Robert J. Low

We investigate a simple a discrete time random process based on the construction of Platonic solids. As is surprisingly often the case with apparently simple processes, we find some intriguing results, in this case requiring a careful consideration of how the probability distributions describing this process behave. The investigation proceeds by a fruitful combination of geometry, probability and elementary real analysis. We conclude with a consideration of the limiting behaviour of the probability distributions describing the size of the solid as the number of steps grows.

We consider an analogue in classical mechanics to Barnsley’s chaos game in which a particle acts as a (possibly damped) harmonic oscillator whose centre of attraction randomly changes at fixed time intervals. This system exhibits a rich behaviour depending on the relationship between the strength of the attractive force and the drag force. We show how this mechanical system includes Barnsley’s (discrete) chaos game, producing for example the Sierpinski gasket, and in addition we observe evolutions which have interesting symmetries not exhibited by the Barnsley game. The system also provides a classical analogue of the quantum chaos game previously investigated by the current authors.

Inspired by the algorithm of Barnsley’s chaos game, we construct an open quantum system model based on the repeated interaction process. We show that the quantum dynamics of the appropriate fermionic/bosonic system (in interaction with an environment) provides a physical model of the chaos game. When considering fermionic operators, we follow the system’s evolution by focusing on its reduced density matrix. The system is shown to be in a Gaussian state (at all time t ) and the average number of particles is shown to obey the chaos game equation. Considering bosonic operators, with a system initially prepared in coherent states, the evolution of the system can be tracked by investigating the dynamics of the eigenvalues of the annihilation operator. This quantity is governed by a chaos game-like equation from which different scenarios emerge.

The natural topology on the space of causal paths of a space-time depends on the topology chosen on the space-time itself. Here we consider the effect of using the path topology on space-time instead of the manifold topology, and its consequences for how properties of space-time are reflected in the structure of the space of causal paths.

The determination of the contact points between two bodies with analytically described boundaries can be viewed as the limiting case of the extremal point problem, where the distance between the bodies is vanishing. The advantage of this approach is that the solutions can be computed efficiently along with the generalized state during time integration of a multibody system by augmenting the equations of motion with the corresponding extremal point conditions. Unfortunately, these solutions can degenerate when one boundary is concave or both boundaries are nonconvex. We present a novel method to derive degeneracy and nondegeneracy conditions that enable the determination of the type and codimension of all the degenerate solutions that can occur in plane contact problems involving two bodies with smooth boundaries. It is shown that only divergence bifurcations are relevant, and thus, we can simplify the analysis of the degeneracy by restricting the system to its one-dimensional center manifold. The resulting expressions are then decomposed by applying the multinomial theorem resulting in a computationally efficient method to compute explicit expressions for the Lyapunov coefficients and transversality conditions. Furthermore, a procedure to analyze the bifurcation behavior qualitatively at such solution points based on the Tschirnhaus transformation is given and demonstrated by examples. The application of these results enables in principle the continuation of all the solutions simultaneously beyond the degeneracy as long as their number is finite.

We see how the relaxation method for Laplace's equation is related to a numerical solution of the heat equation, which is in turn motivated by thinking of the Hamiltonian as the the time evolution operator in quantum mechanics. This gives some physical insight into what the relaxation method is doing. We also use this to suggest investigative work for a student learning about computational methods in physics.