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Taylor MP, Luettmer-Strathmann J. Partition function zeros and finite size scaling for polymer adsorption. J Chem Phys 2014; 141:204906. [PMID: 25429961 DOI: 10.1063/1.4902252] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Mark P. Taylor
- Department of Physics, Hiram College, Hiram, Ohio 44234, USA
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Taylor MP, Aung PP, Paul W. Partition function zeros and phase transitions for a square-well polymer chain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:012604. [PMID: 23944483 DOI: 10.1103/physreve.88.012604] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2012] [Revised: 05/31/2013] [Indexed: 06/02/2023]
Abstract
The zeros of the canonical partition functions for flexible square-well polymer chains have been approximately computed for chains up to length 256 for a range of square-well diameters. We have previously shown that such chain molecules can undergo a coil-globule and globule-crystal transition as well as a direct coil-crystal transition. Here we show that each of these transitions has a well-defined signature in the complex-plane map of the partition function zeros. The freezing transitions are characterized by nearly circular rings of uniformly spaced roots, indicative of a discontinuous transition. The collapse transition is signaled by the appearance of an elliptical horseshoe segment of roots that pinches down towards the positive real axis and defines a boundary to a root-free region of the complex plane. With increasing chain length, the root density on the circular ring and in the space adjacent to the elliptical boundary increases and the leading roots move towards the positive real axis. For finite-length chains, transition temperatures can be obtained by locating the intersection of the ellipse and/or circle of roots with the positive real axis. A finite-size scaling analysis is used to obtain transition temperatures in the long-chain (thermodynamic) limit. The collapse transition is characterized by crossover and specific-heat exponents of φ≈0.76(2) and α≈0.66(2), respectively, consistent with a second-order phase transition.
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Affiliation(s)
- Mark P Taylor
- Department of Physics, Hiram College, Hiram, Ohio 44234, USA.
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Kromer JA, Schimansky-Geier L, Toral R. Weighted-ensemble Brownian dynamics simulation: sampling of rare events in nonequilibrium systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:063311. [PMID: 23848810 DOI: 10.1103/physreve.87.063311] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/12/2013] [Revised: 06/17/2013] [Indexed: 06/02/2023]
Abstract
We provide an algorithm based on weighted-ensemble (WE) methods, to accurately sample systems at steady state. Applying our method to different one- and two-dimensional models, we succeed in calculating steady-state probabilities of order 10(-300) and reproduce the Arrhenius law for rates of order 10(-280). Special attention is payed to the simulation of nonpotential systems where no detailed balance assumption exists. For this large class of stochastic systems, the stationary probability distribution density is often unknown and cannot be used as preknowledge during the simulation. We compare the algorithm's efficiency with standard Brownian dynamics simulations and the original WE method.
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Affiliation(s)
- Justus A Kromer
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany.
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Fonseca JSM, Rizzi LG, Alves NA. Stripe-tetragonal phase transition in the two-dimensional Ising model with dipole interactions: partition function zeros approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:011103. [PMID: 23005364 DOI: 10.1103/physreve.86.011103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2012] [Revised: 06/06/2012] [Indexed: 06/01/2023]
Abstract
We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction δ where the phase characterized by striped configurations of width h = 1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between δ = 0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents ν that are clearly consistent with a single second-order phase transition line, thus excluding such a tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.
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Affiliation(s)
- Jacyana S M Fonseca
- Departamento de Física, FFCLRP, Universidade de São Paulo, Avenida Bandeirantes, 3900 Ribeirão Preto 14040-901, SP, Brazil.
