Mean-field theory of collective motion due to velocity alignment

We introduce a system of self-propelled agents (active Brownian particles) with velocity alignment in two spatial dimensions and derive a mean-field theory from the microscopic dynamics via a nonlinear Fokker-Planck equation and a moment expansion of the probability distribution function. We analyze the stationary solutions corresponding to macroscopic collective motion with finite center of mass velocity (ordered state) and the disordered solution with no collective motion in the spatially homogeneous system. In particular, we discuss the impact of two different propulsion functions governing the individual dynamics. Our results predict a strong impact of the individual dynamics on the mean field onset of collective motion (continuous vs discontinuous). In addition to the macroscopic density and velocity field we consider explicitly the dynamics of an effective temperature of the agent system, representing a measure of velocity fluctuations around the mean velocity. We show that the temperature decreases strongly with increasing level of collective motion despite constant fluctuations on individual level, which suggests that extreme caution should be taken in deducing individual behavior, such as, state-dependent individual fluctuations from mean-field measurements [Yates {\em et al.}, PNAS, 106 (14), 2009].Comment: corrected version, Ecological Complexity (2011) in pres

Swarming and Pattern Formation due to Selective Attraction and Repulsion

We discuss the collective dynamics of self-propelled particles with selective attraction and repulsion interactions. Each particle, or individual, may respond differently to its neighbors depending on the sign of their relative velocity. Thus, it is able to distinguish approaching (coming closer) and moving away individuals. This differentiation of the social response is motivated by the response to looming visual stimuli and may be seen as a generalization of the previously proposed, biologically motivated, escape and pursuit interactions. The model can account for different types of behavior such as pure attraction, pure repulsion, or escape and pursuit depending on the values (signs) of the different response strengths, and provides, in the light of recent experimental results, an interesting alternative to previously proposed models of collective motion with an explicit velocity-alignment interaction. We show the onset of large scale collective motion in a subregion of the parameter space, which corresponds to an effective escape and/or pursuit response. Furthermore, we discuss the observed spatial patterns and show how kinetic description of the dynamics can be derived from the individual based model.Comment: Preprint, 24 pages, submitted to Interface Focu

Self-propelled particles with selective attraction-repulsion interaction - From microscopic dynamics to coarse-grained theories

In this work we derive and analyze coarse-grained descriptions of self-propelled particles with selective attraction-repulsion interaction, where individuals may respond differently to their neighbours depending on their relative state of motion (approach versus movement away). Based on the formulation of a nonlinear Fokker-Planck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of a one-particle density function. This approach allows effective numerical investigation of the stability of possible solutions of the system. The detailed analysis of the interaction integrals entering the equations demonstrates that divergences at small wavelengths can appear at arbitrary expansion orders. Further on, we also derive a hydrodynamic theory by performing a closure at the level of the second Fourier mode of the one-particle density function. We show that the general form of equations is in agreement with the theory formulated by Toner and Tu. Finally, we compare our analytical predictions on the stability of the disordered homogeneous solution with results of individual-based simulations. They show good agreement for sufficiently large densities and non-negligible short-ranged repulsion. Disagreements of numerical results and the hydrodynamic theory for weak short-ranged repulsion reveal the existence of a previously unknown phase of the model consisting of dense, nematically aligned filaments, which cannot be accounted for by the present Toner and Tu type theory of polar active matter.Comment: revised version, 37pages, 11 figure

Phase Transitions and Criticality in the Collective Behavior of Animals -- Self-organization and biological function

Collective behaviors exhibited by animal groups, such as fish schools, bird flocks, or insect swarms are fascinating examples of self-organization in biology. Concepts and methods from statistical physics have been used to argue theoretically about the potential consequences of collective effects in such living systems. In particular, it has been proposed that such collective systems should operate close to a phase transition, specifically a (pseudo-)critical point, in order to optimize their capability for collective computation. In this chapter, we will first review relevant phase transitions exhibited by animal collectives, pointing out the difficulties of applying concepts from statistical physics to biological systems. Then we will discuss the current state of research on the "criticality hypothesis", including methods for how to measure distance from criticality and specific functional consequences for animal groups operating near a phase transition. We will highlight the emerging view that de-emphasizes the optimality of being exactly at a critical point and instead explores the potential benefits of living systems being able to tune to an optimal distance from criticality. We will close by laying out future challenges for studying collective behavior at the interface of physics and biology.Comment: to appear in "Order, disorder, and criticality", vol. VII, World Scientific Publishin

