Long-Lived Giant Number Fluctuations in a Swarming Granular Nematic

Coherently moving flocks of birds, beasts, or bacteria are examples of living matter with spontaneous orientational order. How do these systems differ from thermal equilibrium systems with such liquid crystalline order? Working with a fluidized monolayer of macroscopic rods in the nematic liquid crystalline phase, we find giant number fluctuations consistent with a standard deviation growing linearly with the mean, in contrast to any situation where the central limit theorem applies. These fluctuations are long-lived, decaying only as a logarithmic function of time. This shows that flocking, coherent motion, and large-scale inhomogeneity can appear in a system in which particles do not communicate except by contact.

¹ Center for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India.
² Condensed Matter Theory Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India.
³ Department of Physics, University of Massachusetts, Amherst, MA 01003, USA.

^* To whom correspondence should be addressed. E-mail: vj{at}physics.iisc.ernet.in

Density is a property that one can measure with arbitrary accuracy for materials at thermal equilibrium simply by increasing the size of the volume observed. This is because a region of volume V, with N particles on average, ordinarily shows fluctuations with standard deviation N proportional to , so that fluctuations in the number density go down as . Liquid crystalline phases of active or self-propelled particles (1–4) are different, with N predicted (2–5) to grow faster than and as fast as N in some cases (5), making density an ill-defined quantity even in the limit of a large system. These predictions show that flocking, coherent motion, and giant density fluctuations are intimately related consequences of the orientational order that develops in a sufficiently dense grouping of self-driven objects with anisotropic body shape. This has substantial implications for biological pattern formation and movement ecology (6): The coupling of density fluctuations to alignment of individuals will affect populations as diverse as herds of cattle, swarms of locusts (7), schools of fish (8, 9), motile cells (10), and filaments driven by motor proteins (11–13).

We report here that persistent giant number fluctuations and the coupling of particle currents to particle orientation arise in a far simpler driven system, namely, an agitated monolayer of rodlike particles shown in (14) to exhibit liquid crystalline order. These fluctuations have also been observed in computer simulations of a simple model of the flocking of apolar particles by Chaté et al. (15). The rods we used were cut to a length l = 4.6 ± 0.16 (SEM) mm from copper wire of diameter d = 0.8 mm. The ends of the rods were etched to give them the shape of a rolling pin. The rods were confined in a quasi-two-dimensional cell 1 mm tall and with a circular cross-section 13 cm indiameter. Thecellwas mounted in the horizontal plane on a permanent magnet shaker and vibrated vertically at a frequency f = 200 Hz, with an amplitude, A, between 0.025 and 0.043 mm. The resultant dimensionless acceleration =(4²f ²A)/g, where g is the acceleration due to gravity, varies between =4 and = 7. We varied the total number of particles in the cell, N_total, between 1500 and 2820. N_total in each instance was counted by hand. The area fraction, , occupied by the particles is the total projected area of all the rods divided by the surface area of the cell. varies from 35% to 66%. Our experimental system is similar to those used to study the phase behavior of inelastic spheres (16, 17). Galanis et al. (18) shook rods in a similar setup, albeit with much less confinement in the vertical direction. The particles were imaged with a digital camera (19).

The rods gain kinetic energy through frequent collisions with the floor and the ceiling of the cell. Because the axes of the particles are almost always inclined to the horizontal, these collisions impart or absorb momentum in the horizontal plane. Collisions between particles conserve momentum but also drive horizontal motion by converting vertical motion into motion in the plane. Interparticle collisions as well as particle-wall collisions are inelastic, and all particle motion would cease within a few collision times if the vibrations were switched off. The momentum of the system of rods is not conserved either, because the walls of the cell can absorb or impart momentum. The rods are apolar; that is, individual particles do not have a distinct head and tail that determine fore-aft orientation or direction of motion and can form a true nematic phase. The experimental system thus has all the physical ingredients of an active nematic (1–4).

