Hysteretic dynamics of active particles in a periodic orienting field

Abstract

Active motion of living organisms and artificial self-propelling particles has been an area of intense research at the interface of biology, chemistry and physics. Significant progress in understanding these phenomena has been related to the observation that dynamic self-organization in active systems has much in common with ordering in equilibrium condensed matter such as spontaneous magnetization in ferromagnets. The velocities of active particles may behave similar to magnetic dipoles and develop global alignment, although interactions between the individuals might be completely different. In this work, we show that the dynamics of active particles in external fields can also be described in a way that resembles equilibrium condensed matter. It follows simple general laws, which are independent of the microscopic details of the system. The dynamics is revealed through hysteresis of the mean velocity of active particles subjected to a periodic orienting field. The hysteresis is measured in computer simulations and experiments on unicellular organisms. We find that the ability of the particles to follow the field scales with the ratio of the field variation period to the particles' orientational relaxation time, which, in turn, is related to the particle self-propulsion power and the energy dissipation rate. The collective behaviour of the particles due to aligning interactions manifests itself at low frequencies via increased persistence of the swarm motion when compared with motion of an individual. By contrast, at high field frequencies, the active group fails to develop the alignment and tends to behave like a set of independent individuals even in the presence of interactions. We also report on asymptotic laws for the hysteretic dynamics of active particles, which resemble those in magnetic systems. The generality of the assumptions in the underlying model suggests that the observed laws might apply to a variety of dynamic phenomena from the motion of synthetic active particles to crowd or opinion dynamics.

1. Introduction

The interest to active motion in living or synthetic non-equilibrium systems has recently driven the development of vast interdisciplinary research [1–6]. When one abstracts from the physical mechanisms of propulsion, a self-propelling object is nothing more than an active particle whose motion can be described by laws of physics. The conscious actions, if any, are then mimicked by a set of behavioural rules, adaptive response functions or interaction rules. Even in this simplified set-up, the description of motion of an individual active particle is already a challenging task that might require the introduction of new physical concepts such as adaptive or multi-modal propulsion regimes, or active fluctuations [7–10].

Despite the complexity at the individual level, the collective properties of active particles often look relatively simple. The level of consciousness of the individuals apparently plays a minor role in the large-scale dynamics as the same principles of self-organization that govern the dynamics of groups of animals or cells apply to human social phenomena, traffic, robotics and decision-making [11–17]. One of the best-studied collective dynamics phenomena is the onset of globally aligned motion in active swarms [18–28]. It demonstrates many of the features of the paramagnetic–ferromagnetic phase transition, where the onset of the orientationally ordered ferromagnetic phase is caused by aligning interactions between individual magnetic dipoles of the atoms [19,21,23,29]. In swarms of active agents, often moving at fairly constant propulsion speed, the global alignment is achieved via collective interactions, when the agents adjust their direction of motion to their neighbours. The commonly used order parameter for the swarm, the mean particle velocity, behaves in the same way as the magnetization vector. As these phenomena share the type of symmetry breaking, it is not surprising that they demonstrate similarity in the wide range of macroscopic and statistical properties [19,26,30,31]. It should be noted, however, that the analogy between these two types of systems is far from obvious as we attempt to compare static and dynamic phenomena [5]. In contrast to equilibrium magnetization, the global alignment in swarms is only possible in far-from-equilibrium conditions, where the rate of the external energy supply to the particles is sufficiently high. Moreover, in contrast to molecular systems, the groups of living species are often inhomogeneous, and individual particles can play different roles in the group [15,17,32].

It is tempting to suggest that the similarity between magnets and groups of active particles is not limited to static or steady-state statistical properties but also extends to dynamics. If this is the case, an orienting field can be used to control the direction of motion and ordering similar to the dipole alignment in magnets. While the active particles are unlikely to demonstrate a static hysteresis, at least in the absence of preferred directions in space, there is always room for a dynamic hysteresis controlled by a competition between the external drive and the internal relaxation dynamics. Therefore, exposure of an active group to a periodic field can help to reveal information on the group's orientation relaxation times, field magnitudes or the frequencies required to steer the motion. We can envisage the characterization of the active particle dynamics by dynamic remanence, the ability to preserve the direction of motion in the absence of a field, dynamic coercivity, the magnitude of the field in the opposite direction needed to revert the direction of motion of the group, and susceptibility, the intensity of the response of the group to the action of the field. Moreover, the dynamic hysteresis can provide an insight into the mechanisms of dynamic self-organization in systems with complex interactions, such as animal flocks and herds, human crowds, traffic and social groups.

