Long-Lived Giant Number Fluctuations in a Swarming Granular Nematic
Vijay Narayan1*,
Sriram Ramaswamy1,2 and
Narayanan Menon3
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.
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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.
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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).
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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.
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