The system is in a very dynamic steady state, with particle motion (movie S1) organized in macroscopic swirls. Swirling motions do not necessarily imply the existence of giant number fluctuations (20, 21); however, particle motions in our system generate anomalously large fluctuations in density. Figure 1A shows a typical instantaneous configuration, and the Fig. 1B inset showsthe orientational correlation function G₂(r)= cos2(_i – _j) , where i,j run over pairs of particles separated by a distance r and oriented at angles _i and _j with respect to a reference axis. The angle brackets denote an average over all such pairs and about 150 images spaced 15 s apart in time. The data in the inset show that the systems with N_total = 2500 and N_total = 2820 display quasi–long-ranged nematic order, where G₂(r) decays as a power of the separation, r. On the other hand, the system with N_total = 1500 shows only short-ranged nematic order, with G₂(r) decaying exponentially with r. Details of the crossover between these two behaviors can be found in [Supporting Online Material (SOM) text]. Autocorrelations of the density field as well as of the orientation of a tagged particle decay to zero on much shorter time scales (SOM text), so we expect these images to be statistically independent. To quantify the number fluctuations, we extracted from each image the number of particles in subsystems of different size, defined by windows ranging in size from 0.1l by 0.1l to 12l by 12l. From a series of images we determined, for each subsystem size, the average N and the standard deviation, N, of the number of particles in the window. For any system in which the number fluctuations obey the conditions of the central limit theorem (22), should be a constant, independent of N. Figure 1B shows that, when the area fraction is large, is not a constant. Indeed, for big enough subsystems, the data show giant fluctuations, N, in the number of particles, growing far more rapidly than and consistent with a proportionality to N. For smaller average number density, where nematic order is poorly developed, this effect disappears, and is independent of N, as in thermal equilibrium systems. The roll-off in at the highest values of N is a finite-size effect: For subsystems that approach the size of the entire system, large number fluctuations are no longer possible because the total number of particles in the cell is held fixed.

Fig. 1. Giant number fluctuations in active granular rods. (A) A snapshot of the nematic order assumed by the rods. There are 2820 particles (counted by hand) in the cell (area fraction is 66%) being sinusoidally vibrated perpendicular to the plane of the image, at a peak acceleration of = 5. The sparse region at the top between 10 and 11 o'clock is an instance of a large density fluctuation. These take several minutes to relax and form elsewhere. (B) The magnitude of the number fluctuations (quantified by N and normalized by ) against the mean number of particles, for subsystems of various sizes. The number fluctuations in each subsystem are determined from images taken every 15 s over a period of 40 min (19). The squares represent the system shown in (A). It is a dense system where the nematic order is well developed. The magnitude of the scaled number fluctuations decreases in more dilute systems, where the nematic order is weaker (SOM text). Deviations from the central limit theorem result are still visible at an area fraction 58% (diamonds) but not at an area fraction 35% (circles). (Inset) The nematic-order correlation function as a function of spatial separation. [View Larger Version of this Image (59K GIF file)]

We examined a subsystem of size l by l (i.e., one rod length on a side) and obtained a time series of particle number, N(t), by taking images at a frame rate of 300 frames per s. From this we determined the temporal autocorrelation, C(t), of the density fluctuations. C(t) decays logarithmically in time (Fig. 2), unlike the much more rapid t^–1 decay of random, diffusively relaxing density fluctuations in two dimensions. Thus, the density fluctuations are not only anomalously large in magnitude but also extremely long-lived. Indeed, these two effects are intimately related: An intermediate step in the theoretical argument (5) that predicts giant number fluctuations shows that density fluctuations at a wave number q have a variance proportional to q^–2 and decay diffusively. This leads to the conclusion (SOM text) that in the time regime intermediate between the times taken for a density mode to diffuse a particle length and the size of the system, the autocorrelation function of the local density decays only logarithmically in time. Although the observations agree with the predicted logarithmic decay, we cannot as yet make quantitative statements about the coefficient of the logarithm. We note that the size of subsystem is below the scale of subsystem size at which the standard deviation has become proportional to the mean. In flocks and herds as well, measuring the dynamics of local density fluctuations will yield crucial information regarding the entire system's dynamics and can be used to test the predictions of Toner and Tu (2, 3).

