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Numerical and Analytical Approaches to Modelling 2D Flocking Phenomena

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Numerical and Analytical Approaches to Modelling 2D Flocking Phenomena School of Physics Numerical and Analytical Approaches to Modelling 2D Flocking Phenomena. Jason A. Smith Submitted in total fulfilment of the requirements of the degree of Master of Philosophy 2011 The University of Melbourne ii Abstract In the first section of this thesis, the motion of self-propelled particles (known as boids) in a 2D system with open boundaries is considered using a Lagrangian Individual-Based model. In fact, two different variations of this model are developed, one with a cohesion potential based on the soft core Morse potential and the other based on the hard core Lennard-Jones potential, both well understood in the fields of atomic and molecular physics respectively. The results obtained from these two different variations are then compared with one an- other, as well as with earlier work in the field, in order to determine the effectiveness and applicability of hard-core and soft-core cohesion potentials, the differences in flocking be- haviour they produce, and context in which each one is applicable to real flocking systems. Some of the flocking phases and shapes obtained from these two different models are then compared to a number of specific flocking situations observed in the real world. It is shown that a flocking model with a soft-core cohesion potential is particularly good at modelling the cluster type flocks most commonly seen in bird and fish schools, whilst a hard-core cohesion potential has a tendency to produce a distinctive wavefront type flock which has been well documented in the context of large herds of mammals, particularly wildebeest. It is also found that stable vortex states, observed in systems of bacteria, are only seen with a soft-core cohesive potential. In the second section of this thesis, a novel approach is taken to model a flocking as a gas. An equation of state is derived analytically for a flocking system in 2D using the virial expansion. The relationships obtained from this equation of state are compared to iii the results of a simple numerical simulation in order to establish the accuracy of this new approach to modelling flocking systems. Finally, this new statistical mechanical approach to deriving a flocking model is applied to the Vicsek model, the most well understood flocking model derived from a physics perspec- tive. It is found that the analytical relationships between a number of key variables of the gas system such as pressure, temperature, interaction length, entropy and heat capacity derived from the virial equation of state bear a close comparison to those same variables derived numerically from the standard Vicsek model, thus demonstrating the efficacy and veracity of this new approach to modelling flocking phenomena. iv Declaration This is to certify that: (i) the thesis comprises only my original work towards the MPhil except where indicated in the Statement of Contributions, (ii) due acknowledgment has been made in the text to all other material used, (iii) the thesis is less than 50,000 words in length, exclusive of table, maps, bibliographies, appendices and footnotes. Jason Adam Smith v vi Statement of Contributions This thesis topic and its theory, as presented in Chapters 2, 3 and 4 were formulated and developed in collaboration with my supervisor, Andy Martin. I have been primarily responsible for the code and numerical work presented in Chapter 2 as well as well as the analytical calculations undertaken in Chapters 3 and 4. The ideas and interpretation of the results were developed with input and guidance from my supervisor Andy Martin. Resources obtained from other publications and authors have been cited in the text where appropriate. vii viii Publications During the course of this project, the following article has been published, which is based on the work presented in Chapter 2 of this thesis. It is listed here for reference. J.A. Smith and A.M. Martin., Emergence in Nature: Modelling Flocking Phenomena Through Physics, Poster Presentation, AIP Conference, Adelaide, Australia, December 2008. ix x Acknowledgments I would like to extend a big thank you to my family for their unwavering support and encouragement throughout this long process and to my supervisor Andy Martin for his diligence, enthusiasm and most of all his patience. A big thanks also to all my colleagues at The University of Melbourne that have been there for me throughout my university years, but in particular to Marko Milicevic, Laura Zugno, Steven Petrie, Andrea Ruff and Masum Rab who have always been there to make my time in physics more comfortable and entertaining, and to my close friends Fay Ng and Hollie Pearce for their love, support and encouragement through even the most difficult of times. Finally, I would like to acknowledge the School of Physics and The University of Melbourne for the financial and academic support which made this possible. xi xii Contents 1 Introduction 1 1.1 Early History of Flocking . .3 1.2 Individual-Based Models of Flocking . .6 1.3 Hydrodynamical Models of Flocking . 