Editing Metropolis Monte Carlo
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== Importance sampling == | == Importance sampling == | ||
The importance sampling is useful to evaluate average values given by: | |||
: <math> \langle A(X|k) \rangle = \int dX \Pi(X|k) A(X) </math> | : <math> \langle A(X|k) \rangle = \int dX \Pi(X|k) A(X) </math> | ||
where: | where: | ||
* <math> \left. X \right. </math> represents a set of many variables, | * <math> \left. X \right. </math> represents a set of many variables, | ||
* <math> \left. \Pi \right. </math> is a probability distribution function which depends on <math> X </math> and on the constraints (parameters) <math> k </math> | * <math> \left. \Pi \right. </math> is a probability distribution function which depends on <math> X </math> and on the constraints (parameters) <math> k </math> | ||
* <math> \left. A \right. </math> is an observable which depends on the <math> X </math> | * <math> \left. A \right. </math> is an observable which depends on the <math> X </math> | ||
Depending on the behavior of <math> \left. \Pi \right. </math> we can use to compute <math> \langle A(X|k) \rangle </math> different numerical methods: | Depending on the behavior of <math> \left. \Pi \right. </math> we can use to compute <math> \langle A(X|k) \rangle </math> different numerical methods: | ||
* If <math> \left. \Pi \right. </math> is, roughly speaking, quite uniform: [[Monte Carlo Integration]] methods can be effective | * If <math> \left. \Pi \right. </math> is, roughly speaking, quite uniform: [[Monte Carlo Integration]] methods can be effective | ||
* If <math> \left. \Pi \right. </math> has significant values only for a small part of the configurational space, Importance sampling could be the appropriate technique | * If <math> \left. \Pi \right. </math> has significant values only for a small part of the configurational space, Importance sampling could be the appropriate technique | ||
'''Sketches of the Method:''' | |||
* Random walf over <math> \left. X \right. </math>: | |||
: <math> \left. X_{i+1}^{test} = X_{i} + \delta X \right. </math> | : <math> \left. X_{i+1}^{test} = X_{i} + \delta X \right. </math> | ||
From the configuration at the i-th step | From the configuration at the i-th step we build up a ''test'' configuration by modifying a bit (some of) the variables <math> X </math> | ||
* The test configuration is accepted as the new (i+1)-th configuration with certain criteria (which depends basically on <math> \Pi </math>) | * The test configuration is accepted as the new (i+1)-th configuration with certain criteria (which depends basically on <math> \Pi </math>) | ||
* If the test configuration is not accepted as the new configuration then: <math> \left. X_{i+1} = X_i \right. </math> | * If the test configuration is not accepted as the new configuration then: <math> \left. X_{i+1} = X_i \right. </math> | ||
The procedure is based on the [[Markov chain]] formalism, and on the [[Perron-Frobenius theorem]]. | The procedure is based on the [[Markov chain]] formalism, and on the [[Perron-Frobenius theorem]]. | ||
The acceptance criteria must be chosen to guarantee that after a certain equilibration ''time'' a given configuration appears with probability given by <math> \Pi(X|k) </math> | |||
The acceptance criteria must be chosen to guarantee that after a certain equilibration ''time'' a given configuration appears with | |||
probability given by <math> \Pi(X|k) </math> | |||
== Temperature == | == Temperature == |