Fig. 2. The logarithmic dependence of the local density autocorrelation, C(t)=<(0)(t)> [(t) is the deviation from the mean of the instantaneous number density of particles], is a direct consequence (SOM text), and hence a clear signature, of the large density fluctuations in the system. It is remarkable that such a local property reflects the dynamics of the entire system so strongly. It is seen that increasing shortens the decay time. This is consistent with the fact that the magnitude of the giant number fluctuations grows with the nematic order (SOM text). [View Larger Version of this Image (18K GIF file)]

What are the microscopic origins of the giant density fluctuations? Both in active and in equilibrium systems, particle motions lead to spatial variations in the nematic ordering direction. However, in active systems alone, such bend and splay of the orientation are predicted (5) to select a direction for coherent particle currents. These curvature-driven currents in turn engender giant number fluctuations. We find qualitative evidence for curvature-induced currents in the flow of particles near topological defects (SOM text). In the apolar flocking model of (15), particles move by hopping along their axes and then reorienting, with a preference to align parallel to the average orientation of particles in their neighborhood. Requiring that the hop be along the particle axis was sufficient to produce giant number fluctuations in the nematic phase of the system. It was further suggested in (15) that the curvature-induced currents of (5), although not explicitly put into their simulation, must emerge as a macroscopic consequence of the rules imposed on microscopic motion. This suggestion is substantiated by the work of Ahmadi et al. (23), who started from a microscopic model of molecular motors moving preferentially along biofilaments and showed by coarse-graining this model that the equation of motion for the density of filaments contains precisely the term in (5) responsible for curvature-induced currents.

In our experiments, we found anisotropy at the most microscopic level of single particle motion, even at time scales shorter than the vibration frequency, f. In equilibrium, the mean kinetic energies associated with the two in-plane translation degrees of freedom of the particle are equal, by the equipartition theorem, even if the particle shape is anisotropic. Figure 3 is a histogram of the magnitude of particle displacements over a time corresponding to the camera frame rate [1/300 s, or (2/3)f ^–1]. The displacement along and perpendicular to the axis of the rod are displayed separately, showing that a particle is about 2.3 times as likely to move along its length as it is to move transverse to its length. Because the period of the imposed vibration (f ^–1) sets the scale for the mean free time of the particles, this shows that the motion of the rods is anisotropic even at time scales less than or comparable to the mean free time between collisions.

Fig. 3. The microscopic origin of the macroscopic density fluctuations. The probability distribution of the magnitude of the displacement along and transverse to the particle's long axis over an interval of 1/300 of a second shows that short time motion of the rods is anisotropic even at the time scale of the collision time. This anisotropy is explicitly forbidden in equilibrium systems by the equipartition theorem. [View Larger Version of this Image (31K GIF file)]

We have thus presented an experimental demonstration of giant, long-lived number fluctuations in a two-dimensional active nematic. The particles in our driven system do not communicate except by contact, have no sensing mechanisms, and are not influenced by the spatially varying pressures and incentives of a biological environment. This reinforces the view that, in living matter as well, simple, nonspecific interactions can give rise to large spatial inhomogeneity. Equally important, these effects offer a counterexample to the deeply held notion that density is a sharply defined quantity for a large system.

References and Notes

Received for publication 25 January 2007. Accepted for publication 31 May 2007.

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I. S. Aranson, A. Snezhko, J. S. Olafsen, and J. S. Urbach (2 May 2008)
Science 320 (5876), 612c. [DOI: 10.1126/science.1153456]
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