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Ferreira ALC, Toral R. Projected single-spin-flip dynamics in the Ising model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:011117. [PMID: 17677420 DOI: 10.1103/physreve.76.011117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2007] [Indexed: 05/16/2023]
Abstract
We study transition matrices for projected dynamics in the energy-magnetization, magnetization, and energy spaces. Several single-spin-flip dynamics are considered, such as the Glauber and Metropolis canonical ensemble dynamics, and the Metropolis dynamics for three multicanonical ensembles: the flat energy-magnetization, the flat energy, and the flat magnetization histograms. From the numerical diagonalization of the matrices for the projected dynamics we obtain the subdominant eigenvalues and the largest relaxation times for systems of varying size. Although the projected dynamics is an approximation to the full state space dynamics, comparison with some available results, obtained by other authors, shows that projection in the magnetization space is a reasonably accurate method to study the scaling of relaxation times with system size. For each system size, the transition matrices for arbitrary single-spin-flip dynamics are obtained from a single Monte Carlo estimate of the infinite-temperature transition matrix. This makes the method an efficient tool for evaluating the relative performance of any arbitrary local spin-flip dynamics. We also present results for appropriately defined average tunneling times of magnetization and compare their finite-size scaling exponents with results of energy tunneling exponents available for the flat energy histogram multicanonical ensemble.
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Affiliation(s)
- A L C Ferreira
- Departamento de Física, Universidade de Aveiro, 3810-193 Aveiro, Portugal
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Viana Lopes J, Costa MD, Lopes dos Santos JMB, Toral R. Optimized multicanonical simulations: a proposal based on classical fluctuation theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:046702. [PMID: 17155207 DOI: 10.1103/physreve.74.046702] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2006] [Indexed: 05/12/2023]
Abstract
We propose a recursive procedure to estimate the microcanonical density of states in multicanonical Monte Carlo simulations which relies only on measurements of moments of the energy distribution, avoiding entirely the need for energy histograms. This method yields directly a piecewise analytical approximation to the microcanonical inverse temperature beta(E) and allows improved control over the statistics and efficiency of the simulations. We demonstrate its utility in connection with recently proposed schemes for improving the efficiency of multicanonical sampling, either with adjustment of the asymptotic energy distribution or with the replacement of single spin flip dynamics with collective updates.
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Affiliation(s)
- J Viana Lopes
- Centro de Física do Porto and Departamento de Física, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
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Neuhaus T, Hager JS. Free-energy calculations with multiple Gaussian modified ensembles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:036702. [PMID: 17025781 DOI: 10.1103/physreve.74.036702] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2006] [Indexed: 05/12/2023]
Abstract
We present a Monte Carlo algorithm, which samples free energies of complex systems. Less probable configurations are populated with the help of a multitude of additional Gaussian weights and parallel tempering is used for efficient Monte Carlo moves within phase space. The algorithm is easily parallelized and can be applied to a wide class of problems. We discuss algorithmic performance for the case of low-temperature phase separation in two-dimensional and three-dimensional Ising models, where we determine the magnetic interface tension. Multiple Gaussian modified ensemble simulations, unlike multicanonical ensemble simulations do not require a priori knowledge of the free energy and are of similar efficiency as multicanonical ensemble and Wang-Landau simulations.
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Affiliation(s)
- T Neuhaus
- NIC, Forschungszentrum Jülich, D-52425 Jülich, Germany.
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Gulbahce N, Gould H, Klein W. Zeros of the partition function and pseudospinodals in long-range Ising models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:036119. [PMID: 15089373 DOI: 10.1103/physreve.69.036119] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2003] [Indexed: 05/24/2023]
Abstract
The relation between the zeros of the partition function and spinodal critical points in Ising models with long-range interactions is investigated. We find that the spinodal is associated with the zeros of the partition function in four-dimensional complex temperature/magnetic field space. The zeros approach the real temperature/magnetic field plane as the range of interaction increases.
