Simulation And Theoretical Research Disorder Transition - A Group Phenomenon In Order

Collective Phenomenon is common in nature and society, where simple interactions among participants lead to spontaneous symmetry-breaking in non-equilibrium condition. Vicsek model (VM) provides a minimal model to describe and study these phenomena. To understand the mechanism of the emergence of order in collective motion, we start from the original Vicsek model and study the transition behavior of this model and its variations through both simulation and analytical approaches. The contents of this dissertation are as follows.We first study basic elements of VM and their roles in order-disorder transition of VM and its variations. These elements include but not limited to magnitude of noise, particle speed, average density in the system, position update rules, size of simulation box and boundary conditions etc. We find that the nature of order-disorder transition is a result of the coordination of several key elements. For example, in VM (AN, BU) we can produce discontinuous transition curve by changing key parameters such as average density in system or speed of particles. And ordered structure may emerge under stronger noise when the system is shifting toward thermal dynamic limit. Besides, periodic boundary condition may help to fix the orientation of traveling band of particles, but it is not essential for the emergence of traveling band. Based on these preliminary results and anticipations, we combine simulation and analytical method to systematically study the phase transition behavior of Vicsek Model and its variations.To make the interaction among particles clear, we first use two methods, i.e. size-fixed interaction group and random update rules, to construct homogeneous Vicsek model which exactly agree with the mathematical treatment of Vicsek model in spatial homogeneous condition. In this approach, we find the phase behavior of vectorial noise and angular noise is essentially different. The order-disorder transition in vectorial noise condition is discontinuous when the size of interaction group is larger than4, however, in angular noise the transition is always continuous. We further show that the critical point of transition depends on the size of interaction group, which is critically important for the discontinuous transition emerged in the original Vicsek model.In the transition behavior of original Vicsek model, we notice that traveling band is normally main contributor to order parameter and serve well as a sign of ordered state. Furthermore, the emergence and stabilization of a traveling band bridge disordered and ordered state in most occasions. To study how key parameters and simulation techniques affect this process, we study the dynamical behavior of simulation realization starting from artificially defined initial configuration. When the initial configuration contains a round homogeneous cluster in the vacant background, simulation with different position update rules show different cluster-dispersion rates. Such behavior is predicted by a simple analysis of the update rules using kinetic theory. We further show that when the initial configuration contains a homogeneous rectangle cluster, the resultant spanning probability, i.e. the formation of traveling band, is sensitive to the cluster’s initial density, width and thickness. It implies that localized order structure produced by random fluctuation is able to accumulate particles on its moving path and grows into a traveling band if some critical profile of initial density and shape is achieved. We analyze a large amount of simulation snapshot and find that, first, disorder state or disorder area in order state has non-zero probability to breed clusters of critical profile if system is large enough. Second, in a well-established traveling band, we reveal that particular internal structures and perpendicular diffusion dynamics help the band to maintain a straight and homogeneous body. We also construct a phase map for the discontinuous transitions observed in simulation, which suggest that, in thermal dynamic limit (i.e. the size of the system approaches infinite), the order-disorder transition in Vicsek model is discontinuous and the order is propagated through large scale traveling bands.And finally, we attempt to integrate Vicsek model with Eguiluz-Zimmermann network to study the collective phenomenon widely observed on internet. We modify the coalesce mechanism and let chosen pair of agents calculate the inner product of their opinion vectors to decide if their clusters should be connected. We show spontaneous emergence of periodic behavior of order parameter in the time domain.