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18641 Ahmed Israr Final E-Thesis (Master Copy).Pdf MATHEMATICAL AND COMPUTATIONAL MODELLING OF SOFT AND ACTIVE MATTER by ISRAR AHMED A thesis submitted in partial fulfilment for the requirements for the degree of Doctor of Philosophy at the University of Central Lancashire SEPTEMBER, 2016 Student Declaration Form I declare that while registered as a candidate for the research degree, I have not been a registered candidate or enrolled student for another award of the University or at another academic or professional institution. I declare that no material contained in the thesis has been used in any other submission for an academic award and is solely my own work. Signature of Candidate Type of Award Doctor of Philosophy (PhD) School Physical sciences and computing ACKNOWLEDGEMENTS Firstly, I would like to thank my supervisors, Dr. Dung Ly and Prof Waqar Ahmed for their help and encouragement throughout this research. The work presented in this thesis could not have been done without their inspirational ideas and valuable insights. I should most like to thank my family, for their support and trust. Without their unfailing love and encouragement, it would have proved much more difficult to get through my research. List of publications Ahmed I, Ly D and Ahmed W, “Collective behaviour of self-propelled particles in homogeneous and heterogeneous medium” , International Journal of Modelling, Simulation, and Scientific Computing (submitted). Ahmed I, Ly D and Ahmed W “Collective behaviour of self-propelled particles in the presence of moving obstacles” Physical Review Letter (under process of submission) Abstract The collective motion of organisms such as flights of birds, swimming of school of fish, migration of bacteria and movement of herds across long distances is a fascinating phenomenon that has intrigued man for centuries. Long and details observations have resulted in numerous abstract hypothesis and theories regarding the collective motion animals and organisms. In recent years the developments in supercomputers and general computational power along with highly refined mathematical theories and equations have enabled the collective motion of particles to be investigated in a logical and systematic manner. Hence, this study is focused mathematical principles are harnessed along with computational programmes in order to obtain a better understanding of collective behaviour of particles. Two types of systems have been considered namely homogeneous and heterogeneous systems, which represent collective motion with and without obstacles respectively. The Vicsek model has been used to investigate the collective behaviour of the particles in 2D and 3D systems. Based on this, a new model was developed: the obstacle avoidance model. This showed the interaction of particles with fixed and moving obstacles. It was established using this model that the collective motion of the particles was very low when higher noise was involved in the system and the collective motion of the particles was higher when lower noise and interaction radius existed. Very little is known about the collective motion of self-propelled particles in heterogeneous mediums, especially when noise is added to the system, and when the interaction radius between particles and obstacles is changed. In the presence of moving obstacles, particles exhibited a greater collective motion than with the fixed obstacles. Collective motion showed non-monotonic behaviour and the existence of optimal noise maximised the collective motion. In the presence of moving obstacles there were fluctuations in the value of the order parameter. Collective systems studies are highly useful in order to produce artificial swarms of autonomous vehicles, to develop effective fishing strategies and to understand human interactions in crowds for devising and implementing efficient and safe crowd control policies. These will help to avoid fatalities in highly crowded situations such as music concerts and sports and entertainment events with large audiences, as well as crowded shopping centres. In this study, a new model termed the obstacle avoidance model is presented which investigates the collective motion of self-propelled particles in the heterogeneous medium. In future work this model can be extended to include a combination of a number of motionless and moving obstacles hence bringing the modelling closer to reality. Contents CHAPTER 1 .................................................................................................................................... 1 Introduction and Background ....................................................................................................... 1 1.1 Introduction ........................................................................................................................ 1 1.2 Collective behaviour ........................................................................................................... 2 1.3 Self-propelled particles (SPP) ............................................................................................. 3 1.4 Phase transition .................................................................................................................. 4 1.5 Aim .................................................................................................................................... 10 1.6 Objectives.......................................................................................................................... 10 1.7 Outline of the thesis .......................................................................................................... 11 CHAPTER 2 .................................................................................................................................. 14 Collective behaviour of self-propelled particles - a literature review ........................................ 14 2.1 Introduction ...................................................................................................................... 14 2.2 Collective Behaviour in Homogeneous Medium .............................................................. 15 2.3 Statistical physics of self-propelled particles .................................................................... 19 2.4 Collective Behaviour in Heterogeneous Medium ............................................................. 23 2.4.1 Fixed obstacles ............................................................................................................... 24 2.4.2 Moving Obstacles ........................................................................................................... 26 CHAPTER 3 ................................................................................................................................. 28 Models for Collective Behaviour of Self-Propelled Particles ...................................................... 28 for Homogeneous and Heterogeneous Systems ........................................................................ 28 3.1 Introduction ...................................................................................................................... 28 3.2 Methods for the collective behaviour of self-propelled particles in homogeneous and heterogeneous medium.......................................................................................................... 29 3.2.1 Vicsek model in 2D ......................................................................................................... 29 3.2.2 Vicsek model in 3D ......................................................................................................... 32 3.2.3 Chepizkho model ........................................................................................................... 33 3.2.4 Order of phase transitions ............................................................................................. 36 3.2.5 Limitations of the existing models ................................................................................. 37 3.3 Development of a new improved model for understanding the collective behaviour of self-propelled particles ........................................................................................................... 38 3.3.1 Obstacle avoidance model (OAM) ................................................................................. 38 (i) Moving obstacles .............................................................................................................. 41 3.4 Comparison of OAM and the Chepizkho model ............................................................... 42 3.7 Conclusions ....................................................................................................................... 45 CHAPTER 4 .................................................................................................................................. 46 Simulations Studies using the 2D Vicsek Model ......................................................................... 46 for Self-Propelled Particles ......................................................................................................... 46 4.1 Introduction ...................................................................................................................... 46 4.2 Parameter table ................................................................................................................ 47 4.3 Comparison of calculated results and the simulation results ........................................... 47 4.4 Simulation results ............................................................................................................
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