Throughout the simulation, both the SPPs are kept in the same proportion as N0 =Na=N/2. The overall particle densityρ=N/L2 is fixed asρ= 0.5 for all the
4.4 Results and Discussion
1 R
2
Figure 4.9: The distribution of the SPPs is shown on a square box of size L = 10 where N0 = Na = 25. The orange colour represents usual SPPs and the indigo colour represents the adapters. The arrow associated with an SPP indicates its orientation and here va > v0. An usual SPP present at the center of circle-1 interacts with both SPPs present within the circle of radius R. Whereas, an adapter SPP at the center of circle-2 interacts only with usual SPPs present within the circle of radius R.
observations. One Monte Carlo (MC) time step corresponds to the up-gradation of position and orientation of all the particles. Initial 3×105 MC steps are neglected to achieve the steady-state. An ensemble of size 48×105 is taken for statistical averages (2×105 samples at different times for 24 different initial configurations).
The results are shown with the velocity of the adapters asva > v0 for a fixed velocity of the usual SPPs v0 = 1.0. Variations of the order parameter φ, Binder cumulant U, and order parameter distribution P(φ) are presented in Fig.4.10, varying the angular noise η for two different velocity of adapters as va= 1.2v0 and va= 7.0v0.
4.4.1 Results with v
a= 1.2v
0and v
0= 1.0
First, we present data forφ(andφp),U (andUp), andP(φ) (andP(φp)) respectively in Fig.4.10(a), (b) and (c) on a system of size L = 256 for va = 1.2v0, v0 = 1.0.
In this case, the velocity va of adapters is almost similar as the usual SPPs. The order parameter φ of the whole system and φp, that of the partial systems, are plotted against ηin Fig.4.10(a). There are jumps in the values of φand φp near the transition. The values ofφ,φ0 and φa are almost similar for a given η in this case.
Chapter 4. Effect of orientation adapters on the collective dynamics of self-propelled particles
0 0.1 0.2η0.3 0.4 0.5 0
0.5 1
φ,φ p φ
φ0 φ
0.2 0.3η0.4 0.5 -16
0
dφ/dη
a
0 0.1 0.2η0.3 0.4 0.5 -5
0 0.7
U,U p U
U0 Ua
0 0.15 φ,φp 0.3
0 0.06
P(φ),P(φ p) P(φ)P(φ0)
P(φ )a
(a) (b) (c)
Figure 4.10: For v0 = 1.0, va= 1.2v0: (a) Plot of φand φp versus η. Derivatives of φ and φp with respect to η are shown in the inset. (b) Plot of U and Up versusη. (c) Plot ofP(φ) and P(φp) at η=ηc. System size isL= 256.
(a) (b)
Figure 4.11: For v0= 1.0, va= 1.2v0: System morphology for η≈ηc with (a) ordered phase (density band) and (b) disorderd phase. Orange colour represents usual SPPs and indigo colour represents the adapters.
The derivatives ofφ and φp with respect to η are plotted in the inset of Fig.4.10(a) and sharp minima are observed at ηc ≈ 0.39 (same for the whole and the partial systems). The Binder cumulants U andUp versus η plots are shown in Fig.4.10(b).
The cumulant for the usual SPPs (U0), for adapters (Ua) and for the whole system (U) all shows a sharp negative dip at the transition. It implies the discontinuous transition for the whole as well as for the partial systems. Then, the distributions of order parameters P(φ) and P(φp) at η = ηc are shown in Fig.4.10(c). In this case, all three distributions P(φ), P(φ0) and P(φa) exhibit bimodal distributions which implies the two-phase co-existence in the system [104, 174]. It seems that at the transition point, dense travelling bands of SPPs form and disappear, resulting in the coexistence of two phases in the system, which is shown in Fig.4.11(a) and (b) respectively. Both the adapters and usual SPPs are observed in the density
4.4 Results and Discussion
0 0.1 0.2η0.3 0.4 0.5 0
0.5 1
φ,φ p φ
φ0 φ
0.2 0.3 0.4 0.5η
-20 0
dφ/dη
a
0 0.1 0.2η0.3 0.4 0.5 0.3
0.5 0.7
U,U p
U U0 Ua
0 0.15 φ,φp 0.3
0 0.06
P(φ),P(φ p) P(φ)P(φ
0) P(φ )a
(a) (b) (c)
Figure 4.12: For v0 = 1.0, va= 7.0v0: (a) Plot of φ and φp versus η. Derivatives of φ and φp with respect to η are shown in the inset. (b) Plot of U and Up versusη. (c) Plot ofP(φ) and P(φp) at η=ηc. System size isL= 256.
band, and they move together in a particular direction (shown by arrows). This is expected as the adapters follow the usual SPPs; they move in the density band’s direction. Moreover, their velocity is similar to the usual SPPs; they also form a dense band pattern near the usual SPPs. The situation will be different for a much higher velocity of the adapters with the same velocity of the usual SPPs.
