Long-range interactions in dilute granular systems

In this thesis, on purpose, we focussed on the most challenging, longest ranging potentials. We analyzed granular media of low densities obeying 1/r long-range interaction potentials between the granules. Such systems are termed granular gases and differ in their behavior from ordinary gases by dissipative interactions, i.e., they do not conserve energy. Due to the dissipation, a unique wealth of structures can occur, enhanced or hampered by long-range interactions. Numerics For the analysis of such systems, we developed a soft sphere Molecular Dynamics (MD) method in three dimensions, taking properly into account the interplay between dissipative contacts and long-range interaction forces. Due to the infi- nite range of the 1/r potential, we would have to consider the interaction of all particles with all others, resulting in a computational time effort that scales like O(N2), where N is the particle number. We were able to bypass the pair-wise treatment by exploiting the linked cell structure we use for the neighborhood search such that particles in cells are considered as pseudo particles and grouped together in a hierarchical way. This hierarchical set-up based on linked cells is new to our knowledge. The combination with a multipole expansion of the mass distri- bution inside the pseudo particles gives a reduced number of interaction partners. The implementation of this Hierarchical Linked Cell (HLC) algorithm including periodic or wall boundary conditions shows a scaling behavior as O(NlogN), as confirmed by various simulations. We found that the results of the HLC algo- rithm agree with those of the direct summation code as long as the temperature is higher or about the same as the repulsion/attraction energy barrier. Dilute Homogeneous Granular Systems The second part of the thesis was devoted to the investigation of the cooling behavior of dissipative granular gases in presence of mutual long-range repulsive and attractive forces. In order to obtain reference results, we exclusively treated the particles pair-wise, limiting us to small particle numbers in the simulations. In order to understand the cooling behavior, we applied the pseudo Liouville op- erator theory. Although dealing with soft spheres under the influence of mutual long-range forces, we observed good agreement between theory and simulations in the limit of low densities and weak dissipation. In the case of repulsive long-range forces, the dissipation rate is reduced, taking into account the repulsive energy barrier the relative velocity of two approaching particles must exceed in order to collide. In the case of attractive forces, the dissipation rate is increased due to an escape energy barrier the relative velocity of two separating particles must overcome in order to not experience a collision. Both qualitative effects vanish if we consider the case of vanishing long-range force intensity or density. Our repulsive theory confirms earlier heuristic results [? ], while the attractive theory is new to our knowledge. Even though, the theory works in the dilute limit, for finite densities, the dissi- pation rate observed from simulations changes with increasing density and allows us to empirically provide a predictive analytical correction factor dependent on the density. We performed various elastic simulations with different repulsive and attractive strengths for densities in the range 0.010 0.152. The correction is non-linear dependent on density in the repulsive case and linear dependent in the attractive case, at least for the stable homogeneous cooling state examined. We used the empirical findings for the solution of the equations for the energy evolution with time and obtained an improved prediction, in good quantitative agreement with simulations. Small deviations remain and are supposed to be a consequence of dissipation because they increase with dissipation strength. In the attractive case, the improvement was less successful. Surprisingly, for mod- erate densities and dissipation strengths, the attractive systems show the same dissipation rate as systems without attraction forces. Dilute Ring-Shaped Granular Systems The third part of the thesis contains the investigation of self-gravitating ring- shaped particle systems. We applied the HLC code and tested it for such strongly inhomogeneous systems. It works well and, such as for homogeneous systems, the computational time expense scales as O(NlogN). We found that for sufficiently strong attraction forces, elastic systems show clustering even though a perma- nent shear rate and shear heating are present. Strong dissipation also leads to inhomogeneous rings, formation of clusters, and “planetesimals”. Moreover, we developed and numerically solved an approximate Navier-Stokes hydrodynamic set of equations for the projected density with different Ansatzes for the kinematic viscosity and compared the solutions with our simulation re- sults. The agreement is good only for intermediate times, whereas for short times, besides initial equilibration effects, the rings spread faster and for later times the rings spread slower than predicted by theory.