An efficient algorithm for numerical computations of continuous densities of states

Langfeld, K.; Lucini, Biagio; Pellegrini, R.; Rago, A.

doi:10.1140/epjc/s10052-016-4142-5

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An efficient algorithm for numerical computations of continuous densities of states

K. Langfeld, Biagio Lucini

, R. Pellegrini, A. Rago

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

Swansea University Author: Biagio Lucini

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DOI (Published version): 10.1140/epjc/s10052-016-4142-5

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
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URI:	https://cronfa.swan.ac.uk/Record/cronfa28023
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spelling	2020-08-03T14:12:27.8403269 v2 28023 2016-05-17 An efficient algorithm for numerical computations of continuous densities of states 7e6fcfe060e07a351090e2a8aba363cf 0000-0001-8974-8266 Biagio Lucini Biagio Lucini true false 2016-05-17 SMA 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. Journal Article The European Physical Journal C 76 6 Springer Science and Business Media LLC 1434-6044 1434-6052 Wilson Loop; Order Phase Transition; Importance Sampling; Lattice Gauge Theory; Tunnelling Time 2 6 2016 2016-06-02 10.1140/epjc/s10052-016-4142-5 EPJC is an open-access journal funded by SCOAP3 and licensed under CC BY 4.0 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University SCOAP3 2020-08-03T14:12:27.8403269 2016-05-17T22:08:24.6576121 Faculty of Science and Engineering School of Biosciences, Geography and Physics - Physics K. Langfeld 1 Biagio Lucini 0000-0001-8974-8266 2 R. Pellegrini 3 A. Rago 4 0028023-14102016123454.pdf Lucini2016.pdf 2016-10-14T12:34:54.3930000 Output 1094309 application/pdf Version of Record true This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (CC-BY). true eng http://creativecommons.org/licenses/by/4.0
title	An efficient algorithm for numerical computations of continuous densities of states
spellingShingle	An efficient algorithm for numerical computations of continuous densities of states Biagio Lucini
title_short	An efficient algorithm for numerical computations of continuous densities of states
title_full	An efficient algorithm for numerical computations of continuous densities of states
title_fullStr	An efficient algorithm for numerical computations of continuous densities of states
title_full_unstemmed	An efficient algorithm for numerical computations of continuous densities of states
title_sort	An efficient algorithm for numerical computations of continuous densities of states
author_id_str_mv	7e6fcfe060e07a351090e2a8aba363cf
author_id_fullname_str_mv	7e6fcfe060e07a351090e2a8aba363cf_***_Biagio Lucini
author	Biagio Lucini
author2	K. Langfeld Biagio Lucini R. Pellegrini A. Rago
format	Journal article
container_title	The European Physical Journal C
container_volume	76
container_issue	6
publishDate	2016
institution	Swansea University
issn	1434-6044 1434-6052
doi_str_mv	10.1140/epjc/s10052-016-4142-5
publisher	Springer Science and Business Media LLC
college_str	Faculty of Science and Engineering
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hierarchy_top_id	facultyofscienceandengineering
hierarchy_top_title	Faculty of Science and Engineering
hierarchy_parent_id	facultyofscienceandengineering
hierarchy_parent_title	Faculty of Science and Engineering
department_str	School of Biosciences, Geography and Physics - Physics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Biosciences, Geography and Physics - Physics
document_store_str	1
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description	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.
published_date	2016-06-02T03:34:04Z
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score	11.013371

An efficient algorithm for numerical computations of continuous densities of states

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