Editing Metropolis Monte Carlo
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
| Latest revision | Your text | ||
| Line 31: | Line 31: | ||
== 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 == | ||