Collective motion of active particles in environmental noise
Cite this Article
Chen Qiu-shi, Ji Ming. Collective motion of active particles in environmental noise. Chinese Physics B, 2017, 26(9): 098903 
Permissions
Collective motion of active particles in environmental noise
† Corresponding author. E-mail:
Abstract
We study the collective motion of active particles in environmental noise, where the environmental noise is caused by noise particles randomly diffusing in two-dimensional space. We show that active particles in a noisy environment can self organize into three typical phases: polar liquid, band, and disordered gas states. In our model, the transition between band and disordered gas states is discontinuous. Giant number fluctuation is observed in the polar liquid phase. We also compare our results with the Vicsek model and show that the interaction with noise particles can stabilize the band state to very low noise condition. This band structure could recruit most of the active particles in the system, which greatly enhances the coherence of the system. Our findings of complex collective behaviors in environmental noise help us to understand how individuals modify their self-organization by environmental factors, which may further contribute to improving the design of collective migration and navigation strategies.
1. Introduction
Collective behavior of active matter has attracted many physicists’ attention recently. It displays various fascinating patterns at every scale, down to molecular motors in the cell, up to large animal groups.[1] For example, actin filaments perform persistent random walk, wave-like structures, spirals and swirls,[2–5] the bacillus subtilis grows a peculiar concentric ring-like pattern,[6,7] E. coli in the lab self organizes into a highly ordered phase through growth and division in a dense colony.[8] Similarly, large living organisms such as locusts perform a disorder to order transition,[9] fish schools and bird flocks provide some complex patterns such as travelling band, milling, and cluster.[10–13] The reason why active matters form evolutionary patterns remains unclear. One possible explanation would be that there exists some inherent benefit to overcome environmental perturbations or other distractions.[14–17] However, there is scarce empirical information about the precise interaction rules between the components of active matters because of the technological difficulties.[18] Physicists are trying to find a minimal model to study these unexpected collective properties.[19,20]
Vicsek and collaborators provided a metric interaction self-propelled particles model that exhibits a disorder to order phase transition.[21] In this model, particles move in a constant speed and interact locally with their neighbors within a certain radius to keep alignment with the group members. It is interesting that a collective pattern occurs with large particle density and small noise intensity. Some theoretical description of the dynamics of the flocking behavior for self-propelled particles was proposed by Toner and Tu.[22,23] Beyond the previous investigation on the nature of the original Vicsek model, people proposed many variants to describe other kinds of active matter systems. Most of these models only change the angular interaction rules with their neighbors, and usually these models can exhibit spectacular collective behaviors that are reminiscent to fascinating dynamic patterns.[24–33] Indeed, the collective behavior of organisms is responsive to two kinds of interactions: social interaction with their neighbors and interactions with the surrounding environment.[14,15] However, most previous studies in literature mainly focus on the collective motion in response to nearby neighbors, less concentrating on the environmental factor. Individuals may adopt appropriate moving patterns that facilitate group motion in an environmentally dependent way.[15] For example, under environmental stimuli or threat, bird flocks would align more strongly with their neighbors to keep cohesion.[14] Fish schools form a large size group in the alarm treatment and a small one in the food treatment.[34] Bacteria in colonies forms various patterns on artificial surfaces.[35] In particular, E. coli. employs run and tumble locomotion upon environmental stimuli.[36–38] In summary, environmental factors have a great effect on individual and collective behaviors.
In this paper, we introduce a Vicsek-like model that contains two kinds of particles to investigate the collective motion of the system: active particles that keep alignment with their neighbors and the noise particles randomly diffusing in two dimensional space that lead to the environmental noise introduced here. We study the generic phase behavior of the active particles interacting with the noise particles, in competition with inherent noise and alignment interaction. We show that, under a noisy environment, the system exhibits three typical phases: polar liquid, band and disordered gas states. We also compare our results with the original Vicsek model. In our system, the band state can exist even in a relative low noise region. This effect could help to increase the spatial coherence in collective motion.
2. Methods and models
where 
2 ), v is a constant moving speed of the active particle. For instance, if γ = 0, our model reduces to the original Vicsek model. There is another scenario with η = 0, which corresponds to the Vicsek model with vector-form environmental noise. Finally, we regard the noise particle as a Brownian particle with constant speed v but randomly orientated in discrete time. Our model has two main control parameters: the active particle density ρa=Na/L2 and the noise particle density ρn = Nn/L2, where Na and Nn correspond to the total active particles’ number and noise particles’ number, respectively.
We consider a modified version of the Vicsek model for Na active particles moving off lattice in a two-dimensional space of linear size L, with periodic boundary conditions, interacting via polar alignment with their neighbors[21] in competition with environmental noise. The noise intensity is proportional to the local density of noise particles. The form of environmental noise in the update rule is analogous to the vector noise introduced in Ref. 39 and inherent scalar noise is also considered. We express the evolution of the j-th particle according to
 |
 |
3. Results
Here, we mostly report on the system with v = 0.5, ρa = Na/L2 = 1, R = 1, γ = 5, and time interval Δt = 1. To characterize the global degree of orientational order, we consider the following order parameter, which is defined as 
As we change the noise level η and noise particles’ density ρn, we find three typical phases as shown in Fig.
