Ising_MatLab - Monte Carlo investigation of the Ising model...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 [2]: ...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....
View Full Document

Page1 / 7

Ising_MatLab - Monte Carlo investigation of the Ising model...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online