In this work, we evaluate the characteristics of the dynamics of active particles by steering them with a periodic external field. We perform Langevin dynamics simulation of a two-dimensional system of active Brownian particles with dissipative interactions (ABP-DI) [33–35] and describe the phenomenology of their dynamics. We then introduce the scaling laws governing the response and the ability of the particles to follow the field in the important limiting cases and demonstrate their validity using numerical examples from the ABP-DI model. Finally, we validate the model and one of the scaling laws experimentally, by observing the motion of phototactic unicellular algae confined in a narrow channel under the influence of an alternating light field.

2. Orientational ordering

To study the response of active particles to an alternating external field, we chose a model of active motion with continuous forces, ABP-DI, which can be conveniently studied in a molecular dynamics-type approach. We characterize their motion with the standard polar order parameter—average velocity of individuals [5,14], which turns zero in the disordered state and takes non-zero values that correspond to a net drift of the swarm in the ordered state. In an external field, the group of active (or even passive) particles can demonstrate an induced velocity alignment (similar to the paramagnetic phase) and net drift, so that the motion will have apparent collective features. In contrast to the ferromagnetic state with a global alignment in the absence of a field, where local anisotropy of the lattice may fix the orientation of magnetization in space, the direction of the average velocity in a motile swarm can freely rotate so that the swarm does not follow a straight line at long times but rather performs a persistent random walk. This also implies that in an external orienting field it is only a matter of time until the swarm aligns with the field direction.

The ABP model, as described in our previous work [35] (see also the electronic supplementary material), has been extensively used for studying the motion of microscopic organisms, such as Daphnia or swimming cells [4]. The model is based on the Langevin equation for the velocity V_i of an active particle of mass m

2.1

under the influence of the (i) thrust force (ii) viscous friction (iii) random force mimicking the fluctuations in the environment or noise of other nature (iv) interaction force , and (v) external field The model predicts the existence of a driven motion regime at high energy influx rates, where the particles move with a non-zero preferred speed even in the absence of an external field.

Here, we simulate the ABPs in a two-dimensional narrow linear channel to mimic a generic active motion and also to characterize the motion of motile cells observed in an experiment [35]. The motion of an active particle, as characterized by the velocity vector V, is controlled by a range of parameters, some of which can be directly determined from comparison of the model with experimental systems of interest. The most important parameters for this study are the energy influx rate q, which determines the particle self-propulsion power, the viscous friction coefficient with the environment γ^E, which sets the velocity relaxation time, and the magnitude of velocity fluctuations, which together with the last two parameters determine the particle motion regime and the mean self-propulsion speed. The interaction force provides a mechanism of velocity alignment and speed adjustment between the interacting particles and acts within the interaction zone with radius r_a (figure 1a). It is calculated as pairwise friction that is proportional to the velocity difference between the particles. The smaller interaction zone with radius r_r is responsible for repulsion between the agents (see the electronic supplementary material for a detailed description). In most of the following, except where especially noted, the particle number density is kept constant at Interaction of the particles with channel walls is modelled as a specular reflection (figure 1b). The channel boundaries are periodic in the x-direction.

Figure 1.

Collective dynamics and geometry of a confined group of ABP-DI particles in the absence of the field. (a,b) Interaction parameters in the ABP-DI model: r_r is the radius of repulsion between the particles and between the particles and the walls; r_a is the range of the pairwise friction force, which is responsible for alignment. (c–e) Snapshots from simulations at various energy input rates q. Arrows indicate the direction of motion of the individuals as well as the velocity magnitude. The snapshot in (c) with q = 0 corresponds to a group of passive particles driven by thermal noise only; the snapshot in (d) with q = 1 shows a swarm of active species where the noise competes with the thrust; and in (e) we can see a swarm of particles in the thrust-dominated regime with q = 10. (f,g) A zoom into systems shown in (c,d) at q = 0 and q = 1, respectively. (Online version in colour.)