Collective predator evasion: Putting the criticality hypothesis to the test

According to the criticality hypothesis, collective biological systems should operate in a special parameter region, close to so-called critical points, where the collective behavior undergoes a qualitative change between different dynamical regimes. Critical systems exhibit unique properties, which may benefit collective information processing such as maximal responsiveness to external stimuli. Besides neuronal and gene-regulatory networks, recent empirical data suggests that also animal collectives may be examples of self-organized critical systems. However, open questions about self-organization mechanisms in animal groups remain: Evolutionary adaptation towards a group-level optimum (group-level selection), implicitly assumed in the "criticality hypothesis", appears in general not reasonable for fission-fusion groups composed of non-related individuals. Furthermore, previous theoretical work relies on non-spatial models, which ignore potentially important self-organization and spatial sorting effects. Using a generic, spatially-explicit model of schooling prey being attacked by a predator, we show first that schools operating at criticality perform best. However, this is not due to optimal response of the prey to the predator, as suggested by the "criticality hypothesis", but rather due to the spatial structure of the prey school at criticality. Secondly, by investigating individual-level evolution, we show that strong spatial self-sorting effects at the critical point lead to strong selection gradients, and make it an evolutionary unstable state. Our results demonstrate the decisive role of spatio-temporal phenomena in collective behavior, and that individual-level selection is in general not a viable mechanism for self-tuning of unrelated animal groups towards criticality

Spatial structure and information transfer in visual networks

In human and animal groups, social interactions often rely on the transmission of information via visual observation of the behavior of others. These visual interactions are governed by the laws of physics and sensory limits. Individuals appear smaller when far away and thus become harder to detect visually, while close by neighbors tend to occlude large areas of the visual field and block out interactions with individuals behind them. Here, we systematically study the effect of a group's spatial structure, its density as well as polarization and aspect ratio of the physical bodies, on the properties of the visual interaction network. In such a network individuals are connected if they can see each other as opposed to other interaction models such as metric or topological networks that omit these limitations due to the individual's physical bodies. We study the effect that spatial configuration has on the static properties of these networks as well as its influence on the transmission of information or behaviors which we investigate via two generic models of social contagion. We expect our work to have implications for the study of animal groups, where it could inform the study of functional benefits of different macroscopic states. It may also be applicable to the construction of robotic swarms communicating via vision or for understanding the spread of panics in human crowds

Individuality in Swarm Robots with the Case Study of Kilobots: Noise, Bug, or Feature?

Inter-individual differences are studied in natural systems, such as fish, bees, and humans, as they contribute to the complexity of both individual and collective behaviors. However, individuality in artificial systems, such as robotic swarms, is undervalued or even overlooked. Agent-specific deviations from the norm in swarm robotics are usually understood as mere noise that can be minimized, for example, by calibration. We observe that robots have consistent deviations and argue that awareness and knowledge of these can be exploited to serve a task. We measure heterogeneity in robot swarms caused by individual differences in how robots act, sense, and oscillate. Our use case is Kilobots and we provide example behaviors where the performance of robots varies depending on individual differences. We show a non-intuitive example of phototaxis with Kilobots where the non-calibrated Kilobots show better performance than the calibrated supposedly ``ideal" one. We measure the inter-individual variations for heterogeneity in sensing and oscillation, too. We briefly discuss how these variations can enhance the complexity of collective behaviors. We suggest that by recognizing and exploring this new perspective on individuality, and hence diversity, in robotic swarms, we can gain a deeper understanding of these systems and potentially unlock new possibilities for their design and implementation of applications.Comment: Accepted at the 2023 Conference on Artificial Life (ALife). To see the 9 Figures in large check this repo: https://github.com/mohsen-raoufi/Kilobots-Individuality-ALife-23/tree/main/Figure