{"id":18975748,"url":"https://github.com/babaid/ising-model-simulation","last_synced_at":"2026-01-28T04:33:13.361Z","repository":{"id":167210784,"uuid":"455133067","full_name":"babaid/Ising-Model-Simulation","owner":"babaid","description":"A project containing  simulations of Ising Chains known from theoretical physics. 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In the model ferromagnetism arrises from the interaction of the spins of electrons. These interactions are described by the following Hamiltonian:\n\n$$\nH = -J \\sum_{\\langle i, j \\rangle}s_i s_j - B\\sum_i s_i\n$$\n\nwhere in the sum $ \\sum_{\\langle i, j \\rangle}$ only the interaction between nearest neighbors is taken into account. To deal with the finite dimension of the system simulated, periodic boundary conditions are introduced, so one usually speaks of Ising chains:\n\n$$\ns_{i+N} = s_i\n$$\n\nFirstly we will look at situations with B = 0, since most of the questions we are interested in can still be answered and calculations are a bit simplier.\n\n## Simulation? What and how? - Metropolis-Hatings algorithm\nSince we are speaking in terms of quantum mechanics, statistics is going to be involved and the concept that systems try to occupy the lowest energetic states. The whole procedure involves calculating the energy for a current state, then flipping a random spin, calculating the new energy of the system. If the energy difference is negative then the probability that the spin-flip happens is 1, if its positive one calculates a transition probability for the spin-flip and samples from the according probability distribution, to determine wether the spin will be flipped or not.\nThis method is called Metropolis-Hastings algorithm, which is a pretty combinatorial method, mostly used if one wants to sample from an unknown, very complicated distribution.\nWhich spin will be flipped, is going to be determined by a random walk, wich is probably the simplest, and fastest and computationally inexpensivest way of doing this.\n\nThe observables of interest are going to be calculated using the Monte Carlo technique:\n\n$$\nA = \\int_V dx f(x) = \\lim_{n\\to\\infty}\\frac{1}{N}\\sum_{i=1}^{N}f(x_i)\n$$\n\n\n______________________________________________________________________________\n\nAs of now, the most accurate simulation is the 8. in the plots folder. \n\n_________________________________________________________\n\nThis code base is using the Julia Language and [DrWatson](https://juliadynamics.github.io/DrWatson.jl/stable/)\nto make a reproducible scientific project named\n\u003e Ising-Model-Simulation\n\nTo (locally) reproduce this project, do the following:\n\n0. Download this code base. Notice that raw data are typically not included in the\n   git-history and may need to be downloaded independently.\n1. Open a Julia console and do:\n   ```\n   julia\u003e using Pkg\n   julia\u003e Pkg.add(\"DrWatson\") # install globally, for using `quickactivate`\n   julia\u003e Pkg.activate(\"path/to/this/project\")\n   julia\u003e Pkg.instantiate()\n   ```\n\nThis will install all necessary packages for you to be able to run the scripts and\neverything should work out of the box, including correctly finding local paths.\n","funding_links":[],"categories":[],"sub_categories":[],"project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fbabaid%2Fising-model-simulation","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fbabaid%2Fising-model-simulation","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fbabaid%2Fising-model-simulation/lists"}