A few typical snapshots of the ABP-DI system are presented in figure 1. The distribution of interacting particles along and across the channel and the character of motion are visibly affected by the thrust power. While at low incoming energy rates when compared with the thermal noise power (in our units, q ≪ 1) the particles move diffusively with little correlation in their motion, at a certain minimal energy pumping level (in our system, q > 0.73) their motion spontaneously becomes aligned due to dissipative interactions. Another obvious result of increasing the thrust power is the particle aggregation. Already at q = 1, we observe significant density fluctuations and at q = 10 the particles form large compact clusters and start moving in the same direction. The ordering in the system is described by a polar order parameter, φ = 〈V_x〉, the average particle velocity along the channel. In the absence of the field and at low q, the order parameter vanishes due to isotropy of the particle motion. At high input powers, we observe aligned states with a non-zero φ, such that the swarm is migrating along the channel. In most simulation data below, we use the input power q = 1, which corresponds to globally aligned motion of the ABPs in the absence of the external field [35].

3. Hysteresis of the average velocity in a model swarm

In an alternating external field, the ABP motion becomes oscillatory so that the mean particle velocity oscillates as well. To steer the particle migration, we apply a homogeneous external field with a magnitude that oscillates in time as H(t) = H₀ sin ωt, which exerts a force H(t) on the particles along the channel. In the numerical experiments, we vary the field oscillation amplitude H₀ and frequency f = ω/2π. To compare different systems, we will further present the frequency in dimensionless form, multiplied by the order parameter relaxation time τ. The relaxation time is measured from the order parameter relaxation dynamics towards the steady-state φ_∞ at fixed ρ, q, γ^E and H₀

2.2

Upon application of a step-like signal at time t₀,

2.3

Figure 2 shows the measured values of the orientational order parameter as a function of time together with the corresponding field variation curves. At low frequencies, fτ = 0.1, the average velocity varies nonlinearly with the field although it changes in phase with the latter. At fτ = 1, the variation of φ becomes sinusoidal but now exhibits a phase lag compared with the field variation. At the highest frequency, fτ = 10, the order parameter oscillates with small amplitude about a fixed mean value. The mean value in this case depends on the initial conditions.

Figure 2.

Variation of the order parameter for the ABP-DI system at three different frequencies of the driving field: (a) fτ = 0.1, (b) fτ = 1, (c) fτ = 10. Other parameters: ρ = 0.04, H₀ = 1, q = 1. The order parameter values φ(t) are shown by the solid curves while the field H(t) is shown by the dashed curves. The upper set of curves (a) corresponds to the slow field variation, such that the group has time to relax to the steady state and is always in phase with the field. The lower one corresponds to fast field variation, such that the group cannot follow the field and reorient itself completely. Note that the time axis in each subplot is scaled by the corresponding field oscillation period. (Online version in colour.)

The relationship between the field and the order parameter is best illustrated by hysteresis loops, appearing in the H−φ diagrams as shown in figure 3. All the loop shapes we observe here are quite familiar from the magnetic hysteresis [36–38]. Their width (dynamic coercivity—half-width at middle section) increases with the frequency at fτ < 1 and decreases at fτ > 1. Similarly, the stored alignment (dynamic remanence—the amount of net drift that survives the removal of the field) increases with f at low frequencies but decreases at high ones. Both properties contribute to the integral characteristic of hysteretic systems, the area of the loop, which thus reflects the group's overall dynamic controllability (or rather agreeability in this context). In a perfectly controlled system, such that the mean velocity is always in phase with the external field, the loop area becomes zero. By contrast, a large loop area indicates the ‘amount of disagreement’ between the field and the order parameter.

Figure 3.

Typical H–φ diagrams for the ABP-DI system at different frequencies of the driving field at H₀ = 0.4 and q = 1. The shape of the loops is sigmoidal when the field variation period is longer than the group orientational relaxation time, fτ ≪ 1, symmetric ellipsoidal for fτ ≈ 1 and asymmetric ellipsoidal for fτ ≫ 1. The arrows on the curve fτ = 0.025 indicate the direction of traversal of the loop in time. The position of the ellipsoidal loops at high frequencies depends on the initial condition for the average velocity. (Online version in colour.)

