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Ferreira L, Jorge L, Neto MA, Caparica A. Phase transition in an 1D interacting dimers: an exact and simulational study. Chem Phys 2022. [DOI: 10.1016/j.chemphys.2022.111448] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
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2
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Jorge LN, Ferreira LS, Caparica AA. Three-dimensional Baxter-Wu model. Phys Rev E 2019; 100:032141. [PMID: 31640024 DOI: 10.1103/physreve.100.032141] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2018] [Indexed: 11/07/2022]
Abstract
A classic three-dimensional spin model, based upon the Baxter-Wu scheme, is presented. It is found, by entropic sampling simulations, that the behavior of the energy and magnetization fourth-order cumulants points to a first-order phase transition. A finite-size procedure was performed, confirming that the system scales with the dimensionality d=3, yielding a high-resolution estimate of the transition temperature as T_{c}=11.377485(29).
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Affiliation(s)
- L N Jorge
- Instituto Federal do Mato Grosso, Campus Cáceres, CEP. 78200-000, Cáceres, Mato Grosso, Brazil
| | - L S Ferreira
- Instituto de Física, Universidade Federal de Goiás, Av. Esperança s/n, 74.690-900, Goiânia, GO, Brazil
| | - A A Caparica
- Instituto de Física, Universidade Federal de Goiás, Av. Esperança s/n, 74.690-900, Goiânia, GO, Brazil
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Ferreira LS, Caparica ÁA, Jorge LN, Neto MA. Thermodynamic properties of interacting like-rod chains: Entropic sampling simulations. Chem Phys 2019. [DOI: 10.1016/j.chemphys.2018.10.002] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Lemos CGO, Santos M, Ferreira AL, Figueiredo W. Testing ground for fluctuation theorems: The one-dimensional Ising model. Phys Rev E 2018; 97:042121. [PMID: 29758686 DOI: 10.1103/physreve.97.042121] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2018] [Indexed: 06/08/2023]
Abstract
In this paper we determine the nonequilibrium magnetic work performed on a Ising model and relate it to the fluctuation theorem derived some years ago by Jarzynski. The basic idea behind this theorem is the relationship connecting the free energy difference between two thermodynamic states of a system and the average work performed by an external agent, in a finite time, through nonequilibrium paths between the same thermodynamic states. We test the validity of this theorem by considering the one-dimensional Ising model where the free energy is exactly determined as a function of temperature and magnetic field. We have found that the Jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field applied to the system. We have also determined the probability distribution function for the work performed on the system for the forward and reverse processes and verified that predictions based on the Crooks relation are equally correct. We also propose a method to calculate the lag between the current state of the system and that of the equilibrium based on macroscopic variables. We have shown that the lag increases with the sweeping rate of the field at its final value for the reverse process, while it decreases in the case of the forward process. The lag increases linearly with the size of the chain and with a slope decreasing with the inverse of the rate of variation of the field.
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Affiliation(s)
- C G O Lemos
- Departamento de Física, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, 88040-900, Florianópolis, Santa Catarina, Brazil
| | - M Santos
- Departamento de Física, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, 88040-900, Florianópolis, Santa Catarina, Brazil
| | - A L Ferreira
- Departamento de Física, I3N, Universidade de Aveiro, 3810-193, Aveiro, Portugal
| | - W Figueiredo
- Departamento de Física, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, 88040-900, Florianópolis, Santa Catarina, Brazil and Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, São Paulo, Brazil
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Zhang C, Drake JA, Ma J, Pettitt BM. Optimal updating magnitude in adaptive flat-distribution sampling. J Chem Phys 2017; 147:174105. [PMID: 29117700 PMCID: PMC5669982 DOI: 10.1063/1.5008618] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2017] [Accepted: 10/08/2017] [Indexed: 11/14/2022] Open
Abstract
We present a study on the optimization of the updating magnitude for a class of free energy methods based on flat-distribution sampling, including the Wang-Landau (WL) algorithm and metadynamics. These methods rely on adaptive construction of a bias potential that offsets the potential of mean force by histogram-based updates. The convergence of the bias potential can be improved by decreasing the updating magnitude with an optimal schedule. We show that while the asymptotically optimal schedule for the single-bin updating scheme (commonly used in the WL algorithm) is given by the known inverse-time formula, that for the Gaussian updating scheme (commonly used in metadynamics) is often more complex. We further show that the single-bin updating scheme is optimal for very long simulations, and it can be generalized to a class of bandpass updating schemes that are similarly optimal. These bandpass updating schemes target only a few long-range distribution modes and their optimal schedule is also given by the inverse-time formula. Constructed from orthogonal polynomials, the bandpass updating schemes generalize the WL and Langfeld-Lucini-Rago algorithms as an automatic parameter tuning scheme for umbrella sampling.
