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## Monte Carlo method in statistical physics
## Additional recommended knowledge
## Simple samplingFor simplicity, let us consider the statistical mechanics . It therefore follows that the average value of any quantity that likewise
depends on the states , where the normalizing factor is the . which takes care of the lack of normalization of the probability distribution.
(Indeed, one sees that the average of any constant number < One possible approach is then to exactly enumerate all possible configurations of the system, and calculate averages at will. This is actually done in exactly solvable systems, and in simulations of simple systems with few particles. In realistic systems, on the other hand, even an exact enumeration can be difficult to implement (specially for off-lattice systems). One then may recur to the Monte Carlo method and generate configurations at random. The problem with this approach is that the states that typically contribute with sizable probabilities have a small measure (in the mathematical sense). This method has been likened to calculating the length of the River Nile by considering a set of point on top of a map of Northern Africa (Frenkel and Smit, reference below.) ## Biased samplingTherefore, the Metropolis algorithm is typically employed. The core of the method is the generation of states that obey Boltzmann probabilities. Any average quantity thus sampled is then, by construction, the statistical average that is sought: , where The algorithm employs a Markov chain procedure in order to determine a
A very high value of In order to fully prescribe an algorithm, a basic condition of
## CriticismThe method thus neglects dynamics, which can be a major drawback, or a great advantage. Indeed, the method can only be applied to static quantities, but the freedom to choose moves makes the method very flexible. An additional advantage is that some systems, such as the Ising model, lack a dynamical description and are only defined by an energy prescription; for these the Monte Carlo approach is the only one feasible.
## See also## References- Allen, M.P. and Tildesley, D.J. (1987).
*Computer Simulation of Liquids*. Oxford University Press.__ISBN 0-19-855645-4__. - Frenkel, D. and Smit, B. (2001).
*Understanding Molecular Simulation*. Academic Press.__ISBN 0-12-267351-4__. - Binder, K. and Heermann, D.W. (2002).
*Monte Carlo Simulation in Statistical Physics. An Introduction (4th edition)*. Springer.__ISBN 3-540-43221-3__.
Categories: Computational chemistry | Theoretical chemistry |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Monte_Carlo_method_in_statistical_physics". A list of authors is available in Wikipedia. |