We calculated the loop area for fixed parameters of the active particles and interactions in a broad range of field amplitudes and frequencies. In figure 4a, we show the frequency dependence of the loop area A(f) at three different field magnitudes. The trends confirm our previous observations made from the shape of the loops. All the curves show a maximum at the reduced frequency fτ ≈ 1 and feature a power law decay both on increasing and decreasing oscillation frequency. One can also see that the loop area is larger in the stronger fields.

Figure 4.

(a) The hysteresis loop area A as a function of the scaled oscillation frequency at different field strengths H₀ as obtained from simulation of the ABP-DI model at q = 1. The dotted curve is a fit with equation (3.3), which is valid in the strong field regime. The maximum loop area is observed when the field oscillation period matches the group orientational relaxation time. (b) The hysteresis loop area A for a group of active (q = 1) and a group of passive (q = 0) Brownian particles at two different strengths of the field: H₀ = 0.2 and H₀ = 1. The lines correspond to the systems without interactions, while the symbols denote the systems with interacting particles. The discrepancy between the symbols and the lines seen at fτ < 1 indicates the poorer controllability of the interacting particles' motion with the external field. The greater loop area for active particles also evidences their poorer dynamic controllability when compared with the passive ones. (Online version in colour.)

The variation of the loop shape and the area can be understood from the following simple model of dynamic hysteresis [39–41]. The rate of change of the order parameter is described by a differential equation of the form

3.1

where F(φ) is an odd function of the order parameter. A common example of such a function is

3.2

where λ and b are constants. This form of the function arises in a variety of systems. In the limit of zero frequency (equation (3.1)), the relationship between the order parameter and the field in our model, 0 = −λφ + bφ³ − H₀, coincides with the equation of state of a ferromagnet in the (mean-field) Landau theory [42]. For active systems, the common order parameter is the particle velocity, so that equation (3.1) resembles Newton's second law, where the left-hand side gives the acceleration and the right-hand side the sum of forces acting on the particles. For an active Brownian particle with energy depot (see the electronic supplementary material and [10,41,43]), the limit of small velocities or high dissipation rates gives F = −λφ + bφ³ with and , with V₀ being the steady-state speed of the ABPs in the absence of the field, and the evolution of the ensemble-averaged particle velocity is described by equations (3.1) and (3.2).

The explicit asymptotes for A(f) can be derived for certain forms of F(φ) [39,40]. The simplest possible case corresponds to overdamped motion of a passive particle (q = 0) in a viscous medium, such that F(φ) = −λφ = γ^EV. Similarly, the function takes the same form for an ABP-DI system in the strong field limit with playing the role of effective friction coefficient. In this case, it is possible to derive the closed form for the whole A(f) curve:

3.3

We used this function to fit the frequency dependence of the loop area in figure 4a (dashed curve), treating λ as an unknown parameter (note that λ for a single particle is not the same as that for the average velocity). The fit is indeed very good for the high field magnitude, so the average particle velocity follows the same equation of motion in this regime. At low frequencies, the area of the loop also shows a power law dependence on frequency A ∝ f^α, where α varies with the field magnitude H₀. In strong fields, the area is proportional to f, so α = 1, while upon reduction of H₀ the scaling exponent gradually decreases to 0.75. At high frequencies, fτ ≫ 1, all the curves show universal asymptotic behaviour A ∝ f⁻¹. As can be readily seen, the time derivative of φ becomes then very large and dominates the right-hand side of the equation, so dφ/dt ≈ H₀sinωt and a single integration gives φ ∝ −(1/ω) H₀cosωt. Therefore, one can derive a general high-frequency asymptotic result: In the strong field limit, H₀ ≫ q/V₀, the steady-state speed of the particle is given by V = H₀/γ^E and the loop area scales as , as predicted by equation (3.3), which is also a general relation that applies to both active and passive particles. We also found that at fτ ≫ 1 this scaling law is valid at all field magnitudes. We have extensively studied the scaling laws for the loop area in another publication [41].