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Affiliation(s)
- Cheng Zhang
- Sealy Center for Structural Biology and Molecular Biophysics, The University of Texas Medical Branch, Galveston, Texas 77555-0304, USA
| | - Justin A Drake
- Sealy Center for Structural Biology and Molecular Biophysics, The University of Texas Medical Branch, Galveston, Texas 77555-0304, USA
| | - Jianpeng Ma
- Department of Biochemistry and Molecular Biology, Baylor College of Medicine, Houston, Texas 77030, USA
| | - B Montgomery Pettitt
- Sealy Center for Structural Biology and Molecular Biophysics, The University of Texas Medical Branch, Galveston, Texas 77555-0304, USA
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Zablotskiy SV, Martemyanova JA, Ivanov VA, Paul W. Stochastic approximation Monte Carlo algorithm for calculation of diagram of states of a single flexible-semiflexible copolymer chain. POLYMER SCIENCE SERIES A 2016. [DOI: 10.1134/s0965545x1606016x] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Grzybowski PR, Czekaj Ł, Nogala M, Ścibior A, Chhajlany RW. Minimal energy ensemble Monte Carlo algorithm for the partition function of fermions coupled to classical fields. Phys Rev E 2016; 93:061301. [PMID: 27415201 DOI: 10.1103/physreve.93.061301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2015] [Indexed: 11/07/2022]
Abstract
Models of noninteracting fermions coupled to auxiliary classical fields are relevant to the understanding of a wide variety of problems in many-body physics, e.g., the description of manganites, diluted magnetic semiconductors, or strongly interacting electrons on lattices. We present a flat-histogram Monte Carlo algorithm that simulates a statistical ensemble that allows one to directly acquire the partition function at all temperatures for such systems. The defining feature of the algorithm is that it utilizes the complete thermodynamic information from the full energy spectrum of noninteracting fermions available during sampling of the configuration space of the classical fields. We benchmark the method for the classical Ising and Potts models in two dimensions, as well as the Falicov-Kimball model describing itinerant electrons interacting with heavy ions.
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Affiliation(s)
- Przemysław R Grzybowski
- Faculty of Physics, Adam Mickiewicz University in Poznań, Umultowska 85, 61-614 Poznań, Poland
| | - Łukasz Czekaj
- National Quantum Information Centre of Gdańsk, 81-824 Sopot, Poland
| | - Mariusz Nogala
- Faculty of Physics, Adam Mickiewicz University in Poznań, Umultowska 85, 61-614 Poznań, Poland
| | - Adam Ścibior
- Faculty of Physics, Adam Mickiewicz University in Poznań, Umultowska 85, 61-614 Poznań, Poland
| | - Ravindra W Chhajlany
- Faculty of Physics, Adam Mickiewicz University in Poznań, Umultowska 85, 61-614 Poznań, Poland.,ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
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Belardinelli RE, Pereyra VD. Nonconvergence of the Wang-Landau algorithms with multiple random walkers. Phys Rev E 2016; 93:053306. [PMID: 27301004 DOI: 10.1103/physreve.93.053306] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2015] [Indexed: 12/11/2022]
Abstract
This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and 1/t algorithms. The classical algorithms are modified by the use of m-independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number π by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, t; then, the average over m walkers is performed. It is observed that the error goes as 1/sqrt[m]. However, if the number of walkers increases above a certain critical value m>m_{x}, the error reaches a constant value (i.e., it saturates). This occurs for both algorithms; however, it is shown that for a given system, the 1/t algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value m_{x}, since it does not reduce the error in the calculation. Therefore, the number of walkers does not guarantee convergence.
