Cluster algorithms
WORKING ON THIS PAGE ...
Cluster algorithms in Monte Carlo Simulation.
These algorithms are mainly used in the simulation of Ising-like models. The essential feature is the use of collective motions of particles (spins) in a single Monte Carlo step.
An interesting property of some of these application is the fact that the percolation analysis of the clusters can be used to study phase transitions.
As an introductory example we will discuss the Swendsen-Wang technique (Ref 1) in the simulation of Ising Models.
Sketches of the Swendsen-Wang algorithm
- Consider every pair interacting sites (spins)
In the current configuration the pair interaction can be either negative: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij} = -1 } of positive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij} = + 1 } , depending on the product: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{i} S_{j} } .