As the form of equations (3.1) and (3.2) is quite common, one can easily see that the properties we reported above are not specific to the active systems and to active swarms in particular. For instance, the orientational hysteresis of an ensemble of non-interacting passive particles would follow the same asymptotic laws as can be seen in figure 4b, where we show a comparison between passive and active systems, as well as interacting and non-interacting ones. The curve for q = 0 represents a system of passive Brownian particles with dissipative interactions; in essence, a droplet of a viscous fluid. The A(f) function for passive particles is described by equation (3.3) at all field magnitudes. The curves with q = 1 in figure 4b correspond to an active fluid, where each particle generates thrust. We see that the effort required to reorient the active fluid (swarm), as measured by the loop area, is considerably higher. At the frequencies close to the inverse relaxation time, fτ ≈ 1, the reorientation of the active fluid requires twice as much effort, while at the lower frequencies this ratio is even higher. The main observations from the data in figure 4b are as follows:

• — For passive Brownian particles (q = 0), the mutual friction does not affect the loop area. The areas obtained with or without interactions essentially coincide at all frequencies.

• — For active particles (q = 1), the loop area is greater than that for passive ones at all frequencies. The difference is most pronounced at low frequencies, fτ < 1.

• — The interactions between particles lead to an increase in the loop area at low frequencies, fτ < 1, but hardly affect it at high frequencies, fτ > 1.

• — The effect of interactions is most pronounced in weak fields. In strong fields, the differences between the interacting and non-interacting particles, as well as between the passive and active ones, are small.

One obvious result following from the data in figure 4 is the coincidence of the loop area for interacting and non-interacting particles at high field frequencies. Moreover, we see that for the case of the strong field of high frequency, fτ > 1, the motion of an active interacting swarm is almost indistinguishable from that of a group of completely passive individuals (data not shown). Both limits can be understood from our model. The high-frequency regime corresponds to the field oscillation period shorter than the orientational relaxation time that governs the onset of the collective dynamics. In this situation, the particles have no opportunity to align themselves and develop a collective mode. In the strong field regime, the pulling force due to the field overcomes the thrust of the particles and, therefore, dominates the motion.

4. Hysteresis of the mean velocity of living cells

To validate the ABP model and test some of its predictions experimentally, we performed measurements of motion of living cells under the influence of an external field, as described in the electronic supplementary material. In the selection of the object for the experiments, we were guided by the following considerations: firstly, the objects should be sensitive to an external field, show clear and fast behavioural response to the influence of the field and return to the initial state when the stimulus is removed, thus providing good reproducibility of the experiments. Secondly, the objects should allow collection of good statistics. Moreover, we need full control over the field over the full duration of the experiment. We chose microscopic algae Euglena gracilis as the object for our experiments and light as the external field. Euglena cells are well suited for our goals as they exhibit phototactic reactions: positive phototaxis—the cells sense light intensities and turn towards moderately illuminated regions that are more favourable for photosynthesis, as well as negative phototaxis—the cells escape from strong light, the intensity of which is above the photosynthetic ability of these organisms [44]. We used a negative phototaxis regime for our measurements. The light field was produced by light-emitting diodes of continuous spectrum with dominating wavelength λ = 470 nm.

At the first stage, we collected cell velocity statistics at constant lighting conditions and characterized it using the ABP model. We considered a group of cells confined in a channel and subjected to a homogeneous light field. From fitting the velocity distributions (see the electronic supplementary material), we determined the parameters of the cell motion: input power q = 1.8 × 10⁻¹⁶ W, friction coefficient γ^E = 1.2 × 10⁻⁷ N (m s⁻¹)^–1, the mean cell speed V₀ = 26 µm s⁻¹, as well as the velocity fluctuation amplitude and efficiency of motion under the action of homogeneous illumination with blue light. The characteristics of individual motion show that the cells operate in the weak thrust regime, which is a reflection of the strong directional fluctuations in the cell motion. At these low densities, we do not detect any interaction between the cells and they are merely synchronized by the external field. The overall swimming efficiency of the cells was found to be ε = 3%, which is comparable to the typical numbers for other microswimmers [45].

At the second stage, we measured the cell velocities in an alternating field. Typical experimental curves showing evolution of the order parameter at three different frequencies of the sine field are shown in figure 5. For the lowest frequency (figure 5a), we see that the velocity is not proportional to the field amplitude. The shape of the curve φ(t) changes for higher frequency (figure 5b): it becomes sinusoidal; however, the value of the order parameter drops significantly. For the highest frequency (figure 5c), the value of φ decreases even further. All three curves show significant phase shift with respect to the field variation.

