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The goal of this project is to design a 2D-Ising model using Monte Carlo simulation method.
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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.
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The report contains the analysis of the problem, the methodology used, the python code used, the results, and discussion.
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The Ising model, named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbours. Neighbouring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy, but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.
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This project 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 Metropolis-Hastings’s algorithm.