Global Markov modelling and analysis of the dynamics of granular deformation and flow

Granular deformation and flows are highly complex interconnected dynamical systems that are deterministic, but difficult to predict at a microscopic level over even modest periods of time. One of the most useful mathematical approaches to analyse such microscopically chaotic systems is based upon considering the flow of mesoscopic clusters of particles rather than individual particles. In recent years, these techniques have been honed to offer an efficient means of identifying stable macroscopic structures despite the unpredictability at the microscopic level. The numerical methodology is based around the concept of a transition matrix that describes the dynamics on a user-selected state space. For example, in quasi-static deformation of a dense granular material, the state space may be a continuous space of particle stabilities, or a discrete space describing particle-neighbour interactions, or any one of many other possibilities; the transition matrix describes transitions between different particle states. These methods have previously been applied in a variety of other application settings, including molecular dynamics and physical oceanography.