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.

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

$$ = \frac{1}{n} \sum_{i=m+1}^{m+n} O(\Sigma_i)$$

where the first m steps bring the system into equilbrium.

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

$$\Delta S = S(\Sigma') - S(\Sigma)$$

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