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Affiliation(s)
- Natali Gulbahce
- Department of Physics, Clark University, Worcester, Massachusetts 01610, USA
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Wang F, Landau DP. Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:056101. [PMID: 11736008 DOI: 10.1103/physreve.64.056101] [Citation(s) in RCA: 549] [Impact Index Per Article: 23.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2001] [Revised: 06/27/2001] [Indexed: 05/22/2023]
Abstract
We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy level is visited to produce a flat histogram. By carefully controlling the modification factor, we allow the density of states to converge to the true value very quickly, even for large systems. From the density of states at the end of the random walk, we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature. Using this method, we not only can avoid repeating simulations at multiple temperatures, but we can also estimate the free energy and entropy, quantities that are not directly accessible by conventional Monte Carlo simulations. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. As with the multicanonical Monte Carlo technique, our method overcomes the tunneling barrier between coexisting phases at first-order phase transitions. In this paper, we apply our algorithm to both first- and second-order phase transitions to demonstrate its efficiency and accuracy. We obtained direct simulational estimates for the density of states for two-dimensional ten-state Potts models on lattices up to 200 x 200 and Ising models on lattices up to 256 x 256. Our simulational results are compared to both exact solutions and existing numerical data obtained using other methods. Applying this approach to a three-dimensional +/-J spin-glass model, we estimate the internal energy and entropy at zero temperature; and, using a two-dimensional random walk in energy and order-parameter space, we obtain the (rough) canonical distribution and energy landscape in order-parameter space. Preliminary data suggest that the glass transition temperature is about 1.2 and that better estimates can be obtained with more extensive application of the method. This simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space.
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Affiliation(s)
- F Wang
- Center for Simulational Physics, The University of Georgia, Athens, Georgia 30602, USA
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Alves NA, Felicio JRDD, Hansmann UHE. Partition function zeros and leading-order scaling correction of the 3D Ising model from multicanonical simulations. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0305-4470/33/42/302] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Lee J, Lee KC. Exact zeros of the partition function for a continuum system with double gaussian peaks. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:4558-63. [PMID: 11088995 DOI: 10.1103/physreve.62.4558] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2000] [Indexed: 11/07/2022]
Abstract
We calculate the exact zeros of the partition function for a continuum system where the probability distribution for the order parameter is given by two asymmetric Gaussian peaks. When the positions of the two peaks coincide, the two separate loci of the zeros that used to give a first-order transition touch each other, with the density of zeros vanishing at the contact point on the positive real axis. Instead of the second-order transition of the Ehrenfest classification as one might naively expect, one finds a critical behavior in this limit.
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Affiliation(s)
- J Lee
- Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
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He HX, Hamer CJ, Oitmaa J. High-temperature series expansions for the (2+1)-dimensional Ising model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/23/10/018] [Citation(s) in RCA: 98] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Price PF, Hamer CJ, O'Shaughnessy D. Stochastic truncation for the (2+1)D Ising model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/12/023] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Oitmaa J, Hamer CJ, Weihong Z. Low-temperature series expansions for the (2+1)-dimensional Ising model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/24/12/024] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Rickman JM, Srolovitz DJ. Efficient determination of thermodynamic properties from a single simulation. J Chem Phys 1993. [DOI: 10.1063/1.465675] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Lee KC. Finite-size scaling in the complex temperature plane applied to the three-dimensional Ising model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:3459-3463. [PMID: 9961003 DOI: 10.1103/physreve.48.3459] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Baillie CF, Gupta R, Hawick KA, Pawley GS. Monte Carlo renormalization-group study of the three-dimensional Ising model. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 45:10438-10453. [PMID: 10000948 DOI: 10.1103/physrevb.45.10438] [Citation(s) in RCA: 118] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Lee J, Kosterlitz JM. Finite-size scaling and Monte Carlo simulations of first-order phase transitions. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:3265-3277. [PMID: 9997636 DOI: 10.1103/physrevb.43.3265] [Citation(s) in RCA: 192] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Alves NA, Berg BA, Villanova R. Ising-model Monte Carlo simulations: Density of states and mass gap. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 41:383-394. [PMID: 9992774 DOI: 10.1103/physrevb.41.383] [Citation(s) in RCA: 61] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Ferrenberg AM, Swendsen RH. Optimized Monte Carlo data analysis. PHYSICAL REVIEW LETTERS 1989; 63:1195-1198. [PMID: 10040500 DOI: 10.1103/physrevlett.63.1195] [Citation(s) in RCA: 1545] [Impact Index Per Article: 44.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Berg BA, Villanova R, Vohwinkel C. Correlation length and order of the deconfining phase transition. PHYSICAL REVIEW LETTERS 1989; 62:2433-2435. [PMID: 10039987 DOI: 10.1103/physrevlett.62.2433] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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