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 ($NVT$ )

• Isothermal-Isobaric ensemble ($NpT$)

• Grand canonical ensemble ($\mu VT$)

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 number generators.

MMC 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 $\left.X\right.$ ) 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 $\Pi \left(X|k\right)$, depends on the parameters $k$ (e.g. temperature, pressure)

Example:

$\Pi _{NVT}(X|T)\propto \exp \left[-{\frac {U(X)}{k_{B}T}}\right]$

In most of the cases $\Pi \left(X|k\right)$ exhibits the following features:

• It is a function of many variables
• Only for a very small fraction of the configurational space the value of $\Pi \left(X|k\right)$ is not negligible

Due to these properties, MMC requires the use of Importance Sampling techniques

• To be improved.... hopefully!

Temperature

The temperature is usually fixed in MMC 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 MMC simulations in the microcanonical ensemble (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
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)