Implementation of the Metropolis Monte Carlo Method

A unique random number is generated for each of the three directions x, y and z. is the 

The size of the move at each iteration is governed by the maximum displacement, Sr. This is an adjustable parameter whose value is usually chosen so that approximately 50% of the trial moves are accepted. If the maximum displacement is too small then many moves will be accepted but the states will be very similar and the phase space will only be explored very slowly. Too large a value of Srmax nd many trial moves will be rejected because they lead to unfavourable overlaps. The maximum displacement can be adjusted automatically while the program is running to achieve the desired acceptance ratio by keeping a running score of the proportion of moves that are accepted. Every so often the maximum displacement is then scaled by a few percent if too many moves have been accepted then the maximum displacement is increased too few and is reduced. 

As an alternative to the random selection of particles it is possible to move the atoms sequentially (this requires one fewer call to the random number generator per iteration) Alternatively, several atoms can be moved at once if an appropriate value for the maximum displacement is chosen then this may enable phase space to be covered more efficiently. 

As with a molecular dynamics simulation, a Monte Carlo simulation comprises an equilibration phase followed by a production phase During equilibration, appropriate thermodynamic and structural quantities such as the total energy (and the partitioning of the energy among the various components), mean square displacement and order parameters (as appropriate) are monitored until they achieve stable values, whereupon the production phase can commence. In a Monte Carlo simulation from the canonical ensemble, the temperature and volume are, of course, fixed. In a constant pressure simulation the volume will change and should therefore also be monitored to ensure that a stable system density is achieved. 

The MOD function returns the remainder when the first argument is divided by the second (for example, MOD(14,5) equals 4). If the constants are chosen carefully, the linear congruential method generates all possible integers between 0 and m — 1, and the period (i.e. the number of iterations before the sequence starts to repeat itself) will be equal to 

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