Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review E

Efficient parallel tempering for first-order phase transitions

T. Neuhaus, Martin P. Magiera, Ulrich H. E. Hansmann

Published Oct 9, 2007• 22 citations
Publisher

Abstract

We present a Monte Carlo algorithm that facilitates efficient parallel tempering simulations of the density of states g(E) . We show that the algorithm eliminates the supercritical slowing down in the case of the Q=20 and Q=256 Potts models in two dimensions, typical examples for systems with extreme first-order phase transitions. As recently predicted, and shown here, the microcanonical heat capacity along the calorimetric curve has negative values for finite systems.

DOI
10.1103/physreve.76.045701

Related papers

Accurate first-order transition points from finite-size data without power-law corrections

Wolfhard Janke E

Traditional observables used to locate first-order phase-transition points from Monte Carlo simulations on finite lattices usually exhibit powerlike finite-size scaling behavior. For systems in a box of volume V with periodic boundary conditions, several definitions of finite volume (pseudo-) transition temperatures ${\mathit{T}}_{0}$(V) have recently been proposed which involve only exponential corrections with respect to ${\mathit{T}}_{0}$\ensuremath{\equiv}${\mathit{T}}_{0}$(\ensuremath{\infty}). One of these approaches suggests also a different and efficient way to compute the latent heat. These propositions are tested for the temperature-driven first-order phase transitions in the two-dimensional q-state Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update algorithm and histogram reweighting techniques.

Physical review. B, Condensed matter•Jun 1, 1993•78 citations

Parallel-tempering cluster algorithm for computer simulations of critical phenomena

Elmar Bittner, Wolfhard Janke E

In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.

arXiv:1107.5640•Physical Review E•Sep 1, 2011•15 citations

ROBUST PARAMETER SELECTION FOR PARALLEL TEMPERING

Firas Hamze, Neil G. Dickson, Kamran Karimi

This paper describes an algorithm for selecting parameter values (e.g. temperature values) at which to measure equilibrium properties with Parallel Tempering Monte Carlo simulation. Simple approaches to choosing parameter values can lead to poor equilibration of the simulation, especially for Ising spin systems that undergo first-order phase transitions. However, starting from an initial set of parameter values, the careful, iterative respacing of these values based on results with the previous set of values greatly improves equilibration. Example spin systems presented here appear in the context of Quantum Monte Carlo.

arXiv:1004.2840•International Journal of Modern Physics C•May 1, 2010•16 citations

Exploring transitions in finite-size Potts model: comparative analysis using Wang–Landau sampling and parallel tempering

Fangfang Wang, Wei Liu, Jun Ma +3 more

Abstract This research provides a examination of transitions within the various-state Potts model in two-dimensional finite-size lattices. Leveraging the Wang–Landau sampling and parallel tempering, we systematically obtain the density of states, facilitating a comprehensive comparative analysis of the results. The determination of the third-order transitions location are achieved through a meticulous examination of the density of states using microcanonical inflection-point analysis. The remarkable alignment between canonical and microcanonical results for higher-order transition locations affirms the universality of these transitions. Our results further illustrate the universality of the robust and microcanonical inflection-point analysis of Wang–Landau sampling.

Journal of Statistical Mechanics Theory and Experiment•Sep 9, 2024•4 citations

Finite-size scaling and Monte Carlo simulations of first-order phase transitions

Jooyoung Lee, J. Michael Kosterlitz

We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=${\mathit{U}}_{0}$(${\mathit{tL}}^{\mathit{d}}$)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$${\mathit{U}}_{1}$(${\mathit{tL}}^{\mathit{d}}$) with t=1-${\mathit{T}}_{\mathit{c}}$/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(${\mathit{e}}_{1}$/${\mathit{e}}_{2}$-${\mathit{e}}_{2}$/${\mathit{e}}_{1}$${)}^{2}$/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature ${\mathit{T}}_{\mathit{c}}^{(2)}$(L)-${\mathit{T}}_{\mathit{c}}$=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at ${\mathit{T}}_{\mathit{c}}$ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.

