Metropolis Monte Carlo

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Metropolis Monte Carlo (MMC)

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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle NVT } )
  • Isothermal-isobaric ensemble (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle NpT } )
  • Grand canonical ensemble (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu V T } )

In the case of mixtures, it is useful to consider the so-called:

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 number generators.

Metropolis Monte Carlo makes use of importance sampling techniques.

Configuration

A configuration is a microscopic realisation of the thermodynamic state of the system.

To define a configuration (denoted as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. X \right. } ) 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi \left( X | k \right) } , depends on the parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k } (e.g. temperature, pressure)

Example:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{NVT}(X|T) \propto \exp \left[ - \frac{ U (X) }{k_B T} \right] }

In most of the cases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi \left( X | k \right) } exhibits the following features:

  • It is a function of many variables
  • Only for a very small fraction of the configurational space the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi \left( X | k \right) } is not negligible

Due to these properties, Metropolis Monte Carlo requires the use of Importance Sampling techniques

Importance sampling

The importance sampling is useful to evaluate average values given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle A(X|k) \rangle = \int dX \Pi(X|k) A(X) }

where:

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. X \right. } represents a set of many variables,
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Pi \right. } is a probability distribution function which depends on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } and on the constraints (parameters) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. A \right. } is an observable which depends on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X }

Depending on the behavior of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Pi \right. } we can use to compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle A(X|k) \rangle } different numerical methods:

  • If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Pi \right. } is, roughly speaking, quite uniform: Monte Carlo Integration methods can be effective
  • If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Pi \right. } has significative values only for a small part of the configurational space, Importance sampling could be the appropriate technique


Sketches of the Method:

  • Random walf over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. X \right. } :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. X_{i+1}^{test} = X_{i} + \delta X \right. }

From the configuration at the i-th step we build up a test configuration by modifying a bit (some of) the variables

  • The test configuration is accepted as the new (i+1)-th configuration with certain criteria (which depends basically on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi } )
  • If the test configuration is not accepted as the new configuration then: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. X_{i+1} = X_i \right. }

The procedure is based on the Markov Chain formalism, and on the Perrom-Frobenius theorem.

The acceptance criteria must be chosen to guarantee that after a certain equilibration time a given configuration appears with probability given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi(X|k) }


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 (NVE).

See Monte Carlo in the microcanonical ensemble

Boundary Conditions

The simulation of homogeneous systems is usually carried out using periodic boundary conditions

Initial configuration

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

  • an equilibrated configuration under similar conditions,
  • an ordered lattice structure. For details concerning the construction of such structures see: lattice structures.

Advanced techniques

References

  1. M.P. Allen and D.J. Tildesley "Computer simulation of liquids", Oxford University Press
  2. 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)
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