Figure 5.

Variation of the order parameter in experiments at three different frequencies of the driving field: (a) fτ = 0.5, (b) fτ = 2, (c) fτ = 5.4. The order parameter values φ(t) are shown by the solid lines, while the field H(t) is shown by the dashed curves. The upper set of curves (a) corresponds to the slow field variation while the lower one corresponds to fast field change. The time axis in each subplot is scaled by the corresponding field oscillation period. (Online version in colour.)

The corresponding H–φ diagrams are presented in figure 6a (also see the electronic supplementary material, video). We find that the curves are symmetric with respect to the change of sign of the field; however, the shape of the curves changes upon variation of the frequency. For the lowest frequency, fτ = 0.5, the shape is slightly sigmoidal, while for the other three, fτ = 1.1, 2 and 5.4, the shape is fully ellipsoidal. As we discussed above, the ellipsoidal shapes correspond to the high-frequency mode. Observation of the deeper low-frequency regime, unfortunately, was not feasible for the present system because it required long-time illumination of the cells with light intensity well above their photosynthetic limit, which might reduce the sharpness of the behavioural response.

Figure 6.

(a) The hysteresis loops: mean velocity of Euglena gracilis cells in a microfluidic channel versus light field intensity at different field oscillation frequencies as obtained from experiment. (b) Theoretical hysteresis loops at different field frequencies as predicted by the ABP model with the parameters for Euglena cells. (Online version in colour.)

An analysis of the cell velocity distributions and fit with the ABP model allows us to predict the behaviour of the mean cell velocity of the Euglena cells in an external field. For this purpose, we use the Fokker–Planck formalism as described in the electronic supplementary material. To model the delayed response of the cells to the variation of the field, we added a phase shift of one-eighth of the period to the field term (figure 5). The time lag corresponding to the phase shift is in the range of a few seconds. The predicted hysteresis loops are plotted in figure 6b. Two of them, fτ = 0.5 and 1.1, show sigmoidal features, while two others, fτ = 2 and 5.4, have the same ellipsoidal shape as in the experiment (figure 6a). The mismatch is caused by a too crude assumption for the phase shift used in the calculation.

Finally, we demonstrate that the dependence of the loop area on the field frequency follows the same universal laws as we saw in the previous section. In figure 7, we show A(fτ) data obtained from the experiment and the theoretical model. The experimental data indicate that there might be a peak at fτ ≈ 1, in agreement with the prediction of our generic model and with simulations. For fτ > 1, the system seems to be in the high-frequency regime, where the area decreases upon increase of f. The area reduction roughly follows the predicted law A ∝ f⁻¹, although the amount of data is not great. The model based on the Fokker–Planck equation shows a limited agreement with the experiment. Although we can predict the loop area quite accurately based on the particle velocity distribution (several points from simulation are close to the experimental ones), and it follows the same scaling law on increasing frequency, the model does not demonstrate a peak at fτ ≈ 1, which suggests that a more advanced model for the cell response to the field might be required. This, however, is beyond the scope of this paper.

Figure 7.

The hysteresis loop area A as a function of frequency f as measured in the experiment on Euglena cells and as predicted by the ABP model. (Online version in colour.)

5. Discussion

The above results show that the motion of active particles in an alternating orienting field demonstrates a dynamic hysteresis with the characteristics depending on the individual particle motion regime (ratio of the propulsive power to dissipation power), on the degree of collectivity of their motion, and on the strength of the orienting field. In the simplest limiting cases (weak interactions, strong field), the observed hysteresis phenomenon does not differ qualitatively from that of other dynamic systems such as the ensemble of magnetic dipoles [37,41] or the ensemble of passive Brownian particles. The analogy allows us to abstract from the biological details of the specific system and discuss the generic dynamic properties of an active system with delayed response and dissipation. The loop area is related to both the ‘persistence’ of the group's motion (dynamic remanence—the amount of drift after the field is removed) and its ‘stubbornness’ (dynamic coercivity—the field magnitude required to revert the motion of the group). For example, these observations can be interpreted in terms of opinion dynamics [13,16,46–48], where the consensus can be associated with the aligned state with a non-zero order parameter, while the alternating field would model the frequent contradictory stimuli. Our results suggest that the prominent social (aligning) interactions can enhance the persistence of the group's opinion at relatively low signal frequencies, while frequent contradictory signals prevent the development of a collective response.

