Using Molecular dynamic simulation a cluster of particles was modelled and their behaviour analysed with respect to temperature change. As proof of a specific state i.e. solid, the Mean squared displacement was calculated and illustrated with respect to time. This occurred for varying temperatures. From the internal energy of the system per time step, the heat capacity was determined. Heat capacity was then plotted against temperature in order to view phase transitions. The results show almost independence of heat capacity-temperature curves with respect to the number of particles. That is, for all numbers of particles in said range the heat capacity rises with respect to temperature and then plateaus at temperatures where the cluster changes state. Knowledge of this change of state was found by other means and the shape of these curves was backed by the theory of gradual phase change. There were no visible peaks in these curves which would be an indicator for specific temperatures where state changes occur abruptly. From the simulations and these results it was clear that there was also co-existence of states.
With the birth of Quantum mechanics came uncertainty at the particle level and it began to dominate the microscopic world. Nonetheless, today a lot still needs to be learnt from the classical world. The numerical solutions of N-body problems are something that Molecular dynamics solves elegantly. Molecular Simulations then allowed for these particles to be viewed in certain environments without real experimentation. This investigation was done on clusters as it was clear from that start that the results obtained would be beneficial. This is in the sense that, although there have been many studies done on clusters, with respect to state transition there is not a definite conclusion. Thus this investigation was undergone with the intent to better understand this area within molecular dynamics. There are those that believe the three states co-exist at the same point in time. The other view point is that there are sharp transitions between states. However, the sharp transitions in state have been found at lower particle numbers. This then begs the question of how the state can be so well defined at such sizes and the accuracy of results could also be disputed at this size. In the past research that has shown these sharp transitions also have some clusters whereby the heat-capacity curves do not show these properties at all. This is then said to be a result of Magic numbers. Whereby certain numbers of particles are part of this magic number sequence. Of which the structure of these clusters are said to be more stable. This may be due to the similarity with icosehedral shape sequence. Nonetheless, sharp transitions are said to be visible in clusters under the umbrella of magic numbers. It must be said that the sequence of magic numbers is said to reach into the size 50 to 100 particles. Thus the results from this investigation would be obtained from these clusters. At very low temperatures it is well known that clusters form a solid structure with harmonic-like vibrations and no transitions between states. Also, the energy approaches the global minimum potential energy. On the other hand, at high temperatures the particles become gaseous and this is known as dissociation. The ideal gas limits would be approached by both the internal energy and the heat capacity. The data obtained is in Lennard Jones units and within those units a few things are certain. For an inert gas like that of the one modelled, the triple point is at a temperature of 0.7. This meant that at this temperature it was possible to run a simulation in the knowledge that a cluster should be present. Thus any other simulations were with respect to that one when it came making sure the particles were of a certain...