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Affiliation(s)
- R E Belardinelli
- Instituto de Física Aplicada (INFAP)-CONICET, San Luis, Argentina.,Departamento de Física, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina
| | - V D Pereyra
- Departamento de Física, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina
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Kwak W, Jeong J, Lee J, Kim DH. First-order phase transition and tricritical scaling behavior of the Blume-Capel model: A Wang-Landau sampling approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022134. [PMID: 26382370 DOI: 10.1103/physreve.92.022134] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2015] [Indexed: 06/05/2023]
Abstract
We investigate the tricritical scaling behavior of the two-dimensional spin-1 Blume-Capel model by using the Wang-Landau method of measuring the joint density of states for lattice sizes up to 48×48 sites. We find that the specific heat deep in the first-order area of the phase diagram exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. The first-order transition curve is systematically determined by employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. At the tricritical point, we characterize the tricritical exponents through finite-size-scaling analysis including the phenomenological finite-size scaling with thermodynamic variables. Our estimation of the tricritical eigenvalue exponents, yt=1.804(5), yg=0.80(1), and yh=1.925(3), provides the first Wang-Landau verification of the conjectured exact values, demonstrating the effectiveness of the density-of-states-based approach in finite-size scaling study of multicritical phenomena.
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Affiliation(s)
- Wooseop Kwak
- Department of Physics, Chosun University, Gwangju 61452, Korea
| | - Joohyeok Jeong
- Department of Physics and Photon Science, School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
| | - Juhee Lee
- Department of Physics and Photon Science, School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
| | - Dong-Hee Kim
- Department of Physics and Photon Science, School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
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Caparica AA. Wang-Landau sampling: a criterion for halting the simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:043301. [PMID: 24827359 DOI: 10.1103/physreve.89.043301] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2013] [Indexed: 06/03/2023]
Abstract
In this work we propose a criterion to finish the simulations of the Wang-Landau sampling. Instead of determining a final modification factor for all simulations and every sample size, we investigate the behavior of the temperature of the peak of the specific heat during the simulations and finish them when this value varies below a given limit. As a result, different runs stop at different final modification factors. We show that in place of the temperature of the peak of the specific heat one can adopt alternatively the integrated heat transfer as a reference quantity. We apply this technique to the two-dimensional Ising model and a homopolymer. We verify that for the Ising model the mean order of the final modification factors is roughly the same for all lattice sizes, but for the homopolymer the order of the final modification factors increases with increasing polymer sizes. The results show that the simulations can be halted much earlier than is conventional in Wang-Landau sampling, but manifold finite-size simulations are required in order to obtain accurate results. A brief application to the three-dimensional Ising model is also available.
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Affiliation(s)
- A A Caparica
- Instituto de Física, Universidade Federal de Goiás. C.P. 131, CEP 74001-970, Goiânia, GO, Brazil
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Koh YW, Lee HK, Okabe Y. Dynamically optimized Wang-Landau sampling with adaptive trial moves and modification factors. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:053302. [PMID: 24329374 DOI: 10.1103/physreve.88.053302] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2013] [Revised: 09/16/2013] [Indexed: 06/03/2023]
Abstract
The density of states of continuous models is known to span many orders of magnitudes at different energies due to the small volume of phase space near the ground state. Consequently, the traditional Wang-Landau sampling which uses the same trial move for all energies faces difficulties sampling the low-entropic states. We developed an adaptive variant of the Wang-Landau algorithm that very effectively samples the density of states of continuous models across the entire energy range. By extending the acceptance ratio method of Bouzida, Kumar, and Swendsen such that the step size of the trial move and acceptance rate are adapted in an energy-dependent fashion, the random walker efficiently adapts its sampling according to the local phase space structure. The Wang-Landau modification factor is also made energy dependent in accordance with the step size, enhancing the accumulation of the density of states. Numerical simulations show that our proposed method performs much better than the traditional Wang-Landau sampling.
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Affiliation(s)
- Yang Wei Koh
- Bioinformatics Institute, 30 Biopolis Street, no. 07-01, Matrix, Singapore 138671
| | - Hwee Kuan Lee
- Bioinformatics Institute, 30 Biopolis Street, no. 07-01, Matrix, Singapore 138671
| | - Yutaka Okabe
- Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
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