20 1.4 Hard-Core and Soft-Core Flocking Potentials in Free Space . 24 1.5 Flocking and the Virial Expansion . 28 2 Modelling Flocking in Free Space 33 2.1 Introduction to Individual-Based Flocking . 33 2.2 Formulation of a computational flocking model . 35 2.3 Computational Methodology . 42 2.4 Results and Discussion . 44 2.5 Summary . 55 3 The Virial Expansion for a Flocking Gas 59 3.1 The Virial Expansion Introduced . 59 3.2 Flocking and the Virial Expansion . 64 3.3 Deriving a Flocking Equation of State . 69 3.4 Comparison to Computational Flocking System . 76 3.5 Discussion and Limitations . 81 4 A New Approach to the Vicsek Model 85 4.1 The Vicsek Model and the Virial Expansion . 85 4.2 Deriving a Virial Equation of State for the Vicsek Model . 92 4.3 Numerical Comparison . 98 4.4 Order, Disorder and Phase Transitions in the Vicsek Gas Model . 105 xiii 4.5 Discussion and Limitations . 118 5 Summary and Future Directions 121 5.1 Hard-Core and Soft-Core Free Space Flocking . 121 5.2 Flocking as a Simple Gas . 123 5.3 State of Play . 126 5.4 Future Directions in Flocking Research . 129 5.5 Final Remarks . 131 References 133 A Deriving the equation of state for alternate potential forms. 141 (2) A.1 Quadratic Potential (Vij (φ)).......................... 142 (3) A.2 Gaussian Potential (Vij (φ)) .......................... 143 (4) A.3 Triangular Potential (Vij (φ)).......................... 145 B Calculating the fugacity for a system of indistinguishable particles. 147 C Mathematica Notebooks for thermodynamic quantities in Chapter 4. 151 xiv List of Figures 1.1 Basic behaviour regions of the Huth and Wissel flocking model. .5 1.2 4 different behaviours observed in the original Vicsek model. .8 1.3 An example of a flocking potential of the form specified by Y. Tu . 10 1.4 Phase diagram for the Y.Tu model in a − b space. 12 1.5 Chaotic flock geometry in D'Orsogna model. 15 1.6 A typical flock reconstructed in 3D by STARFLAG. 16 1.7 Phase like transition in prey flock as predator attacks from critical angle θc. 17 1.8 Information propagation through a school of fish. 19 1.9 A typical hydrodynamic model of a flocking system . 23 1.10 Example of boundary conditions in a number of typical flocking models. 25 2.1 Specification of parameters. 36 2.2 (a) Lennard-Jones (red line) and Morse (blue line) Potentials. 40 2.3 Examples of 4 different coherent states observed for the MP. 46 2.4 Examples of breakdown modes in MP and LJP flocks. 47 2.5 Phases observed in nature . 50 2.6 Phase Diagram for the MP in the moving regime. 53 2.7 Phase Diagram for the LJP in the moving regime. 54 3.1 Parameters in a Flocking Gas . 65 3.2 Specification of interaction potentials. 68 3.3 Comparison of analytic and computational system properties. 77 3.4 Comparison of predicted Pressure vs Time behaviour from virial equation of state (dashed lines) and observed Pressure vs Time from numerically simulated system (solid lines). 79 3.5 The behaviour of the f(a) term with respect to a............... 80 xv 4.1 A diagrammatic representation of the Vicsek model. 87 4.2 Analytic potentials used to approximate the Vicsek system . 89 4.3 Set of alignment potentials compared. 91 4.4 Plots of the pressure vs β for the different 4 potential forms. 97 4.5 Plots for the 4 different interaction potentials for pressure vs interaction length. 99 4.6 Theoretical plot of pressure vs β based on the virial equation compared to computational data . 102 4.7 Theoretical plot of pressure vs interaction length based on the virial equa- tion of state compared to computational data. 104 4.8 The effective fugacity λe plotted against kBT ................. 109 4.9 The fugacity λ plotted against inverse temperature β ............. 111 4.10 The specific heat CV plotted against kBT for various values of interaction R scales L ....................................... 113 4.11 Entropy, S, plotted against inverse temperature β............... 116 xvi Introduction 1 Flocking is happening all around us, each and every day of our lives. One has only to look to the sky on a clear day to see groups of birds wheeling in unison, seemingly of a single mind, flying as if belonging to a single body, united in action and in purpose. Many of us have been diving, and swam amongst the shoals of fish which behave almost as a single en- tity, their movements occurring in a strange, synchronous harmony. We have experienced the throngs of people leaving a football stadium on a Saturday evening or the jostling motion as our fellow humans thronging to catch the peak hour train.
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