Editing Metropolis Monte Carlo

The '''Metropolis Monte Carlo''' technique (Ref. 2) is a variant of the original [[Monte Carlo]] method proposed by [[Nicholas Metropolis]] and [[Stanislaw Ulam]] in 1949 (Ref. 1).
== Main features ==
Metropolis Monte Carlo simulations can be carried out in different ensembles. For the case of one-component systems the usual ensembles are:
* [[Canonical ensemble]] (<math> NVT </math> )
* [[Isothermal-isobaric ensemble]] (<math> NpT </math>)
* [[Grand canonical ensemble]] (<math> \mu V T </math>)
In the case of mixtures, it is useful to consider the so-called [[Semi-grand ensembles]].
The purpose of these techniques is to sample representative configurations of the system at the corresponding
thermodynamic conditions. The sampling techniques  make use the so-called pseudo-[[Random numbers |random number]] generators.

== Configuration ==

A configuration is a microscopic realisation of the ''thermodynamic state'' of the system.
To define a configuration (denoted as <math> \left. X \right. </math> ) we usually require:
*The position coordinates of the particles
*Depending on the problem, other variables like volume, number of particles, etc.
The probability of a given configuration, denoted as <math> \Pi \left(  X | k \right)  </math>,
depends on the parameters <math> k </math>  (e.g. [[temperature]], [[pressure]])

Example: 

:<math> \Pi_{NVT}(X|T) \propto \exp \left[ - \frac{ U (X) }{k_B T} \right] </math>

In most of the cases <math> \Pi \left(  X | k \right)  </math> exhibits the  following features:
* It is a function of many variables
* Only for a very small fraction of the configurational space the value of <math> \Pi \left(  X | k \right)  </math> is not negligible. 
Due to these properties, Metropolis Monte Carlo requires the use of '''Importance Sampling''' techniques

== Importance sampling ==

Importance sampling is useful to evaluate average values given by:

: <math> \langle A(X|k) \rangle = \int dX \Pi(X|k) A(X) </math>

where:
* <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. 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:
* 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
====Outline of the Method====
* Random walk over <math> \left. X \right. </math>: 

: <math> \left. X_{i+1}^{test} = X_{i} + \delta X \right. </math> 

From the configuration at the i-th step one builds up a ''test'' configuration by slightly modifying 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>)
* 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 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 ==

The [[temperature]] is usually fixed in Metropolis Monte Carlo simulations, since in classical statistics the kinetic degrees of freedom (momenta) can be generally, integrated out. However, it is possible to design procedures to perform Metropolis Monte Carlo simulations in the [[Microcanonical ensemble| microcanonical ensemble]] (NVE).

See [[Monte Carlo in the microcanonical ensemble]]

== Boundary Conditions ==

The simulation of homogeneous systems is usually carried out using [[boundary conditions |periodic boundary conditions]].

== Initial configuration ==

The usual choices for the initial configuration in fluid simulations are:

* an equilibrated configuration under similar conditions (for example see Ref. 3)

* an ordered lattice structure. For details concerning the construction of such structures see: [[Lattice Structures | lattice structures]].
==Interesting reading==
*[http://links.jstor.org/sici?sici=0162-1459%28194909%2944%3A247%3C335%3ATMCM%3E2.0.CO%3B2-3 Nicholas Metropolis and S. Ulam "The Monte Carlo Method", Journal of the American Statistical Association '''44''' pp. 335-341 (1949)]
*[http://library.lanl.gov/cgi-bin/getfile?00326886.pdf Herbert L. Anderson "Metropolis, Monte Carlo, and the MANIAC", Los Alamos Science '''14''' pp. 96-107 (1986)]
*[http://library.lanl.gov/cgi-bin/getfile?00326866.pdf N. Metropolis "The Beginnning of the Monte Carlo Method"  Los Alamos Science '''15''' pp. 125-130 (1987)]
*[http://cise.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=CSENFA000002000001000065000001&idtype=cvips&prog=normal      Isabel Beichl and Francis Sullivan "The Metropolis Algorithm",  Computing in Science & Engineering '''2''' Issue 1 pp. 65-69 (2000)]
*[http://dx.doi.org/10.1063/1.1632112 Marshall N. Rosenbluth "Genesis of the Monte Carlo Algorithm for Statistical Mechanics", AIP Conference Proceedings '''690'''  pp. 22-30 (2003)]
*[http://dx.doi.org/10.1063/1.1632114 Marshall N. Rosenbluth "Proof of Validity of Monte Carlo Method for Canonical Averaging", AIP Conference Proceedings '''690'''  pp. 32-38 (2003)]
*[http://dx.doi.org/10.1063/1.1632115 William W. Wood "A Brief History of the Use of the Metropolis Method at LANL in the 1950s", AIP Conference Proceedings '''690'''  pp. 39-44 (2003)]
*[http://dx.doi.org/10.1063/1.1632124 David P. Landau "The Metropolis Monte Carlo Method in Statistical Physics", AIP Conference Proceedings '''690'''  pp. 134-146 (2003)]

== Advanced techniques ==
* [[Configurational bias Monte Carlo]]
* [[Gibbs-Duhem integration]]
* [[Cluster algorithms]]
*[[Computing the Helmholtz energy function of solids]]
*[[Wang-Landau method]]
== References ==
#[http://links.jstor.org/sici?sici=0162-1459%28194909%2944%3A247%3C335%3ATMCM%3E2.0.CO%3B2-3 Nicholas Metropolis and S. Ulam "The Monte Carlo Method", Journal of the American Statistical Association '''44''' pp. 335-341 (1949)]
#[http://dx.doi.org/10.1063/1.1699114  Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller and Edward Teller, "Equation of State Calculations by Fast Computing Machines", Journal of Chemical Physics '''21''' pp.1087-1092  (1953)]
#[http://dx.doi.org/10.1016/j.cpc.2005.02.006  Carl McBride, Carlos Vega and Eduardo Sanz "Non-Markovian melting: a novel procedure to generate initial liquid like phases for small molecules for use in computer simulation studies", Computer Physics Communications  '''170''' pp. 137-143 (2005)]
[[Category: Monte Carlo]]