Dokument

Summary:
This thesis addresses the investigation of the transition from order to chaos in two different systems. In this context, both numerical simulations and theoretical considerations are applied. Popular examples of such transitions are, among others, the melting of a crystal or the transition from a laminar flow to turbulence. They have in common that the variation of an external parameter, for example temperature, results in an abrupt change in the properties of the system: The ordered, well-defined structure of a crystal transforms to the unordered, random configuration of a liquid. In the first part of this thesis, we investigate the influence of a linear, periodic shear on a system of mutually repelling particles. This can be considered as a model for a well known phenomenon, the mixing of a blob of dye in a liquid confined between two concentric cylinders. If the cylinders are rotated back and forth slowly enough so that the flow remains in the laminar regime, a demixing is possible and after one period we retrieve the original blob. Something similar occurs in the simple two-dimensional model system that we investigate: For small shear rates, we observe a self-organization of the particles, ending up in a lattice configuration. Above a critical shear rate, an abrupt change takes place, and the system is found - and remains - in an unordered, chaotic state. We are able to associate the transition with a loss of stability of the sheared lattice. In the chaotic regime, the spatially resolved correlations reveal more details and allow us to extract phase information. Additionally, they provide a possible explanation why diffusion parallel to the shear is enhanced beyond the known advection-diffusion coupling. In the further course of this work, we turn toward a transition known from everyday life, the melting of a solid. There, we again restrict ourselves to a two-dimensional model similar to the one in the first part. We are especially interested in the microscopic processes which eventually result in the melting of the crystal. Therefore, we perform molecular-dynamics (MD) simulations which confirm a two-step process of melting. Furthermore, the computed trajectories allow us deeper insights into the dynamics of the system. In the critical temperature range, we initially observe isolated localized processes where several particles exchange their positions. With the help of a projection on the energetically lowest configuration, these transitions can be identified as hopping events on the hexagonal lattice. As temperature is increased, more processes occur simultaneously, and eventually secondary, more complex transitions are stimulated which result in the melting of the solid. On this account, we investigate the melting transition in view of a rate activated process induced by localized reorganizations of a few particles. We identify possible transitions and the corresponding transition states which comprise up to 18 particles and only affect up to approximately 25 particles, implying that the states are well localized. We characterize the states, both by their arrangement in configuration space as well as by their thermodynamically relevant properties such as the energy barriers. We determine the dependence of the transition rates on the temperature, and compare them to the melting temperature of the system. Apparently, the rates are too low to explain melting on their own. Nevertheless, this is in accordance with previous observations in the simulations, where localized reorganizations lead to secondary transitions which break up the lattice structure and hence initiate the melting process. In the course of our studies, we also investigate the elastic properties of the system. The crystal, consisting of individual particles, can be described by the elastic constants of a continuous solid body. We show that the displacement field induced by the local disturbance of the transition state can be approximated by a superposition of several displacement fields of singular forces acting on an elastic medium. The screening of the potential not only gives rise to a rescaling of the energy of the system, but alters its elastic properties as well. This is partly reflected in the transition states. Though their basic configuration remains unaffected, energy barriers and displacement fields change considerably. We once again refer to the rate model in order to determine the transition rate at the critical temperature. A concluding comparison with results from MD-simulations and other predictions reveals that the model captures the dependence of the melting temperature on the screening parameter of the potential very well.

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