Physical review. B, Condensed matter•Feb 1, 1991•316 citations

Efficiency of Parallel Tempering for Ising Systems

Stephan Burkhardt

The efficiency of parallel tempering Monte Carlo is studied for a two-dimensional Ising system of length L with N=L^2 spins. An external field is used to introduce a difference in free energy between the two low temperature states.\nIt is found that the number of replicas R_opt that optimizes the parallel tempering algorithm scales as the square root of the system size N. For two symmetric low temperature states, the time needed for equilibration is observed to grow as L^2.18. If a significant difference in free energy is present between the two states, this changes to L^1.02.\nIt is therefore established that parallel tempering is sped up by a factor of roughly L if an asymmetry is introduced between the low temperature states. This confirms previously made predictions for the efficiency of parallel tempering. These findings should be especially relevant when using parallel tempering for systems like spin glasses, where no information about the degeneracy of low temperature states is available prior to the simulation.

•Aug 15, 2023•0 citations
We present a Monte Carlo algorithm that facilitates efficient parallel tempering simulations of the density of states g(E) . We show that the algorithm eliminates the supercritical slowing down in the case of the Q=20 and Q=256 Potts models in two dimensions, typical examples for systems with extreme first-order phase transitions. As recently predicted, and shown here, the microcanonical heat capacity along the calorimetric curve has negative values for finite systems.
Traditional observables used to locate first-order phase-transition points from Monte Carlo simulations on finite lattices usually exhibit powerlike finite-size scaling behavior. For systems in a box of volume V with periodic boundary conditions, several definitions of finite volume (pseudo-) transition temperatures T0{\mathit{T}}_{0}T0​(V) have recently been proposed which involve only exponential corrections with respect to T0{\mathit{T}}_{0}T0​\ensuremath{\equiv}T0{\mathit{T}}_{0}T0​(\ensuremath{\infty}). One of these approaches suggests also a different and efficient way to compute the latent heat. These propositions are tested for the temperature-driven first-order phase transitions in the two-dimensional q-state Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update algorithm and histogram reweighting techniques.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
This paper describes an algorithm for selecting parameter values (e.g. temperature values) at which to measure equilibrium properties with Parallel Tempering Monte Carlo simulation. Simple approaches to choosing parameter values can lead to poor equilibration of the simulation, especially for Ising spin systems that undergo first-order phase transitions. However, starting from an initial set of parameter values, the careful, iterative respacing of these values based on results with the previous set of values greatly improves equilibration. Example spin systems presented here appear in the context of Quantum Monte Carlo.
Abstract This research provides a examination of transitions within the various-state Potts model in two-dimensional finite-size lattices. Leveraging the Wang–Landau sampling and parallel tempering, we systematically obtain the density of states, facilitating a comprehensive comparative analysis of the results. The determination of the third-order transitions location are achieved through a meticulous examination of the density of states using microcanonical inflection-point analysis. The remarkable alignment between canonical and microcanonical results for higher-order transition locations affirms the universality of these transitions. Our results further illustrate the universality of the robust and microcanonical inflection-point analysis of Wang–Landau sampling.
We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U0{\mathit{U}}_{0}U0​(tLd{\mathit{tL}}^{\mathit{d}}tLd)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$U1{\mathit{U}}_{1}U1​(tLd{\mathit{tL}}^{\mathit{d}}tLd) with t=1-Tc{\mathit{T}}_{\mathit{c}}Tc​/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(e1{\mathit{e}}_{1}e1​/e2{\mathit{e}}_{2}e2​-e2{\mathit{e}}_{2}e2​/e1{\mathit{e}}_{1}e1​)2{)}^{2})2/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature Tc(2){\mathit{T}}_{\mathit{c}}^{(2)}Tc(2)​(L)-Tc{\mathit{T}}_{\mathit{c}}Tc​=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc{\mathit{T}}_{\mathit{c}}Tc​ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.
The efficiency of parallel tempering Monte Carlo is studied for a two-dimensional Ising system of length L with N=L^2 spins. An external field is used to introduce a difference in free energy between the two low temperature states.\nIt is found that the number of replicas R_opt that optimizes the parallel tempering algorithm scales as the square root of the system size N. For two symmetric low temperature states, the time needed for equilibration is observed to grow as L^2.18. If a significant difference in free energy is present between the two states, this changes to L^1.02.\nIt is therefore established that parallel tempering is sped up by a factor of roughly L if an asymmetry is introduced between the low temperature states. This confirms previously made predictions for the efficiency of parallel tempering. These findings should be especially relevant when using parallel tempering for systems like spin glasses, where no information about the degeneracy of low temperature states is available prior to the simulation.