Periodic behaviours emergent in discrete systems with random dynamics

Pickton, John-Nathan Edward (2017) Periodic behaviours emergent in discrete systems with random dynamics. PhD thesis, University of Nottingham.

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Abstract

Periodic behaviours in continuous media can be described with great power and economy using conceptual machinery such as the notion of a field. However periodic effects can also be `observed' in collections of discrete objects, be they individuals sending emails, fire-flies signalling to attract mates, synapses firing in the brain or photons emerging from a cavity. The origin of periodic behaviours becomes more difficult to identify and interpret in these instances; particularly for systems whose individual components are fundamentally stochastic and memoryless. This thesis describes how periodic behaviour can emerge from intrinsic fluctuations in a fully discrete system that is completely isolated from any external coherent forcing.

This thesis identifies the essential elements required to produce naturally emerging periodic behaviours in a collection of interacting `particles' which are constrained to a finite set of `states', represented by the nodes of a network. The network can be identified with a type of a spatial structure throughout which particles can move by spontaneously jumping between nodes. The particles interact by affecting the rate at which other particles jump. In such systems it is the collective ensemble of particles, rather than the individual particles themselves, that exhibit periodic behaviours. The existence or non-existence of such collective periodic behaviours is attributed to the structure of the network and the form of interaction between particles that together describe the microscopic dynamics of the system. This thesis develops a methodology for deriving the macroscopic description of the ensemble of particles from the microscopic dynamics that govern the behaviour of individual particles and uses this to find key ingredients for collective periodic behaviour.

In order for periodic behaviours to emerge and persist it is necessary that the microscopic dynamics be irreversible and hence violate the principle of detailed balance. However such a condition is not sufficient and irreversibility must also manifest on the macroscopic level. Simple systems that admit collective periodic behaviours are presented, analysed and used to hypothesise on the essential elements needed for such behaviour. Important general results are then proven. It is necessary that the network have more than two nodes and directed edges such that particles jump between states at different rates in both directions. Perhaps most significantly, it is demonstrated that collective periodic behaviours are possible without invoking `action at a distance' - there need not be a field providing a mechanism for the interactions between particles.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Hopcraft, Keith
Jakeman, Eric
Avitabile, D.
Keywords: Periodic Behaviour, Markov Processes, Emergent Behaviour Coherence
Subjects: Q Science > QA Mathematics > QA276 Mathematical statistics
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 43209
Depositing User: Pickton, John
Date Deposited: 26 Jul 2017 12:13
Last Modified: 13 Oct 2017 00:20
URI: http://eprints.nottingham.ac.uk/id/eprint/43209

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