Collective motion as a function of time

The order parameter as a function of time was investigated. The noise level and the density of the particles were varied and the curve shows the evolution of collective motion with time. The speed of the particles was 1, time steps were 3000, and the box length was equal to 20. In the first four figures the number of particles was N = 3000 and in the next

0

Figure 0.23Figure 5.24 First order phase transitions at

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two figures the density of the particles was varied to investigate the effect on the collective motion.

(a) 𝜂 = 0 (b) 𝜂 = 0.4

Figure 5.25 Collective motion as a function of time at different noise values

Figure 0.24Figure 5.25 Collective motion as a function of time at different noise values

(a) 𝜂 = 0.8 (b) 𝜂 = 1.0

Figure 5.26 Collective motion as a function of time for strong noise values

Figure 0.25Figure 5.26 Collective motion as a function of time for strong noise values

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Collective behaviour is plotted as a function of time for different noise values at 𝜂 = 0.0, 0.4, 0.8, and 1.2 (figures 5.25 to 5.26). At the initial time steps, the particles had random direction and positions but after several time steps they began to align with each other. At 𝜂 = 0, the particles showed very smoothing behaviour, see figure 5.25(a). The value of the order parameter was consistent and remained approximately equal to 1, suggesting that there was a higher alignment in the direction of the particles. At 𝜂 = 0.4, the value of the order parameter decreased. The curve showed smaller fluctuations, see figure 5.25(b). Most of the time, the value of the order parameter remained between 0.8 and 0.9. The noise value increased from 0.4 to 0.8. There was greater impact of this value on the collective motion of the particles. There were more fluctuations in the collective motion than in the two previous cases of noise, see figure 5.26(a). The value of the order parameter remained between 0.5 and 0.6 and this value suggested that collective motion of the particles existed but on a smaller scale. The noise was further increased to 1.0.

There was a huge disturbance in the direction of the particles due to the stronger noise in the system. The impact of this noise can be seen in Figure 5.26(b). The system was in a complete state of disorder.

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(a) 𝑁 = 200 (b) 𝑁 = 4000

Figure 5.27 Collective motion as a function of time for 200 and 4000 particles

Figure 0.26Figure 5.27 Collective motion as a function of time for 200 and 4000 particles

Particle density was varied in the system and the results are shown in figure 5.27. Noise was fixed to 0.1 while other parameter values were the same as had been used in other results of this section. It can be clearly seen in figure 5.27(a) that the order parameter at the initial time step was smaller, but after some time steps it increased. There were also fluctuations in the system due to the smaller number of particles. The number of particles was further increased from 200 to 4000. This is a very large number and the results of this simulation are shown in Figure 5.27(b) with a smooth curve. The larger number of particles thus exhibited more collective motion than the smaller number of particles.

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5.3 Conclusions

The three-dimensional self-propelled particles model was studied in detail and the effects of the different parameters investigated. These are: speed, interaction radius, noise, and the density of the particles, and it was observed that at the first time step the order parameter obtained a value approximately equal to zero and the particles showed randomness. Similar behaviour was also observed in the case of higher noise when the density of the particles was equal to 0.87.

Particles had a loss of cohesion with a small particle density of 0.019. At a lower noise level of 0.1 and a higher density of 2.87, the particles showed ordered motion. . For a larger number of particles such as N3800, along with lower noise of 0.1, particles showed alignment in the direction of the particles. In the case of r0.5and in the absence of noise, however, where there were 3000 particles simulated, the results showed group formation in the system which was due to the smaller radius.

It was observed that with an increase in speed the collective motion of the particles increased. The effect of the interaction radius was also investigated, showing that at r0, the system was in a complete state of disorder; the increment in the value of the radius parameter brought the collective motion to a larger scale. Variations in the noise parameter had a significant effect on the collective motion of the particles. At zero noise, collective motion was higher and with a gradual increase in the noise, collective motion started decreasing; at  2.0,collective motion did not exist. The order of phase transition was also investigated, and with noise level 1.0, the system showed second order phase transition.

105 there was first order phase transition in the system. For collective motion as a function of time at various noise values and particles it was observed that for zero noise and a large number of particles, collective motion was higher; whereas for higher noise levels, collective motion was decreased.

The main difference seen in 3D compared to 2D is that in 3D there is a better view of the particle. 2D is flat and has only two dimensions, while 3D has depth and rotation. The 3D model exhibits a more realistic collective dynamic. In 3D more time steps are required for the particles to interact with each other; particles show less collective motion than in 2D. In 3D, more smoothness appears in the noise graphs compared to 2D. In 3D there is less collective motion observed than in the 2D, which can be seen in Figures 5.1-5.4.

At the initial time step in both cases the particles showed random motion. In the 2D for

25

L , and N = 300, there was group formation, but in 3D for the same parameter values,

no group formation occurred; an example can be seen in figures 4.2 and 5.2. At L = 5, 1

.

0

 , and N300, there was perfect alignment in the particles in the 2D system, whereas in 3D there was alignment but it was not as high as it was in 2D.

There was another difference which appeared in the 3D compared to 2D, that in the 3D second order phase, transitions existed for noise 1.0, whereas in the case of 2D, first order phase transition existed in the system for 1.0.

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CHAPTER 6

Simulations using new Obstacle Avoidance Model

6.1 Introduction

This chapter focuses on the model that has been developed in this study. As mentioned earlier, this model is termed the obstacle avoidance model (OAM). Even though the collective behaviour of particles has been investigated using previous models, no work has been carried out on investigating collective motion with the presence of a variety of obstacles along with various parameters. There are many examples available in the environment where the dynamics of particles is given in the presence of obstacles.

Bacteria show complex collective behaviours, for example, swarming in a heterogeneous environment such as soil, or highly complex tissues in a gastrointestinal tract; herds of mammals travel long distances crossing rivers and forests [115].

The results obtained from the obstacle avoidance model are discussed in this chapter.

Particles move in the presence of static obstacles. First of all, the simulation results are presented for various system sizes. Collective motion was plotted as a function of time.

The effects of interaction radius, noise and speed on the collective motion of the particles was also investigated in both homogeneous and heterogeneous media. The order of phase transition was also investigated for large numbers of particles.

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6.2 Parameter table

The key parameters used in this model for simulation studies are summarised in Table 6.1. These were discussed in Chapter 3 (see the equations developed for the obstacle avoidance model).

Table 6.1 Parameters used in the simulation

Table 9Table 6.1 Parameters used in the simulation

Symbol Description

L Length of box

N b Number of particles N o Number of obstacles

t Time step

 Noise amplitude

R o Interaction radius between the particle and the obstacles

r Interaction radius between the particles vo Absolute velocity

o Particle’s turning speed when it interacts with obstacle

t Time interval

w Collective motion parameter (order parameter)

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