Monte Carlo method in statistical mechanics

Monte Carlo in statistical physics refers to the application of the Monte Carlo method to problems in statistical physics, or statistical mechanics. 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.

It employs a Markov chain procedure in order to determine a new state for a system from a previous one. According to its stochastic nature, this new state is accepted at random, with an acceptance probability that depends on the Boltzmann factor of the energy increment. The 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.

A basic condition of balance must be satisfied in order equilibrium be properly described, but detailed balance, a stronger condition, is usually imposed when designing new algorithms.

The great success of this method in statistical mechanics has led to various generalizations such as the method of simulated annealing for optimization, in which a fictitious temperature is introduced and then gradually lowered.

See also

• Quantum Monte Carlo
• Monte Carlo molecular modeling

References

• Allen, M.P. and Tildesley, D.J. (1987). Computer Simulation of Liquids. Oxford University Press. ISBN 0-19-855645-4.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Frenkel, D. and Smit, B. (2001). Understanding Molecular Simulation. Academic Press. ISBN 0-12-267351-4.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Binder, K. and Heermann, D.W. (2002). Monte Carlo Simulation in Statistical Physics. An Introduction (4th edition). Springer. ISBN 3-540-43221-3.{{cite book}}: CS1 maint: multiple names: authors list (link)