Monte Carlo technique for very large ising models (original) (raw)

Monte Carlo methods in sequential and parallel computing of 2D and 3D ising model

Because of its complexity, the 3D Ising model has not been given an exact analytic solution so far, as well as the 2D Ising in non zero external field conditions. In real materials the phase transition creates a discontinuity. We analysed the Ising model that presents similar discontinuities. We use Monte Carlo methods with a single spin change or a spin cluster change to calculate macroscopic quantities, such as specific heat and magnetic susceptibility. We studied the differences between these methods. Local MC algorithms (such as Metropolis) perform poorly for large lattices because they update only one spin at a time, so it takes many iterations to get a statistically independent configuration. More recent spin cluster algorithms use clever ways of finding clusters of sites that can be updated at once. The single cluster method is probably the best sequential cluster algorithm. We also used the entropic sampling method to simulate the density of states. This method takes into account all possible configurations, not only the most probable. The entropic method also gives good results in the 3D case. We studied the usefulness of distributed computing for Ising model. We established a parallelization strategy to explore Metropolis Monte Carlo simulation and Swendsen-Wang Monte Carlo simulation of this spin model using the data parallel languages on different platform. After building a computer cluster we made a Monte Carlo estimation of 2D and 3D Ising thermodynamic properties and compare the results with the sequential computing. In the same time we made quantitative analysis such as speed up and efficiency for different sets of combined parameters (e.g. lattice size, parallel algorithms, chosen model).

A simulation of the ising model

The Ising Model provides an entirely new understanding of how phase transitions in various systems take place and gives us a bet- ter idea of the magnetic behavior/properties of certain systems. This project aims at analyzing phase transitions and magnetic properties of some systems through Monte Carlo simulations based on the 2D Ising Model.

Monte Carlo Simulations of a Disordered Binary Ising Model

International Journal of Modern Physics C, 2012

A disordered binary Ising model, with only nearest-neighbor spin exchange interactions J > 0 on the square lattice, is studied through Monte Carlo simulations. The system consists of two di®erent particles with spin-1/2 and spin-1, randomly distributed on the lattice. We found the critical temperatures for several values of the concentration x of spin-1/2 particles, and also the corresponding critical exponents. Int. J. Mod. Phys. C Downloaded from www.worldscientific.com by UNIVERSIDADE FEDERAL DE MATO GROSSO on 07/23/12. For personal use only. 1240015-3 Monte Carlo Simulations of a Disordered Binary Ising Model Int. J. Mod. Phys. C Downloaded from www.worldscientific.com by UNIVERSIDADE FEDERAL DE MATO GROSSO on 07/23/12. For personal use only.

Theory and Simulation of the Ising Model

2021

We have provided a concise introduction to the Ising model as one of the most important models in statistical mechanics and in studying the phenomenon of phase transition. The required theoretical background and derivation of the Hamiltonian of the model have also been presented. We finally have discussed the computational method and details to numerically solve the twoand threedimensional Ising problems using Monte Carlo simulations. The related computer codes in both Python and Fortran, as well as a simulation trick to visualize the spin lattice, have also been provided.

Monte Carlo study of growth in the two-dimensional spin-exchange kinetic Ising model

Physical Review B, 1988

Results obtained from extensive Monte Carlo simulations of domain growth in the twodimensional spin-exchange kinetic Ising model with equal numbers of up and down spins are presented. Using difterent measures of domain sizeincluding the pair-correlation function, the energy, and circularly-averaged structure factorthe domain size is determined (at T =0.5T,) as a function of time for times up to 10 Monte Carlo steps. The growth law R(t)= A +Bt'/ is found to provide an excellent fit (within 0.3%) to the data, thus indicating that at long times the classical value of-, for the exponent is correct. It is pointed out that this growth law is equivalent to an effective exponent for a1I times (as given by Huse) n, z(t)=-, '-, 'C/8 (t). No evidence for logarithmic behavior is seen. The self-averaging properties of the various measures of domain size and the variation of the constants A and 8 with temperature are also discussed. In addition, the scaling of the structure factor and anisotropy efrects due to the lattice are examined.

Three-dimensional ising model in the fixed-magnetization ensemble: A monte carlo study

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000

We study the three-dimensional Ising model at the critical point in the fixed-magnetization ensemble, by means of the recently developed geometric cluster Monte Carlo algorithm. We define a magnetic-field-like quantity in terms of microscopic spin-up and spin-down probabilities in a given configuration of neighbors. In the thermodynamic limit, the relation between this field and the magnetization reduces to the canonical relation M(h). However, for finite systems, the relation is different. We establish a close connection between this relation and the probability distribution of the magnetization of a finite-size system in the canonical ensemble.

A Monte Carlo Sampling Scheme for the Ising Model

Journal of Statistical Physics, 2000

In this paper we describe a Monte Carlo sampling scheme for the Ising model and similar discrete state models. The scheme does not involve any particular method of state generation but rather focuses on a new way of measuring and using the Monte Carlo data.

Study of 3 dimensional Ising model using Monte Carlo, Metropolis-Hastings Algorithm

In this project, I use Monte Carlo technique, particularly the Metropolis algorithm, to computationally realise the 3 dimensional Ising model, and use it to study various properties of the lattice, for example, the variation of its heat capacity with temperature, as it cools, or to see how average magnetisation varies with time as we reach equilibrium, etc. An interesting property of the lattice, which is otherwise not realisable in real world experiments, namely, the finite size limit, will also be studied.

Simulations of the low-dimensional Ising and Heisenberg models

1996

Thc critical coupling a,nd spontaneous magnetiza,tiott cttrvc for the 2ËIsing model arc ca,lc.ulated using the eÍIcctivc-Íicld rnethod with correlations. An application of thc nrcthod to thc qrrantutr'r .9=á 1rl-Heiscnberg nrodcl is prcscnted and rcliablc lowtcntpera,turc estimatcs nf thc specifrc ltcat a,re evalua,tcd' We <levelop l;he effective-field mel;hod rvith correlal,ions propcrsed in [l]. For a cla.ssical svsl,em rvil:h short; range int,eracl;ions the lattice ca,n alu'avs be <livided into a finil,e clrrster Í1. its borrntlary áO and O, the-complem:ent of O U dO, in sttclt a rvay tha't its Hamiltonian ca.n lre rvrit,l,en as-flTí:11o(O,AA) + H1(AA,O). The idea is l.o correct for l,he neglect of