4.4.2 Results with v
a= 7.0v
0and v
0= 1.0
Next, the situation is going to be drastic if the velocity va of adapters is much higher than the usual SPPs. We present the dataφ(andφp),U (andUp), andP(φ) (and P(φp)) respectively on a system of size L = 256 for va = 7.0v0, v0 = 1.0 in Fig.4.12(a), (b) and (c) respectively. The order parameters φ and φp, are plotted againstηin Fig.4.12(a). The order of the usual SPPs (φ0) is higher than the order of adpters (φa) in the lowηregion in this case. However, near the transition their values are similar. The values of φ and φp are decreasing smoothly to zero asη increases.
The derivatives ofφ and φp with respect to η are plotted in the inset of Fig.4.12(a) and minima of the plots at the transition noise ηc ≈0.37 (same for the whole and partial systems) are observed. In Fig.4.12(b), U andUp are plotted against η. Both U andUp remain positive over the whole range ofη. In Fig.4.12(c), the distribution of order parameters P(φ) and P(φp) are plotted at η = ηc. All the distributions are unimodal. The positive Binder cumulants and unimodal distributions of order parameters indicate a continuous transition in the whole system as well as in the partial systems for the case of va = 7.0v0 and v0 = 1.0.
The system morphologies for the ordered phase withη= 0.30 and for disordered phase with η = 0.42 are shown in Fig.4.13(a) and (b) respectively for the case of
Chapter 4. Effect of orientation adapters on the collective dynamics of self-propelled particles
-π 0 θ π
0 0.06 0.12
P(θ)
v0
va
-π 0 θ π
I II
(a) η= 0.30 (b) η = 0.42 (c) θ-distribution
Figure 4.13: For v0 = 1.0, va = 7.0v0: System morphology for (a) ordered phase η= 0.30 and (b) disorderd phaseη= 0.42 for the system of size L= 256. (c) P(θ) of the morphology (a) and (b) are plotted in I and II. Orange colour represents usual SPPs and indigo colour represents the adapters.
va = 7.0v0 and v0 = 1.0. Below the transition, at η = 0.30 shown in Fig.4.13(a), less dense large clusters of usual SPPs are observed with small clusters of adapters, and they are directed in the same direction (shown by arrow). Whereas, above the transition, at η = 0.42 shown in Fig.4.13(b) they move randomly. The orientation distribution of these two morphologies are shown in Fig.4.13(c)I and II, respectively.
In this case, adapters move with much higher velocity and cannot flock with the usual SPPs. Moreover, they adopt the alignment information from usual SPPs, which are distant. It seems that as the usual SPPs interact with the adapters, randomness enters in the alignment of the usual SPPs. Hence, the correlation between the usual SPPs gets destroyed to form a dense band structure.
Hysteresis study: Simulations are carried out to measure the instantaneous order parameter (φt) by either gradually increasing or decreasing η with a fixed ramp rate as discussed earlier. The simulation results of hysteresis for the whole system with different values of adapter’s velocity as va= 1.2v0, 4.0v0 and 7.0v0 for a fixed v0 = 1.0, are presented in Fig.4.14(a),(b) and (c) respectively. Observations show that when va is relatively less as va = 1.2v0 with v0 = 1.0, there arise an abrupt jump in theφt and the positions of the jump by the implementation of the forward and backward simulations schemes are different. So the emergence of a hysteresis loop indicates the irreversibility of transition [104], shown in Fig.4.14(a). Hysteresis loop is still exist for a moderately high value of va = 4.0v0 shown in Fig.4.14(b), however, it is less prominent than the va = 1.2v0 case. Then, for a large enough value of va = 7.0v0, the change of the order parameter versus η becomes reversible,