Let us begin with the simulated annealing. In this method an initial configuration, a random or a completely ordered one, is chosen and the system is heated up to a high temperature . Then the system is cooled down in a very slow rate, which depends on specific system under investigation. The Monte-Carlo algorithm is used to equilibrate and to sample the system during this cooling process.
The sampling of the system is performed in the following way:
1.) A random atom selected and displaced randomly, therefore a new configuration of the atoms is generated. This displacement is called a trial move.
2.) The potential energy of the new configuration of the system is calculated and compared with the potential energy of the system before the trial move .
3.) If the energy of the system after the trial move is lowered : the trial move accepted unconditionally.
4.) But, if the energy of the system is increased, than this trial move could be accepted with a certain probability, i.e. we generate a random number between and compare it with .
If this expression larger than the trial move accepted, otherwise it is rejected.
5.) This procedure repeats again and again.
This method allows us to escape from local minima, while we are sampling the phase space and looking for the global minimum. However, we can not be confident in general, that using this method we will not be trapped in some local minimum, due to limited time of our Monte-Carlo run ( and therefore we can not prove that the the global minimum is necessary found). If we are trapped in a local minimum most of the simulation time, then only a small part of the phase space will be explored, and hence the physical quantities will not be calculated with appropriate accuracy (pure statistics).
One of effective way to overcome those difficulties is to perform simulations
in a so called generalized (extended) ensemble,
where the probability to cross energy barriers (get away from local minima ) could be
increased in an artificial way, thus MC simulations can be sufficiently accelerated.
The simulated tempering method, which belongs to the class of MC accelerated algorithms ,
has been applied in studying of point defects configurations at room temperature.
This methods was originally invented by Lubartsev et al. ,
and independently about at the same time by Marinary and Parisi .
In simulated tempering we perform a random walk in the energy space,
like in standard Monte-Carlo method, as well as in the temperature space.
The temperature space, unlike the energy space,
is chosen in a special way, i.e. it has a discrete structure:
4.) Otherwise, pick up a random number from
The only question is how to determine function?
Well, unfortunately, there is no prescription how to find this function in general.
Actually, for each system there is its own special function, which reflects the system properties.
This function has to be chosen very carefully,
if we want to be efficient, otherwise the system gets stuck around of some,
usually uncontrolled value of , and there is no good sampling in the temperature space,
and we return to a standard Monte-Carlo scheme.
The simplest way to choose is following: