We study a two-dimensional ferromagnetic Ising model in which spins are updated using modified versions of the Metropolis and Glauber algorithms. These update rules do not obey the detailed balance condition. The steady-state behavior of the model is studied using molecular field theory and Monte Carlo simulations. This model is found to exhibit a nonequilibrium phase transition from a "paramagnetic" state with zero magnetization to a "ferromagnetic" state with nonzero magnetization as the variable that plays the role of temperature in the spin updates is decreased. From detailed Monte Carlo simulations using the modified Metropolis algorithm, we demonstrate explicitly the nonequilibrium nature of the transition and show that it cannot be described as an equilibrium transition with an effective temperature different from the temperature used in the spin updates. The critical exponents that characterize the singular behavior near this continuous phase transition are calculated from finite size scaling of specific heat, magnetization, susceptibility, and correlation length. We find that the values of these exponents are the same (within error bars) as those of the equilibrium Ising model in two dimensions.
We study a two-dimensional ferromagnetic Ising model in which spins are updated using modified versions of the Metropolis and Glauber algorithms. These update rules do not obey the detailed balance condition. The steady-state behavior of the model is studied using molecular field theory and Monte Carlo simulations. This model is found to exhibit a nonequilibrium phase transition from a "paramagnetic" state with zero magnetization to a "ferromagnetic" state with nonzero magnetization as the variable that plays the role of temperature in the spin updates is decreased. From detailed Monte Carlo simulations using the modified Metropolis algorithm, we demonstrate explicitly the nonequilibrium nature of the transition and show that it cannot be described as an equilibrium transition with an effective temperature different from the temperature used in the spin updates. The critical exponents that characterize the singular behavior near this continuous phase transition are calculated from finite size scaling of specific heat, magnetization, susceptibility, and correlation length. We find that the values of these exponents are the same (within error bars) as those of the equilibrium Ising model in two dimensions.
Many problems in statistical physics resist exact solution once the number of interacting particles becomes large, and the two-dimensional Ising model is a classic case where numerical simulation fills that gap. This paper reports a Monte Carlo study of the two-dimensional Ising model using the Metropolis algorithm, undertaken to examine how magnetization, internal energy, specific heat, and magnetic susceptibility change with temperature. Spin configurations were generated on a square lattice under periodic boundary conditions, with each proposed spin flip accepted or rejected according to the Boltzmann weight of the resulting energy change. The simulation reproduces the expected transition from an ordered ferromagnetic state at low temperature to a disordered paramagnetic state at high temperature, with magnetization falling sharply and both specific heat and susceptibility peaking near a critical temperature of Tc ≈ 2.3–2.4 (in units of J/kB), consistent with the exact value obtained analytically for this system. These results confirm that the Metropolis algorithm, despite its simplicity, remains a dependable tool for locating second-order phase transitions and extracting thermodynamic quantities in lattice spin systems, and the same computational framework can be adapted to more elaborate models used in condensed matter and materials research.
Phase transitions are widespread physical phenomena in nature and the phase transition of ferromagnetic substances which can be described by Ising model is an important part of physics. This paper discusses the phase transition of the low-dimensional Ising model. For one-dimensional Ising model, the exact solution is given and no phase transition is found when there is no external magnetic field. For two-dimensional Ising model, Monte Carlo method along with importance sampling, careful balance conditions and the Metropolis algorithm is introduced to numerically simulate it. The trends of order parameters like magnetization, specific heat and magnetic susceptibility are analyzed and the mutation of some order parameters is found around critical temperature. The influences of external magnetic field and the scale of system are discussed and critical temperature of phase transition is explored by several methods, which is eventually found around 2.269 ( kBT/J). Lastly, compared with theoretical values, the disadvantages are pointed out and optimization suggestions are raised for improvement.
We simulate the Ising Model with the Monte Carlo method and use the algorithms of Metropolis to update the distribution of spins. We found that, in the specific case of the three-dimensional Ising Model, methods of Metropolis are efficient. Studying the system near the point of phase transition, we observe that the magnetization goes to zero. In our simulations we analyzed the behavior of the magnetization and magnetic susceptibility to verify the phase transition in a paramagnetic to ferromagnetic material. The behavior of the magnetization and of the magnetic susceptibility as a function of the temperature suggest a phase transition around KT/J ≈ 4.5 and was evidenced the problem of finite size of the lattice to work with large lattice.
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.
Abstract Phase transitions play a central role in statistical physics and are well understood in systems at thermal equilibrium. However, their nonequilibrium counterparts remain less explored because of the lack of a unified theoretical framework. In this study, we examine the potential of deep learning, specially convolutional neural networks (CNN), to detect nonequilibrium phase transitions (NEPT) in a 2D Ising model-like square lattice ferromagnetic system under effective parameter h . Without this parameter ( h = 0 ), we generate equilibrium spin configurations using the standard Metropolis algorithm as usual. With h ≠ 0 , we generate nonequilibrium spin configurations using an effective Glauber update rule which breaks detailed balance. While the former is used to train CNN, the later is not subsequently described by the usual Boltzmann-Gibbs distribution rather used to test the model. The study shows that the trained CNN can distinguish between ordered and disordered nonequilibrium steady states and accurately predict the critical temperature of the system undergoing NEPT. Finite size scaling analysis of the model’s output (perdition) reproduces critical exponents that agree with theoretical expectations, indicating that this method provides a strong, data driven approach for examining nonequilibrium critical phenomena.
This paper presents the investigation of convolutional neural network (CNN) prediction successfully recognizing the temperature of the nonequilibrium phase transitions in two-dimensional (2D) Ising spins on a square lattice. The model uses image snapshots of ferromagnetic 2D spin configurations as an input shape to provide the average output predictions. By considering supervised machine learning techniques, we perform Metropolis Monte Carlo (MC) simulations to generate the configurations. In the equilibrium Ising model, the Metropolis algorithm respects detailed balance condition (DBC), while its nonequilibrium version violates DBC. Violating the DBC of the algorithm is characterized by a parameter −8<ε<8. We find the exact result of the transition temperature Tc(ε) in terms of ε. If we set ε=0, the usual single spin-flip algorithm can be restored, and the equilibrium configurations generated with such a set up are used to train our model. For ε≠0, the system attains the nonequilibrium steady states (NESS), and the modified algorithm generates NESS configurations (test dataset). The trained model is successfully tested on the test dataset. Our result shows that CNN can determine Tc(ε≠0) for various ε values, consistent with the exact result.