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

arXiv:physics/0308070 · International Journal of Modern Physics C

THE SELF-ORGANIZED MULTI-LATTICE MONTE CARLO SIMULATION

Denis Horváth, Martin Gmitra

Published Nov 1, 2004• 4 citations
arXivHTMLPDF

Abstract

Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.

DOI
10.1142/s0129183104006674

Related papers

Critical Behavior Simulation of the Two-Dimensional Ising Model—A Computational Physics Study Based on the Monte Carlo Method

佳钢 李

To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.

Modern Physics•Jan 1, 2026•0 citations

Monte Carlo simulations in a disordered binary Ising model

Dorilson Silva Cambui, A.S. de Arruda, Maurício Godoy

In this work we study a disordered binary Ising model on the square lattice. The model system consists of two different particles with spin-1/2 and spin-1, which are randomly distributed on the lattice. It has been considered only spin nearest-neighbor exchange interactions with $J>0$. This system can represent a disordered magnetic binary alloy $A_{x}B_{1-x}$, obtained from the high temperature quenching of a liquid mixture. The results were obtained by the use of Monte Carlo simulations for several lattice sizes $L$, temperature $T$ and concentration $x$ of ions $A$ with spin-1/2. We found its critical temperature, through the reduced fourth-order Binder cumulant for the several values of the concentration $x$ of particles (spin-1/2, spin-1), and also the magnetization, the susceptibility and the specific heat as a function of temperature $T$.

arXiv:1109.3797•Sep 17, 2011•1 citation

Monte Carlo-Based Modeling of 2-D Ising Systems Using Metropolis Algorithm, Simulation Techniques, Thermodynamic Behavior and Magnetization Patterns

Adama Gaye, Omolola Dorcas Atanda

This study presents a comprehensive Monte Carlo simulation of the two-dimensional (2-D) Ising model using the Metropolis algorithm to investigate critical phenomena and thermodynamic behavior in spin-lattice systems. The model, implemented in MATLAB with periodic boundary conditions, explores equilibrium properties such as magnetization, internal energy, specific heat, and susceptibility across a range of temperatures. By employing various initial spin configurations—ordered and random—the simulations demonstrate the system's ergodicity and convergence to thermal equilibrium. Key results include a sharp decline in magnetization and a pronounced peak in specific heat near the critical temperature, consistent with second-order phase transition behavior. The simulation captures microscopic domain evolution, highlighting the transition from ferromagnetic to paramagnetic phases as thermal fluctuations increase. The study further evaluates algorithmic efficiency, discusses the impact of lattice size on statistical accuracy, and proposes improvements using advanced cluster algorithms and parallel computing frameworks. The findings validate theoretical predictions from Onsager's solution and underscore the versatility of Monte Carlo techniques in modeling collective behavior in magnetic systems. The simulation framework offers a robust foundation for analyzing critical dynamics and extends its relevance to broader applications in material science, computational physics, and complex systems modeling.

•May 20, 2025•1 citation

Monte Carlo studies of finite-size scaling in the Gibbs ensemble at criticality for the two-dimensional Ising model

K. K. Mon

In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.

Physical review. B, Condensed matter•Mar 1, 1993•8 citations

Introduction to Monte Carlo methods for an Ising Model of a Ferromagnet

Jacques Kotze

This discussion serves as an introduction to the use of Monte Carlo simulations as a useful way to evaluate the observables of a ferromagnet. Key background is given about the relevance and effectiveness of this stochastic approach and in particular the applicability of the Metropolis-Hastings algorithm. Importantly the potentially devastating effects of spontaneous magnetization are highlighted and a means to avert this is examined. An Ising model is introduced and used to investigate the properties of a two dimensional ferromagnet with respect to its magnetization and energy at varying temperatures. The observables are calculated and a phase transition at a critical temperature is also illustrated and evaluated. Lastly a finite size scaling analysis is undertaken to determine the critical exponents and the Curie temperature is calculated using a ratio of cumulants with differing lattice sizes. The results obtained from the simulation are compared to exact calculations to endorse the validity of this numerical process. A copy of the code used, written in C++, is enclosed and is freely available for use and modification under the General Public License.

arXiv:0803.0217•Mar 3, 2008•16 citations

Magnetic Properties of a Square Lattice Ising Ferromagnetic System for Different Spins with Monte Carlo Simulation

