Metropolis Monte Carlo: Difference between revisions
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The probability of a given configuration, denoted as <math> \Pi \left( X | k \right) </math>, | The probability of a given configuration, denoted as <math> \Pi \left( X | k \right) </math>, | ||
depends on the parameters <math> k </math> (e.g. temperature, pressure) | depends on the parameters <math> k </math> (e.g. temperature, pressure) | ||
Example: | |||
<math> \Pi_{NVT}(X|T) \propto \exp \left[ - \frac{ U (X) }{k_B T} \right] </math> | |||
In most of the cases <math> \Pi \left( X | k \right) </math> exhibits the following features: | In most of the cases <math> \Pi \left( X | k \right) </math> exhibits the following features: | ||
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Due to these properties, MMC requires the use of '''Importance Sampling''' techniques | Due to these properties, MMC requires the use of '''Importance Sampling''' techniques | ||
* To be improved.... hopefully! | |||
== Importance sampling == | == Importance sampling == | ||
Revision as of 18:24, 23 February 2007
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:
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 ) 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 , depends on the parameters (e.g. temperature, pressure)
Example:
![{\displaystyle \Pi _{NVT}(X|T)\propto \exp \left[-{\frac {U(X)}{k_{B}T}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/236c44d391db957491afdb8af9f85e75d5e8d27a)
In most of the cases exhibits the following features:
- It is a function of many variables
- Only for a very small fraction of the configurational space the value of is not negligible
Due to these properties, MMC requires the use of Importance Sampling techniques
- To be improved.... hopefully!
Importance sampling
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
References
- M.P. Allen and D.J. Tildesley "Computer simulation of liquids", Oxford University Press
- 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)