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Unformatted text preview: Monte Carlo investigation of the Ising model Tobin Fricke December 2006 1 The Ising Model The Ising Model is a simple model of a solid that exhibits a phase transition resembling ferromagnetism. In this model, a spin direction is assigned to each vertex on a graph. The standard Hamiltonian for an Ising system includes only nearest-neighbor interactions and each spin direction may be either up (+1) or down (-1), though generalized models may include long-range interactions and more choices for spin direction. The standard Hamiltonian is: H =- J X neighbors S i S j (1) In this note I consider an Ising system on a square grid, where each spin interacts directly with four neighbors (above, below, to the right, and to the left). 2 The Metropolis Algorithm We wish to draw sample configurations of Ising systems in thermal equilibrium at a given temperature. The standard method of acquiring these sample configurations is via the use of the Metropolis algorithm, or a variant, in a Monte Carlo loop. The Metropolis algorithm is explained succinctly in the original paper : ...the method we employ is actually a modified Monte Carlo scheme, where, instead of choosing configurations randomly, then weighting them with exp(- E/kT ), we choose configurations with a probability exp(- E/kT ) and weight them evenly. This we do as follows: We place the N particles in any configuration, for example, in a regular lattice. Then we move each of the particles in succession... We then calculate the change in 1 energy of the system E , which is caused by the move. If E < 0, i.e., if the move would bring the system to a state of lower energy, we allow the move and put the particle in its new position. If E > 0, we allow the move with probability exp(- E/kT )... Then, whether the move has been allowed or not, i.e., whether we are in a different configuration or in the original configuration, we consider that we are in a new configuration for the purpose of taking our averages. In essense, the Metropolis algorithm consists of a guided random walk through phase space. The distri- bution of states visited on this walk are consistent with the canonical ensemble at the given temperature....
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