Monte Carlo Simulations

A Monte Carlo (MC) simulation is a method commonly used in thermodynamics and statistical mechanics.  While in general MC is just a sampling scheme, it is normally used for importance sampling.  The MC method starts from a given 'position' and takes a random walk, returning information from wherever it travels.  With importance sampling, MC does not take a purely random walk, but instead will take a random step and then check the final position against some requirement. If the final position meets the requirements, the information for that position is recorded as data and a new step is tried. Otherwise, the simulation goes back and takes a new random step.



MC is useful for thermodynamics/statistical dynamics because in such situations the item of interest is the partition function.  In theory, all possible configurations of, say, the atoms in the air of a room includes having them all hexagonally packed in one corner.  However, the energy required for such a configuration is very high and its probability is correspondingly low, despite whatever entropy it may have (the number of ways to have that same configuration, aka its degeneracy).  The MC method can be used to take some particles in a box with some type of energy distribution, such as short-range attraction between particles with hard cores (no overlap allowed), and find minimal-energy configurations in the following way: take a random particle, move it a given short distance in a random direction, check that it has not overlapped any other particle, and calculate the change in energy between the old and new configuration. If the energy decreases, the move is accepted. If the energy increases, a probability function is used to determine whether the move is accepted (Boltzmann distribution, for instance) where the greater the energy increase, the less likely the move.  Given enough steps, the overall energy will decrease to some stable or meta-stable state.  Depending on the particle interactions and the system's density, a MC simulation could reach an end state of any phase or some phase coexistence.  This type of simulation is great for modeling the phase diagram for a given substance, provided a model for the intermolecular forces.

One thing to remember about MC simulations is that they are not dynamical: there is no consideration of momentum, and steps do not correspond with any kind of 'time' that could be compared to an experiment. A MC simulation simply finds low-energy, high-entropy states. Since in general materials head to such states naturally, the end result will be similar to that of an experiment, but the paths may be very different.