Metropolis Monte Carlo

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 ($NVT$ )

• 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.

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 $\left.X\right.$ ) 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, Metropolis Monte Carlo requires the use of Importance Sampling techniques

Importance sampling

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

$\langle A(X|k)\rangle =\int dX\Pi (X|k)A(X)$

where:

• $\left.X\right.$ represents a set of many variables,
• $\left.\Pi \right.$ is a probability distribution function which depends on $X$ and on the constraints (parameters) $k$
• $\left.A\right.$ is an observable which depends on the $X$

Depending on the behavior of $\left.\Pi \right.$ we can use to compute $\langle A(X|k)\rangle$ different numerical methods:

• If $\left.\Pi \right.$ is, roughly speaking, quite uniform: Monte Carlo Integration methods can be effective
• If $\left.\Pi \right.$ 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 $\left.X\right.$:

$\left.X_{i+1}^{test}=X_{i}+\delta X\right.$

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

• The test configuration is accepted as the new (i+1)-th configuration with certain criteria (which depends basically on $\Pi$)
• If the test configuration is not accepted as the new configuration then: $\left.X_{i+1}=X_{i}\right.$

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 $\Pi (X|k)$

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).

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)