6.1. Conclusions
The analysis of a two particle system with internal, vibrational degrees of freedom in Ch. 2 resulted in an explicit solution of the equations of motion which allowed for a detailed investigation into the properties of the probability density for the co-efficient of restitution in two-particle collisions with unspecified (“thermal”) initial conditions. Though no exact analytical expression could be found, an approximation was constructed which fulfilled all of the known exact properties of the probability density and which could very well be fitted to results from Monte Carlo simulations.
Using this analytic, albeit approximate, probability density, many particle simula-tions were performed in Ch. 3. As a result, it was shown that the system does not run into an inelastic collapse, so that the method is well suited to long simulation runs. Complex cluster dynamics take place during the approach to equilibrium in a system with total energy conservation (the equilibrium being a state where the total energy is uniformly distributed among all degrees of freedom, translational and vibrational). The equilibrium state itself was characterised by its dynamic struc-ture factor and complete agreement with hydrodynamical calculations was observed, somewhat surprising for a one-dimensional system but explained by the stochastic nature of the collision which destroys memory and asserts the molecular-chaos as-sumption which is a central prerequisite for hydrodynamic theories. The approach to equilibrium is dominated by the slowest decaying mode which is the one with the largest wavelength. Finally, a system with damped vibrations was simulated which showed that the final state of such a system without total energy conservation is one where basically all particles are clumped together and all kinetic energy is lost.
The dynamics of this system consisted of a formation of clusters in the initial stage, followed by “coarsening” in the sense that colliding clusters interacted completely inelastically, thus forming one larger cluster, until only one big cluster was left. The energy decay could of course not be described by anything like Haff’s law [Haf83] but instead followed a succession of steps, each corresponding to a cluster collision.
Starting from the one-dimensional model including only a finite but arbitrary number of vibrational modes, a solution of the equations of motion in the hard-core limit was given by two methods: The first involved the explicit solution of a
non-6. Conclusions and outlook
linear integral equation, the second arrived at the same result using simple energy conservation arguments. While the first is more convincing from a fundamental point of view, the latter is much simpler and could also be generalised to two and three dimensions. This generalisation allowed for a relatively efficient numerical simulation of a complete collision process of two elastic spheres. The simulations showed that the production of vibrations for collidingidentical spheres is weak but noticeable for impact velocities around 1/10 of the transversal sound velocity. Comparison with Hertz’ theory showed qualitative agreement for almost quasistatic collisions. Quan-titative agreement could not be achieved due to computational limitations on the number of vibrational modes used. Collisions of spheres of unequal size but with the same impact velocity as before, however, are qualitatively different: The qua-sistatic assumption breaks down, collisions are unsymmetric in time, and excitation of vibrations becomes important. Thus one central question that was posed in the introduction, namely if Rayleigh’s estimate [Ray06] of the relative unimportance of vibrations remains valid for the case of unequal spheres, can be answered negatively.
6.2. Outlook
While the one-dimensional system may not be of chief experimental relevance, it provides a simple testing ground for various theories. Since the cluster geometry as seen in the simulations is naturally simple in one dimension, it might be possible to make progress by trying to analyse the coarsening of clusters as it was described above. Thus one might arrive at a theory describing the energy decay in a situation which is far away from the homogeneous cooling state for which Haff’s law applies (see also [BE98a, vNE99]). Other possible extensions of the one-dimensional systems are investigations of particles with a different set of vibrational modes; this is made possible (at least numerically) by the general solution of Ch. 4. One could for in-stance consider particles which do not only vibrate longitudinally but which also have transversal vibrations which could become weakly excited upon slightly non-central collisions. Numerous other variations are conceivable, e.g. particles consisting of a small number of “atoms” connected by springs, or particles containing defects. The main question would be if the modified microscopic details result in different macro-scopic behaviour or if the macromacro-scopic system is independent of such modifications and is universal in this respect. The same question could be addressed by using different expressions for the probability density pβ() than the one given here.
The general method for collisions of three-dimensional objects being set up, one could ask a large number of further questions; simulations could be done for all conceivable values of the parameters. However, there are also some more fundamental questions that one might attempt to answer:
• It might be possible to solve the collision process of two spheres under special conditions exactly, using only a limited set of vibrational modes. An example
6.2. Outlook
where this is possible under certain conditions was given in Sec. 4.7. There it was argued that for slow velocities two colliding spheres can be regarded as if they were in a (half-sided) harmonic potential. Connected with this is the question if it is possible to carry out the limit of infinitely many modes in order to obtain a closed expression (but containing memory terms) for the three-dimensional system like Eq. (2.18) for the one-dimensional case. While such an expression would most likely still be unsolvable in practice (just as Eq. (2.18)), perhaps one could derive some exact results from it.
• In experiments it is found that elastic vibrations of solids are always damped by internal friction through various different mechanisms (see e.g. [Kol63], Ch. 5).
These effects could be incorporated into the theory and simulations and thus one could try to reach quantitative agreement with experiments and other the-ories of viscoelastic impact [Pao55, KK87, HSB95, BSHP96]. This amounts to combining two of the three loss mechanisms (plastic deformation and fracture, viscoelastic behaviour, and elastic vibrations) into one general framework.
• Simulations of many particle systems using stochastic coefficients of restitution (normal and tangential) based on the model presented here can be performed.
It is expected that they do not show inelastic collapse (for the same reason that it doesn’t appear in one dimension) which would make them good candidates for long runs into the clustering regime, while at the same time being based on a microscopic model. Comparisons of this method with other methods (e.g. the TC model from [LM98] or the rotation of rebound velocities by a small random angle as proposed in [DB97]) could help decide whether such differences are important for macroscopic properties.