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Perturbations in Flocking Systems

Final Report Summary - PIFS (Perturbations in Flocking Systems)

Active matter is composed by a large number of active particles, each of which consumes energy in order to move or to exert mechanical forces, a situation common for instance to many biological systems. Due to energy consumption, these systems are inherently out of thermodynamic equilibrium, and the study of active matter collective properties is nowadays fast-emerging interdisciplinary research field, which links out-of-equilibrium statistical physics with biological as well as engineering-related topics such as swarming robots.
Biological examples range from flocks of birds and school of fish to bacteria colonies, migrating cellular tissues and self-organising bio-polymers such as microtubules and actin, both of which are part of the cytoskeleton of living cells.

In the course of this project, we investigated the response of moving groups to perturbations, a question of both of theoretical interest and of practical importance in order to be eventually able to control active matter systems.

The most obvious origin of these perturbations is obviously exogenous, that is perturbations due to external environmental stimuli such as attacking predators, interactions with competing species or the perception of non-homogeneous landscapes.
In a first work, we have characterized – through analytical and numerical means – the long-time response of flocks to small perturbations. We have established a scaling relation that connects the response to the magnitude of the external perturbation, thus effectively extending classic equilibrium linear response theory results to far-from-equilibrium flocking systems.

Further work has been carried on the short-time response. Our results suggest that the response right after a perturbation (in the form of a small external field) is applied seem to satisfy, at least to a good approximation, a generalised fluctuation-dissipation relation: The system response being proportional to the innate, internal fluctuations of the system. Moreover, the proportionality constant can be thought of as an ‘effective temperature’; there appear to be two different such temperatures, one in the transversal direction to the mean orientation of the flock and one in the longitudinal. This is to be expected in out of equilibrium systems where the uniqueness of such an effective temperature is not guaranteed.

But perturbations may also be of endogenous nature: even in absence of external stimuli, individuals may suddenly and spontaneously switch their behavioral patterns, subsequently inducing changes in the group behaviour at the collective level.
We have investigated an example of this latter case in a concrete example ethological interest. In collaboration with a French group, we studied the behavior of large groups of Merino sheep, a highly gregarious social animal. While grazing, these sheep must balance two competing needs: (i) the maximization of individual foraging space and (ii) the protection from predators offered by a large dense group. Our results indicate that they resolve this conflict
by alternating slow foraging phases—during which the group spreads out—with fast packing events triggered by an individual level behavioral shift. This leads to an intermittent collective
dynamics with large density oscillations triggered by packing events on all accessible scales: a quasi-critical state. We have also developed an explicit model with individual behavioral shifts and strong allelomimetic properties that well accounts for our field observations.

During the course of this project, we have realized that in order to better understand the response to localized and finite perturbations applied to finite flocks (that is, realistic flock models in which the flock interacts with the external world through its boundaries), a better characterization of finite models with cohesion was needed. To this end, we have performed a first numerical study of the stability of finite flocking models in three spatial dimensions by introducing proper surface tension terms. Interestingly, our results indicate that model flocks are characterized by faster boundary oscillations than the ones typically observed in, say, equilibrium droplets. These fluctuations propagate to the bulk, affecting flock properties such as the degree of internal correlation (which are therefore different with respect to the one predicted by standard hydrodynamic theories for non-bounded, infinite flocks). This is an interesting and partially unexpected new result that will probably result in a new future research line.
Altogether, our numerical results show that the dynamical information inflow from the flock
boundary (due to spontaneous fluctuations and/or external perturbations) gives rise to strong bird-to-bird correlations akin to the ones observed in real flocks, thus confirming one of the underlying hypothesis of this project. Analogous results have also been obtained in a two dimensional Vicsek model analysed by an Aberdeen physics student in her undergraduate thesis (honours project).

Ancillary research work funded or partially funded by this project includes the development of a dynamical maximum entropy approach for flocking systems, that is, a new method to infer
individual dynamics from observation at the collective flock level of collectively moving flocks. Application of this method to actual flock data provided strong indications that bird orientations are in a state of local quasi-equilibrium over the individual-to-individual interaction length scale. This result provides firm ground for the applicability of statistical physics techniques and results to these systems.


Finally, we also developed a general theoretical framework, the Boltzmann-Ginzburg-Landau approach, to derive continuous equations for systems of active particles. Such approach provides a clearer link between microscopic models and the continuous hydrodynamic theories we have employed, for instance, in the study of the asymptotic response to external perturbations.

Interestingly, in a recent collaboration with an Italian experimental group, we have also been able to experimentally test for the first time hydrodynamic theory results in an in vitro confluent monolayer of epithelial cells which shows flocking. Our results confirm theoretical predictions such as the celebrated giant number fluctuations which are a cornerstone of collective motion theory.


To summarize, the results of this research project have contributed in increasing our theoretical understanding of flocking systems and active matter. In particular, our work has highlighted the importance of boundaries and external perturbation when dealing with realistic finite system which are inevitably in interaction with a complex external world.

Moreover, this project also contributed to the training of Nikos Kyriakoupolos, an early stage researcher co-funded by this grant. Dr. Kyriakoupolos successfully defended his thesis and received a PhD in physics from the University of Aberdeen in February 2016.