 | Fig. 1. (color online) Phase diagram in the (ρnη) plane. The red region corresponds to the polar liquid state. The blue region corresponds to the band state. The green region corresponds to the disordered gas state. |
 | Fig. 2. (color online) (a) The snapshot of polar liquid state with ρn = 0.1 and η = 0.1. The snapshots of band state for different parameters: (b) ρn = 0.1 and η =0.3, (c) ρn = 1 and η = 0.1, (d) ρn = 1 and η = 0.3, respectively. The black dots represent the active particles, while the red dots represent the noise particles. |
We study the phase transition between different phases on the phase diagram in detail. Firstly, we show the phase transition from the band state to the disordered gas state by increasing noise η. The transition could be well characterized by the measurement of the polar order parameter. At the low noise region, as the noise intensity increases, the order parameter decreases slowly. Further increasing the noise intensity, when it is close to the transition point, the order parameter φ drops sharply to zero, as shown in Fig.
 | Fig. 3. (color online) (a) The time-averaged order parameter φ vs noise strength η for various densities of noise particles. (b) Binder cumulant value G. (c) Piece of order parameter time series around the transition point (ρn = 0.3 and η = 0.4). (d) Order parameter distribution around the transition point ρn = 0.3). |
We turn our attention to the question whether the order to disorder phase transition is discontinuous, the same as that in the Vicsek model. A direct method to distinguish first-order phase transition and second-order phase transition is to measure Binder cumulant value G, which is defined as 
We also show the time series of the order parameter around the transition point in Fig.
Secondly, we study the phase transition from the polar ordered liquid state to the band state on the phase diagram. It is difficult to observe this transition in the measurement of the polar order parameter, because both of these states are of high orientational order. The main difference between these two phases is the existence of the well organized band structures. A clear travelling band is observed in the band state in Figs. 
 | Fig. 4. (color online) (a) The time-averaged variances of density profiles as a function of noise strength η. (b) The time-averaged variances of order parameter profiles as a function of η. (c) The density profile in the band state. (d) Local order parameter profiles for panel (c). Only active particles are shown. Parameters: L = 128, ρn = 1, and η = 0.3. |
For larger noise strength near η = 0.33, robust band structures lead to strong spatial inhomogeneity. As shown in Figs.
As we further increase the noise intensity η, the bands vanish, leaving a spatially homogeneous disordered phase (Fig. 
 | Fig. 5. (color online) (a) Giant number fluctuations that the root mean square ∆ n as a function of particles n contained in boxes of various linear sizes. The green line is a power-law of slope 0.75 ρn = 0.1 and η = 0.1. (b) PDF of coarse-grained density |
Finally, we consider the limiting case when ρn = 0. In this case our model reduces to the original Vicsek model. Recently, the order–disorder transition in the original Vicsek model could be understood as a liquid–gas transition rather than an order–disorder phase transition.[40] Our findings are in agreement with such a scenario that as the noise intensity η increases, the system exhibits three phases: polar liquid at low noise, micro-phase separation with band structure at mediate noise, disordered gas at high noise. In our simulation, we show a phase transition from a disordered state to an ordered one. This order-disorder transition is also observed in experiments on locusts and fish schools. As the density of noise particles increases, there is a rapid transition from highly synchronized behavior to disordered state. Our model is useful for a qualitative understanding of such phenomena. For a low ρn condition, each particle only interacts with a small number of noise particles. Thus, their behaviors mainly depend on the alignment interaction. As the number of noise particles increases, each particle has more “noise” neighbors rather than “active” neighbors. The noise leads to a lower alignment order. In the band state, the particles form some clusters and rapidly aggregate into ordered bands, travelling in a disordered background. We also show that the band state could be extended into a finite ρn region, but the polar liquid phase as defined above would shrink with the increase of ρn. In general, the system may stay in the coherent moving band state at very low noise, with the introduction of noise particles. While in the original Vicsek model, the band state at the low noise condition already disappears because of the transition to a polar liquid state.
4. Conclusion
In this paper, we study the collective motion of self-propelled particles in environmental noise. We explore the (ρn, η) parameters plane and show three typical phases: polar liquid, band, and disordered states. When ρn approaches zero, the system reduces to the original Vicsek model. In comparison with the phase transition in the Vicsek model, we study the phase transition in noise particles condition and find that the transition from order to disorder state is strongly discontinuous, which is in agreement with the Vicsek model. For finite ρn, the disorder region becomes larger because the noise particle can be regarded as a noise source. At the same time, the transition point from band state to liquid phase also shifts to the low noise region. If η is low enough, for finite density of noise particles, active particles can recruit most of the particles into the band structure. This greatly enhances the spatial coherence of the system in the low η condition. However, in the original Vicsek model, the system is spatially homogeneous because of the transition to the polar liquid state. Our findings of complex collective behaviors in environmental noise help us to understand how individuals modify their self-organization by environmental factors, which may further contribute to improving the design of collective migration and navigation strategies.
Reference
| [1] | |
| [2] | |
| [3] | |
| [4] | |
| [5] | |
| [6] | |
| [7] | |
| [8] | |
| [9] | |
| [10] | |
| [11] | |
| [12] | |
| [13] | |
| [14] | |
| [15] | |
| [16] | |
| [17] | |
| [18] | |
| [19] | |
| [20] | |
| [21] | |
| [22] | |
| [23] | |
| [24] | |
| [25] | |
| [26] | |
| [27] | |
| [28] | |
| [29] | |
| [30] | |
| [31] | |
| [32] | |
| [33] | |
| [34] | |
| [35] | |
| [36] | |
| [37] | |
| [38] | |
| [39] | |
| [40] | |
| [41] |