Metropolis Monte Carlo

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 significant 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 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 $\Pi (X|k)$

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 (for example see Ref. 3)

• an ordered lattice structure. For details concerning the construction of such structures see: lattice structures.

Interesting reading

1. Nicholas Metropolis and S. Ulam "The Monte Carlo Method", Journal of the American Statistical Association 44 pp. 335-341 (1949)
2. Herbert L. Anderson "Metropolis, Monte Carlo, and the MANIAC", Los Alamos Science 14 pp. 96-107 (1986)
3. N. Metropolis "The Beginnning of the Monte Carlo Method" Los Alamos Science 15 pp. 125-130 (1987)
4. Marshall N. Rosenbluth "Genesis of the Monte Carlo Algorithm for Statistical Mechanics", AIP Conference Proceedings 690 pp. 22-30 (2003)
5. Marshall N. Rosenbluth "Proof of Validity of Monte Carlo Method for Canonical Averaging", AIP Conference Proceedings 690 pp. 32-38 (2003)
6. 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)
7. David P. Landau "The Metropolis Monte Carlo Method in Statistical Physics", AIP Conference Proceedings 690 pp. 134-146 (2003)
8. Isabel Beichl and Francis Sullivan "The Metropolis Algorithm", Computing in Science & Engineering 2 Issue 1 pp. 65-69 (2000)

Advanced techniques

• Configurational bias Monte Carlo
• Gibbs-Duhem integration
• Cluster algorithms
• Computing the Helmholtz energy function of solids
• Wang-Landau method

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)
3. 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)