A. Azdouh, Said Harir, Anouar Elidrysy +3 more

Using the Monte Carlo Simulation (MCS) method combined with the Metropolis algorithm, we explore the magnetic properties of a two-dimensional (2D) spin system with spins 1, 2 and 3. The adapted simulation to the Ising model aims to study the impact of the correction term a of the interaction with second neighbors and the crystal field D on the critical temperature [Formula: see text] of the 2D anisotropic square lattice for different integer values of spins. The obtained results demonstrate that the critical temperature [Formula: see text] is significantly affected by the competing interactions between the anisotropy parameter, second-neighbor interaction and the crystal field. The considered model exhibits a critical phase transition regardless of the spin S value where the critical temperature increases with the value of S. Note that this phase transition occurs under the condition [Formula: see text]. Moreover, the obtained results show that the complex interactions give rise to a variety of phase diagrams, thereby revealing the richness of possible magnetic behaviors in the studied system.

SPIN•Feb 12, 2025•1 citation
Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
In this work we study a disordered binary Ising model on the square lattice. The model system consists of two different particles with spin-1/2 and spin-1, which are randomly distributed on the lattice. It has been considered only spin nearest-neighbor exchange interactions with J>0J>0J>0. This system can represent a disordered magnetic binary alloy AxB1−xA_{x}B_{1-x}Ax​B1−x​, obtained from the high temperature quenching of a liquid mixture. The results were obtained by the use of Monte Carlo simulations for several lattice sizes LLL, temperature TTT and concentration xxx of ions AAA with spin-1/2. We found its critical temperature, through the reduced fourth-order Binder cumulant for the several values of the concentration xxx of particles (spin-1/2, spin-1), and also the magnetization, the susceptibility and the specific heat as a function of temperature TTT.
This study presents a comprehensive Monte Carlo simulation of the two-dimensional (2-D) Ising model using the Metropolis algorithm to investigate critical phenomena and thermodynamic behavior in spin-lattice systems. The model, implemented in MATLAB with periodic boundary conditions, explores equilibrium properties such as magnetization, internal energy, specific heat, and susceptibility across a range of temperatures. By employing various initial spin configurations—ordered and random—the simulations demonstrate the system's ergodicity and convergence to thermal equilibrium. Key results include a sharp decline in magnetization and a pronounced peak in specific heat near the critical temperature, consistent with second-order phase transition behavior. The simulation captures microscopic domain evolution, highlighting the transition from ferromagnetic to paramagnetic phases as thermal fluctuations increase. The study further evaluates algorithmic efficiency, discusses the impact of lattice size on statistical accuracy, and proposes improvements using advanced cluster algorithms and parallel computing frameworks. The findings validate theoretical predictions from Onsager's solution and underscore the versatility of Monte Carlo techniques in modeling collective behavior in magnetic systems. The simulation framework offers a robust foundation for analyzing critical dynamics and extends its relevance to broader applications in material science, computational physics, and complex systems modeling.
In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.
This discussion serves as an introduction to the use of Monte Carlo simulations as a useful way to evaluate the observables of a ferromagnet. Key background is given about the relevance and effectiveness of this stochastic approach and in particular the applicability of the Metropolis-Hastings algorithm. Importantly the potentially devastating effects of spontaneous magnetization are highlighted and a means to avert this is examined. An Ising model is introduced and used to investigate the properties of a two dimensional ferromagnet with respect to its magnetization and energy at varying temperatures. The observables are calculated and a phase transition at a critical temperature is also illustrated and evaluated. Lastly a finite size scaling analysis is undertaken to determine the critical exponents and the Curie temperature is calculated using a ratio of cumulants with differing lattice sizes. The results obtained from the simulation are compared to exact calculations to endorse the validity of this numerical process. A copy of the code used, written in C++, is enclosed and is freely available for use and modification under the General Public License.
Using the Monte Carlo Simulation (MCS) method combined with the Metropolis algorithm, we explore the magnetic properties of a two-dimensional (2D) spin system with spins 1, 2 and 3. The adapted simulation to the Ising model aims to study the impact of the correction term a of the interaction with second neighbors and the crystal field D on the critical temperature [Formula: see text] of the 2D anisotropic square lattice for different integer values of spins. The obtained results demonstrate that the critical temperature [Formula: see text] is significantly affected by the competing interactions between the anisotropy parameter, second-neighbor interaction and the crystal field. The considered model exhibits a critical phase transition regardless of the spin S value where the critical temperature increases with the value of S. Note that this phase transition occurs under the condition [Formula: see text]. Moreover, the obtained results show that the complex interactions give rise to a variety of phase diagrams, thereby revealing the richness of possible magnetic behaviors in the studied system.