The New Geometry of Computer Modeling

Originally, Greek philosophers thought that the universe was continuous and that the world could be described by lines, areas, and volumes, according to the geometry (literally earth-measurement ), set down by Euclid, for example, around 300 BC. It became evident a few centuries ago that shapes are not continuous but are composed of similar but smaller shapes as they become more magnified. Thus a tree looks more complex the more it is studied on a finer scale, as shown in Fig. 5.7. This concept is the basis of fractal geometry which has been described by Mandelbrot.  

The pioneering studies of this type of model, now known as molecular dynamics, were carried out by Alder and Wainwright in 1957 and the following decade. Wood and Jacobson, at the same time, obtained similar results by a slightly different approach. Berni Alder was working at the Lawrence Radiation Laboratories in California trying to bounce theoretical, nonadhering spheres around inside a computer. Since computers were rather limited in power at that time, a million times less potent than today s machines, he could only model small numbers of spheres from 4 to 32 up to a maximum of 500. He found that, when the spheres had a lot of space to move around, then the motion was disordered like a gas, as in Fig. 5.8(a). 

But at high packing, as shown in Fig. 5.8(b), the spheres jostled together to produce an ordered structure which was crystalline in nature, although the spheres were still essentially fluid in their ability to move freely without any adhesion. This structuring was eventually proved to be a first-order phase transition and happens to be one of the few discoveries made by computer. 

Carl Stainton has repeated those early computer calculations of Alder and Wainwright using a modern desktop machine to show the detailed nature of the phase transition. Like Alder, he used periodic boundaries so that the spheres did not encounter obstacles in the model (Fig. 5.9(a)). As a particle left the cell during the calculation, it was assumed to re-enter from the other side. The particles obeyed Newton s laws of motion in three dimensions and bounced perfectly off one another with no energy loss. When the particles were reduced in diameter to lower the packing density in the cell, the phase transition shown in Fig. 5.9(b) was found. 

The phase transition appeared suddenly, as the close-packed sphere stracture was being diluted by shrinking each sphere diameter, when the spheres occupied 49% of the volume, i.e. at 0.49 packing fraction. The pressure was calculated by working out the impacts of the spheres on one square meter of the cell in one second. As the volume of the spheres was reduced during the computation, the 

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