Metropolis Monte Carlo

Metropolis Monte Carlo (MMC)

Main features

MMC 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 } )

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

MMC makes use of importance sampling tecniques

Configuration

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

To define a configuration (denoted as $\left.X\right.$ ) 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: 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)

Temperature

The temperature is usually fixed in MMC simulations, since in clasical statistics the kinetic degrees of freedom (momenta) can be generally, integrated out.

However, it is possible to design procedures to perform MMC simulations in the microcanonical ensembe (NVE).

Boundary Conditions

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

Advanced techniques

• Configurational bias Monte Carlo

• Gibbs-Duhem Integration

• Cluster algorithms

References

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