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

[edit] Simple sampling

For simplicity, let us consider the statistical mechanics $N V T$ ensemble. The internal energy $U$ of the system is given as a function of the states of all the particles in the system ${r i}$ (this can be the positions for a fluid, or the spin variables in a magnet). If the system obeys Boltzmann statistics, the probability that it will be in some state is proportional to an inverse exponential factor:

$P(\{r_i\}) \propto \exp [ - U(\{r_i\}) / k_B T ]$ .

It therefore follows that the average value of any quantity that likewise depends on the states $A = A ({r i})$ can be calculated as

$<A>=\frac{ \sum_{\{r_i\}} A(\{r_i\}) \exp [ - U(\{r_i\}) / k_B T ] } {Z}$ ,

where the normalizing factor is the partition function

$Z= \sum_{\{r_i\}} P(\{r_i\}) = \sum_{\{r_i\}} \exp [ - U(\{r_i\}) / k_B T ]$ .

which takes care of the lack of normalization of the probability distribution. (Indeed, one sees that the average of any constant number $< a > = a$ , as is sensible).

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 (especially for off-lattice systems). One then may defer 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 points on top of a map of Northern Africa (Frenkel and Smit, reference below.)

[edit] Biased sampling

Therefore, 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:

$<A>=\frac{ \sum_j A_j } {N}$ ,

where $A j$ is the value of $A$ at step $j$ , and $N$ is the total number of steps simulated.

The algorithm uses a Markov chain procedure 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. i.e., if the trial state has an energy $U n e w$ , and the previous had an energy $U o l d$ , the criterion of acceptance depends on

$p = exp[ - (U n e w - U o l d) / k B T]$

A very high value of $U n e w$ will then lead to a very small $p$ , whereas a very small (negative) one will lead to a high $p$ .

To fully prescribe an algorithm, a basic condition of balance must be satisfied for equilibrium to be properly described, but detailed balance, a stronger condition, is usually imposed. A typical choice is

automatic acceptance if $p > 1$ (going downhill in energy)
otherwise, acceptance if a random number $r$ (between 0 and 1) happens to be smaller than $p$ .

[edit] Applicability

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.

[edit] Generalisations

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.

[edit] See also

[edit] 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.