Hard drives make up a system of many particles, so we can approach the dynamics from a statistical point of view. The following model is based on the game of stones, let us consider a large square-shaped court, suppose we throw a stone with our eyes closed and in a random direction, there are two possibilities, the first is that the stone falls inside the court, in that case, we move to that position, take out a new stone and throw it. The second possibility is that the stone falls outside the court, in that case we stay in the current position, we ask someone to bring us the stone, we stack it in that place (along with the stone that is already in those coordinates) and we throw another different stone. After a certain time, if we keep track of our trajectory, we would observe a chain of segments linked by stones or clusters of them. These same rules can be applied to hard drives. As the positions are chosen randomly, (This is carried out computationally by selecting random numbers in a bounded interval, and for which the probability distribution is uniform), the sampling is based on the Monte Carlo method, and Since each launch depends on the previous launch (the landing position), the path traveled forms a Markov Chain. This dynamic is known as MCMC for its acronym in English (Markov Chain Monte Carlo).
If we increase the number of disks, keeping the volume of the box fixed, the density is increased. The study of the variation of densities shows that at low densities, the system behaves like a liquid and as the density increases, the liquid properties are lost and those of a solid predominate. The phase change in these systems is not well defined, however; It is clear that when the density increases, the packing phenomenon occurs, this consists of the piling up or confinement of solids, in this case of spheres, and the way in which it occurs depends on the algorithm used. Interesting questions arise around packaging, for example; What is the densest way to pack spheres of the same size? It is one of the questions in problem number 18 of the list of 23 problems compiled by the German mathematician David Hilbert, one of the most influential in the history of mathematics. Kepler's conjecture is that the best packing is hexagonal. After 400 years, the American mathematician Thomas Hales computationally demonstrated in 1998 that the densest way to pack spheres in three dimensions is effectively through a hexagonal pattern, filling approximately 74% of the total space. This is the usual way oranges are stacked in the supermarket.