The Wolff algorithm, named after Ulli Wolff, is an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

for

Monte Carlo simulation Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...

of the

Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...

and

Potts model In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenom ...

in which the unit to be flipped is not a single spin (as in the

heat bath In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, radiation, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-at ...

or Metropolis algorithms) but a cluster of them. This cluster is defined as the set of connected spins sharing the same spin states, based on the Fortuin-Kasteleyn representation. The Wolff algorithm is similar to the

Swendsen–Wang algorithm The Swendsen–Wang algorithm is the first non-local or cluster algorithm for Monte Carlo simulation for large systems near criticality. It has been introduced by Robert Swendsen and Jian-Sheng Wang in 1987 at Carnegie Mellon. The original a ...

, but different in that the former only flips one randomly chosen cluster with probability 1, while the latter flip every cluster independently with probability 1/2. It is shown numerically that flipping only one cluster decreases the

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...

time of the spin statistics. The advantage of Wolff algorithm over other algorithms for magnetic spin simulations like single spin flip is that it allows non-local moves on the energy. One important consequence of this is that in some situations (e.g. ferromagnetic Ising model or fully frustrated Ising model), the scaling of the Multicanonic simulation is

N^2

, better than

N^

, where z is the exponent associated with the critical slowing down phenomena.

References

* * *

External links

''Cluster Algorithms''
at

Netlib Netlib is a repository of software for scientific computing maintained by AT&T, Bell Laboratories, the University of Tennessee and Oak Ridge National Laboratory. Netlib comprises many separate programs and libraries. Most of the code is written in ...

*Implementation in Julia: https://github.com/cossio/SquareIsingModel.jl Monte Carlo methods Statistical mechanics {{statisticalmechanics-stub