-Landau type algorithms, Monte-Carlo updates are performed with respect to the density of states, which is iteratively refined during simulations. The partition function and thermodynamic observables are then obtained by standard integration. In this work, our recently introduced method in th...

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Published in: The European Physical Journal C
ISSN: 1434-6044 1434-6052
Published: Springer Science and Business Media LLC 2016
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa28023
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Abstract: In Wang-Landau type algorithms, Monte-Carlo updates are performed with respect to the density of states, which is iteratively refined during simulations. The partition function and thermodynamic observables are then obtained by standard integration. In this work, our recently introduced method in this class (the LLR approach) is analysed and further developed. Our approach is a histogram free method particularly suited for systems with continuous degrees of freedom giving rise to a continuum density of states, as it is commonly found in Lattice Gauge Theories and in some Statistical Mechanics systems. We show that the method possesses an exponential error suppression that allows us to estimate the density of states over several orders of magnitude with nearly-constant relative precision. We explain how ergodicity issues can be avoided and how expectation values of arbitrary observables can be obtained within this framework. We then demonstrate the method using Compact U(1) Lattice Gauge Theory as a show case. A thorough study of the algorithm parameter dependence of the results is performed and compared with the analytically expected behaviour. We obtain high precision values for the critical coupling for the phase transition and for the peak value of the specific heat for lattice sizes ranging from 84 to 204. Our results perfectly agree with the reference values reported in the literature, which covers lattice sizes up to 184. Robust results for the 204 volume are obtained for the first time. This latter investigation, which, due to strong metastabilities developed at the pseudo-critical coupling of the system, so far has been out of reach even on supercomputers with importance sampling approaches, has been performed to high accuracy with modest computational resources. This shows the potential of the method for studies of first order phase transitions. Other situations where the method is expected to be superior to importance sampling techniques are pointed out.
Item Description: EPJC is an open-access journal funded by SCOAP3 and licensed under CC BY 4.0
Keywords: Wilson Loop; Order Phase Transition; Importance Sampling; Lattice Gauge Theory; Tunnelling Time
College: Faculty of Science and Engineering
Issue: 6