The loop area as a function of the scaled frequency shows a characteristic strong peak at fτ ≈ 1, which emphasizes the central role of the orientational relaxation time. At frequencies lower than 1/τ, both passive and active particles have enough time to align with the field and, therefore, the group follows the field direction more obediently and can develop high mean velocities (figure 2a). If the low-frequency field is applied to a passive system, one would observe synchronized rather than self-organized collective motion. By contrast, at high frequencies, fτ ≫ 1, the velocity variation is never too large (figure 2c). The mean position of the hysteresis loop depends on the initial conditions. If the group was already in the aligned state before the field was applied, the high-frequency field cannot change its predominant orientation but can cause only small oscillation about the mean value of the velocity. The particles are merely accelerating or decelerating but progressing along their way all the same. If the group is in the disordered state (non-interacting particles, passive particles or a noisy interacting system) when the high-frequency field is switched on, it would never be able to acquire a high speed but would rather wiggle about zero velocity and about the same position in space. The motion would resemble a somewhat synchronized spinning. In this regime, the velocity follows the field direction only half of the time, which is reflected in the small loop area. The high-frequency behaviour, however, cannot be that easily interpreted in terms of controllability. By contrast, at fτ ≈ 1, the order parameter is in antiphase with the field most of the time, which is reflected in the large loop area (figure 2b). This can be understood as low dynamic controllability or lack of ‘obedience’ of the group.

In the low-frequency regime, fτ ≪ 1, the hysteresis can probe the collective dynamics of the active agents, as the particles have time to develop the ordered phase (consensus). In figure 4b, we see a significantly larger area A for the interacting active swarms at q = 1 than for the non-interacting counterpart (note the logarithmic scale of the graph). At the same time, interactions in the system of passive particles (q = 0) do not reveal any dependence of the loop area A on the interaction. So, we see that in all cases the exponent β (A ∝ f^β) less than unity appears in systems with interactions [41] and can therefore serve as an indicator of self-organized collective behaviour. By contrast, in the high-frequency regime, fτ ≫ 1, the particles have no chance to self-organize, at least on a larger scale, so that the group behaves as a set of disconnected individuals, which are merely perturbed by the field, no matter whether they are able to interact or not. The curves A(f) at fτ ≫ 1 show no difference between the interacting and non-interacting systems. We therefore can speak of a breakdown of collective behaviour under the action of orienting signals of high frequency.

6. Conclusion

We studied the dynamics of active particles in external fields using the method and ideas from condensed matter physics. Active particles in an alternating external field exhibit a dynamic hysteresis, which is qualitatively identical to that observed in magnets. We measured the orientational hysteresis of the mean velocity for groups of simulated active Brownian particles with dissipative interactions as a function of field oscillation frequency. The response of an active group to the field variation depends on the ratio of the orientational relaxation time to the field oscillation period. At high field frequencies and/or high field amplitudes, the collective component of the behaviour becomes negligible and the group behaves as a collection of independent individuals, while at low frequencies the particles can develop collective dynamics. The active particles are least disposed to follow the field direction in the intermediate frequency regime where the field variation period is close to their relaxation time. We discuss the theoretical foundations of this behaviour using a simple dynamic model. Experiments performed on living motile cells support the theory predictions. Moreover, the generality of the assumptions of the theory suggests that the described dynamic features may be observed in groups of active agents of any nature.

Data accessibility

The datasets supporting this article are available from http://dx.doi.org/10.5061/dryad.bh52d.

Authors' contributions

M.R. and V.L. developed the computational model; M.R. performed the simulations and experiments; V.L. and M.R. processed the experimental results and did the model parametrization; D.S. developed the experimental methodology, prepared the Euglena cells and set up the particle-tracking experiment; M.R. designed and assembled the lighting and electronic scheme; V.L. supervised the project. All authors developed the concept and wrote the paper.

Competing interests

We declare we have no competing interests.

Funding

Financial support from the Irish Research Council for Science, Engineering and Technology (IRCSET) is gratefully acknowledged. M.R. was also supported through European Research Council grant IDCAB 220/104702003 to D.J.T. Sumpter.