Title:

Statistical properties of a randomly excited granular fluid

In this thesis we describe numerical simulations performed in one and twodimensions of a theoretical granular model called the Random Force Model. The properties of nonequilibrium steady state granular media, which this model is a simple example of, are still hotly debated. We begin by observing that the onedimensional Random Force Model manifest multiscaling behaviour brought on by the clustering of particles within the system. For high dissipation we find that the distribution of nearest neighbour distances are approximately renormalisable and devise a geometrical method that accounts for some of the structural features seen in these systems. We next study twodimensional systems. The structure factor, S(k), is known to vary, for small k, as a powerlaw with an exponent D_f, referred to as the fractal dimension. We show that the value of the D_f is unchanged with respect to both dissipation and particle density and that the powerlaw is different from that given in any previous study. These structural features influence the long distance behaviour of individual particles by affecting the distances travelled by particles between consecutive collision. The velocity distribution, P(v), is known to strongly deviate away from MaxwellBoltzmann statistics and we advocate that the velocity distributions have asymptotic shape which is universal over a range of dissipation and particle densities. This invariance in behaviour of the largescale structure and velocity properties of the twodimensional Random Force Model leads us to develop a new selfconsistent model based around the motion of single high velocity particles. The background mass of low velocity particles are considered to be arrange as a fractal whereby the high velocity particles move independently in ballistic trajectories between collisions. We use this description to construct the high velocity tail of P(v), which we find to be approximately exponential. Finally we propose a method of structure formation for these systems that builds selfsimilarity into the system by consecutively fracturing the system into smaller parts.
