Glassy behaviour in simple systems
In this thesis we study several different models which display glassy behaviour. Firstly, we investigate a simple, purely topological, cellular model for which the Hamiltonian is non-interacting but the dynamics are constrained. We find a non-thermodynamic transition to a glassy phase in which the energy fails to reach the equilibrium value below a characteristic temperature which is dependent on the cooling rate. This model involves activated processes and displays two-step relaxation in both the energy and the correlation functions; the latter also exhibit signs of aging. The relaxation time can be well-fitted at all temperatures by an offset Arrhenius law. Some predictions of Mode-coupling Theory are tested with some agreement found, but no convincing evidence that this description is the most fitting. By defining a suitable response function, we find that the equilibrium Fluctuation-Dissipation Theorem (FDT) is upheld for all but very short waiting-times, despite the fact that the system is not in equilibrium. This topological model is simplified to a hexagonally-based spin model, which also displays glassy behaviour, involves activated processes and exhibits two-step relaxation. This is a consequence of reaction-diffusion processes on two different time-scales, one temperature-independent and the other an exponential function of inverse temperature. We study two versions of this model, one with a single absorbing ground state, and the other with a highly degenerate ground state. These display qualitatively similar but quantitatively distinct macroscopic behaviour, and related but different microscopic behaviour. We extend this work to a square lattice, and find that the geometry of the lattice has a considerable impact on the behaviour, and to three dimensions, which provides support for the reaction-diffusion classification of the early behaviour. We find observable-dependent FDT plots; the observable can be chosen such that FDT is upheld for a region whilst the system is out of equilibrium — this observation is supported by some preliminary results for one-dimensional kinetically-constrained Ising chains.