Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes
This thesis discusses scaling and critical behavior of two different models. One model describes Ising dipoles, originates in condensed matter physics and depicts equilibrium critical phenomena. The other model, taken from the earth sciences, describes faulting instabilities and the resulting earthqnakes, and involves self-organized criticality — a nonequilibrium phenomenon. Both models are characterized by long range interactions, with a resulting sensitivity to boundary conditions. The ordering properties of Ising dipoles on lattices are studied in a mean field theory and by Monte Carlo simulations. The mean field theory is manifestly shape independent in zero external field. In the case of dipoles on a diluted lattice the mean field theory predicts a critical concentration above which the low temperature phase is ferroelectric (or anti-ferroelectric depending on the lattice structure). Extensive Monte Carlo simulation results are in agreement with those of mean field theory. We propose a finite size scaling form that includes logarithmic corrections for systems at the critical dimensionality. In the case of dipoles on a body centered tetragonal lattice we found that the finite scaling form significantly improved the data collapse over the scaling form with mean field exponents. With lattice parameters appropriate to the Ising ferromagnetic compound LiHoF4,we obtain a ferromagnetic transition temperature Tc= 1.51K in excellent agreement with experiment. This indicates that the material LiHoF4 is dominated by the dipole-dipole interaction; since in the simulations we only include dipole-dipole interactions. For dipoles on the simple cubic lattice, the ordered state is made up of anti-ferromagnetic rows. The critical exponents obtained by finite size scaling are ß~1/7,y ~ 8/7 and a ~4/7. These results are in good agreement with those of high temperature series expansions. A model of self—organized ruptures in an elastic medium is developed; and applied to earthquakes. In the model the local ruptures are represented by double couples to be consistent with elastic theory. The explicit form of this double couple source is derived. The system is driven by slowly increasing the shear stress. The model evolves towards a self-organized critical state in which the earthquake distribution follows the Gutenberg Richter law with an exponent in agreement with observational data. By modeling the local static fatigue for the rocks, we also obtained Omori’s law for the rate of aftershocks. The effects of annealing are investigated.
This thesis discusses scaling and critical behavior of two different models. One model describes Ising dipoles, originates in condensed matter physics and depicts equilibrium critical phenomena. The other model, taken from the earth sciences, describes faulting instabilities and the resulting earthqnakes, and involves self-organized criticality — a nonequilibrium phenomenon. Both models are characterized by long range interactions, with a resulting sensitivity to boundary conditions. The ordering properties of Ising dipoles on lattices are studied in a mean field theory and by Monte Carlo simulations. The mean field theory is manifestly shape independent in zero external field. In the case of dipoles on a diluted lattice the mean field theory predicts a critical concentration above which the low temperature phase is ferroelectric (or anti-ferroelectric depending on the lattice structure). Extensive Monte Carlo simulation results are in agreement with those of mean field theory. We propose a finite size scaling form that includes logarithmic corrections for systems at the critical dimensionality. In the case of dipoles on a body centered tetragonal lattice we found that the finite scaling form significantly improved the data collapse over the scaling form with mean field exponents. With lattice parameters appropriate to the Ising ferromagnetic compound LiHoF4,we obtain a ferromagnetic transition temperature Tc= 1.51K in excellent agreement with experiment. This indicates that the material LiHoF4 is dominated by the dipole-dipole interaction; since in the simulations we only include dipole-dipole interactions. For dipoles on the simple cubic lattice, the ordered state is made up of anti-ferromagnetic rows. The critical exponents obtained by finite size scaling are ß~1/7,y ~ 8/7 and a ~4/7. These results are in good agreement with those of high temperature series expansions. A model of self—organized ruptures in an elastic medium is developed; and applied to earthquakes. In the model the local ruptures are represented by double couples to be consistent with elastic theory. The explicit form of this double couple source is derived. The system is driven by slowly increasing the shear stress. The model evolves towards a self-organized critical state in which the earthquake distribution follows the Gutenberg Richter law with an exponent in agreement with observational data. By modeling the local static fatigue for the rocks, we also obtained Omori’s law for the rate of aftershocks. The effects of annealing are investigated.
