Information theoretic measures of transitions to collective behaviour

Flocks of birds are awe-inspiring in their scale, grandeur and grace. They are but one of the countless displays of collective behaviour in nature, including---although certainly not limited to---schools of fish, herds of sheep, bacteria colonies, to even the arrangement of particles in a magnet or other materials.

The study of these collective behaviours is an active field, with many different models and variations replicating the behaviours and systems seen in nature. These models provide insight into how system dynamics change as parameters, such as temperature, are varied. Thus it is possible to observe the system in different states or phases---whether a bird flock is coherent and aligned or disparate and disordered---as well as how the system transitions between phases.

Information Theory has been applied to a wide variety of fields since its proposal in the mid-twentieth century, especially in communications and computing. Within the last few decades it has also been applied to collective behaviour. Information Theory quantifies uncertainty, with metrics like Mutual Information (MI) and Transfer Entropy (TE) quantifying reductions in uncertainty given system observations. MI measures the instantaneous reduction of uncertainty of one element given knowledge of another element, and can be described as "information sharing". TE, however, is a temporal quantity measuring the reduction of uncertainty in the future of one variable when the current and past states of both variables are known, giving the "information flow" from one to the other.

Recently TE has been extended to measure the information flow from an entire system to individual elements, giving Global Transfer Entropy (GTE). Thus far GTE has only been applied to the Ising model|an equilibrium system modelling ferro-magnetism via binary states. While MI and TE peak precisely at the phase transition, GTE was shown to peak on the disordered side of the transition, allowing it to act as a predictor for an impending transition to an ordered phase. This dissertation extends this work, applying MI and GTE to two other models: the Potts and Vicsek models.

The Potts model generalises the 2-state Ising system to q-states, which changes the nature of the phase transition as q increases. The Vicsek model models bird flocks as point particles and is an example of a far-from-equilibrium, "active matter" system, whose elements are continuous rather than discrete as in the Potts and Ising models. Two variants of the Vicsek model are considered, differing on how nearby particles are chosen: all particles within a radius, or only the closest k particles.

Application of GTE to both models encountered difficulties around computational feasibility which required development of novel reductions. These techniques reduce the problem dimensionality, allowing sufficient statistics to be collated in a timely manner for analysis.

It is revealed here that the information theoretic behaviours in the Potts model differ from the Ising model, with both metrics peaking away from the transition in finite systems, converging to the transition as system size increases. Furthermore, in the thermodynamic limit, N to infinity, both quantities become undefined at the transition, peaking infinitesimally above the transition, with a discontinuous jump as the system goes from order to disorder. Meanwhile, the Vicsek variants display radically differing behaviour with MI diverging to positive infinity as flocks become increasingly ordered, while GTE fails to peak and the model instead experiences maximal information flow for the entire ordered regime.