User:Jackliddle/Metropolis algorithm

The Metropolis Algorithm is used in the Monte Carlo methods for importance sampling probablity distributions.

The algorithm is widely used in lattice gauge theory.

For generating a sequence of field configurations ${\displaystyle \Sigma }$ distributed with a probability density ${\displaystyle e^{-S}}$. Expectation values of Quantum Mechanical observables can then be calculated

${\displaystyle <O>={\frac {1}{n}}\sum _{i=m+1}^{m+n}O(\Sigma _{i})}$

where the first m steps bring the system into equilbrium.

If we change one link in the field configuration ${\displaystyle \Sigma }$ giving us a new configuation ${\displaystyle \Sigma '}$ there is a corresponding change in the action.

${\displaystyle \Delta S=S(\Sigma ')-S(\Sigma )}$

using Wikicode

• Metropolis-Hastings algorithm