The ferromagnetic Ising model with antiferromagnetic dipole interactions is investigated by means of Monte Carlo simulations, focusing on the characterization of the phase transitions between the tetragonal liquid and stripe of width h phases. The dynamic evolution of the physical observables is analyzed within the short-time regime for 0.5≤δ≤1.3, where δ is the ratio between the short-range exchange and the long-range dipole interaction constants. The obtained results for the interval 0.5≤δ≤1.2 indicate that the phase transition line between the h=1 stripe and tetragonal liquid phases is continuous. This finding contributes to clarifying the controversy about the order of this transition. This controversy arises from the difficulties introduced in the simulations due to the presence of long-range dipole interactions, such as an important increase in the simulation times that limits the system size used, strong finite size effects, as well as to the existence of multiple metastable states at low temperatures. The study of the short-time dynamics of the model allows us to avoid these hindrances. Moreover, due to the fact that the finite-size effects do not significantly affect the power-law behavior exhibited in the observables within the short-time regime, the results could be attributed to those corresponding to the thermodynamic limit. As a consequence of this, a careful characterization of the critical behavior for the whole transition line is performed by giving the complete set of critical exponents.
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 chapter one, we explain briefly the continuum limit, scaling, and high temperature expansion of critical phenomena, Monte Carlo algorithms and fitting. In chapter two, different continuum limits of the Ising model in dimensions (D) 2, 3 and 4 are investigated numerically. The data indicate that triviality occurs for D = 4 and fails for D < 4 in each limit. In chapter three, a relation between the critical exponents of the leading and confluent scaling terms is derived using the finite size scaling argument. We also determine the new scaling variable of the 4D Ising model based on a new Monte Carlo simulation data. In chapter four, a Monte Carlo study of two dimensional diluted Ising systems is reported. It is shown that regular dilution does not affect critical exponents, while a random one does, with critical exponents varying continuously with impurity concentration. The importance of fluctuations in producing such effects is emphasized. In chapter five, a different point of view regarding the critical exponent of the specific heat of the 3D Ising model is presented. Based on the analysis of high temperature expansion, finite size scaling and Monte Carlo data in the symmetric phase of the 3D Ising model, it is shown that logarithmic scaling behavior of specific heat is more consistent than power scaling behavior.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behavior of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form rd+σ1, with σ=0.75. The critical temperature, as well as the critical exponents, is evaluated from the power-law behavior of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different cutoffs of the interaction range. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν) are in good agreement with Renormalization Group (RG) predictions and previously reported numerical data obtained under equilibrium conditions. It is found that the dynamic exponent z is different for quenching and annealing experiments, most likely due to the influence of the Kosterlitz-Thouless transition occurring at relatively similar algebraic decay of the interactions with σ=1. However, for annealing experiments the measured exponent z is close to the RG predictions. On the other hand, the relevant exponents of the dynamic behavior (z and θ) are slightly different than the RG predictions, most likely due to the fact that they may depend on the especific dynamics used (Metropolis in the present paper).
Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).
Critical behavior of Ising ferromagnetic systems with the competition between short- and long-range interactions near the point of a second-order phase transition has been investigated. The effects of long-range interaction lead to the need to calculate the influence of next-nearest spins, which lie inside the sphere of a specific radius. The study was carried out for a power law of decreaseing the exchange integral with distance for a three-dimensional Ising model with a cubic lattice. A comparison with the results of a field-theoretical approach and data of real experiments has been carried out. It has been shown that the critical temperature grows linearly with an increase in the parameters of long-range interaction. Critical exponents have been calculated at different parameters of long-range interaction. It is shown that at small dimensions of a system it is the short-range interaction that exerts the dominant effect on the critical behavior of the system. With an increase in the linear dimension of the system, long-range forces begin to dominate.