Technical Comments

Response to Comment on "Long-Lived Giant Number Fluctuations in a Swarming Granular Nematic"

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Science  02 May 2008:
Vol. 320, Issue 5876, pp. 612
DOI: 10.1126/science.1154685

Abstract

On the basis of experiments using monolayers of spherical grains, Aranson et al. suggest that the giant number fluctuations we observed in active granular rods may be explained by static inhomogeneity or inelastic clustering. We refute these alternative explanations and underline the evidence that the fluctuations originate in nematic ordering.

We recently documented the presence of large, persistent, dynamic inhomogeneities in vibration-fluidized rods (1). Study of critical phenomena in an abundance of systems shows that pinning of large fluctuations by small stray fields is a hallmark of a system intrinsically susceptible to such fluctuations. Aranson et al. (2) raise the question of whether the fluctuations we report are intrinsic or whether they could be produced by an imperfection in the experimental system. To address this issue, we show in Fig. 1A a coarse-grained, time-averaged number density [similar to figure 2 in (2)] with the locations of maximum and minimum density indicated for 12 different realizations of our system, each with different particle number and different acceleration, but all taken in the same cell and with the cell and drive in the same orientation. Although the time-average density is not uniform in any given realization, the locations of the maxima and minima vary considerably from one realization to the next. Thus, the spatial variation in density is not a consequence of an “imperfection of the drive” or of the cell.

Fig. 1.

Coarse-grained density maps in active granular rods. (A) Red and blue symbols indicate the location of maximum and minimum time-averaged density, respectively, in 12 different realizations of the experiment at varying densities and excitation strengths. In each case, the duration of the experiment was 45 min and the vibration frequency was 200 Hz. (B) For each realization of the experiment (rows correspond to N = 1500, 2000, 2520, and 2800, and columns to dimensionless acceleration of Γ = 5, 6, 7), we show in each coarse-grained element the highest density achieved at that location during the run. There are multiple local maxima and minima distributed around the cell, showing that the density fluctuations are very mobile.

To demonstrate that even within a given realization the spatial density variation is not correctly described as a static inhomogeneity, we show in Fig. 1B the maximum density achieved at every coarse-grained location for several experimental runs. It is evident that the high-density region visits all parts of the cell; indeed, there are several local maxima in each of these plots. Thus, the density inhomogeneities are profoundly mobile.

If the inhomogeneities are mobile, then long time series should yield a spatially uniform time-averaged density (3). However, the second major finding of our study (1) was that even local density fluctuations relax extremely slowly. As shown in the Supporting Online Material (SOM) of (1), the time scale over which this slow relaxation gets cut off grows with the length scale of observation. The practical consequence of this slow relaxation is that a completely ergodic density, with a globally uniform time-averaged value, was not experimentally achieved even in long runs such as those shown in Fig. 1, each of which was 45 min long (5.4 × 105 vibration cycles).

Figure 1 in (1) shows that the giant number fluctuations closely track nematic order. Figs. S2 to S4 of the SOM of (1) supports this observation: As the acceleration of the cell is increased at fixed number density, the nematic order undergoes a complex and non-monotonic evolution. The number fluctuations track the nematic order through this evolution, whereas one would expect inelastic clustering to increase monotonically with density and decrease as kinetic granular temperature is increased.

Furthermore, collisions with the plate play a different role in systems of rods and spheres. Collisions of rods with the plate invariably involve contact with the extremities of the rod, and can efficiently transfer in-plane momentum to the rods, typically along the rod axis. Collisions with other rods serve mainly to enforce packing constraints. This strongly suggests that density fluctuations in our experiment are dominated by issues of packing rather than inelasticity, and we do not see any evidence to the contrary in (2). For systems of spheres, however, energy derived from the vertical momentum imparted by the driving mechanism can only be transferred to inplane momentum by inelastic collisions between spheres. Because of the direct and local transfer of horizontal momentum, rods in dense regions (1) show considerable relative motion, unlike in systems of spheres, where particles in inelastically condensed clumps move together or are stationary with respect to the plate (2, 4).

Turning to the analysis in figure 1 of Aranson et al. (2), we emphasize that only the procedure labeled P1 [which was used in (1)] is appropriate when the experimental observation time is less than the ergodicity time. Because P1 and P2 differ greatly for the relevant data in figure 1 in (2), this implies that this is just as true for the data of Aranson et al. as it is for our data. P2 will generate spurious anomalous density fluctuations because the time-averaged density is different in different parts of the cell. We note that the assertion by Aranson et al. that the density for the homogeneous gas is highly uniform is irrelevant; this phase should be compared not to the giant-number-fluctuation regime but to the isotropic phase. Thus, the only data in figure 1 in (2) that are relevant to the discussion are the inverted triangles in the upper and lower panels. Of these two data sets, the triangles in the lower panel show only a very small deviation from σNN1/2 and are not close to σNN.

The difference in spatial character of the density fluctuations in rod systems and the fluid-crystal coexistence pictured in the upper panel of figure 1 in (2) is clearly evident. Our data (1), simulations of active nematics (5, 6), and simulations of closely related models of fluctuation-dominated phase separation (7), do show transient separation between dense and dilute “phases.” However, these phases are not separated by a sharp interface and appear at all values of wave vector. The difference between active nematics and the inelastic clustering shown in the lower panel of figure 1 in (2) is less obvious. The deviations from N1/2 behavior shown by the inverted triangles in that portion of the figure are less than compelling. However, the theoretical and simulational literature on inelastic clustering (8) includes predictions of power-law tails in the structure factor (9, 10), just as in the mechanism driven by nematic fluctuations. To our knowledge, there is no clear experimental evidence of these power-law tails, and only short-range correlations have been found.

A possible dynamical signature of the difference between giant number fluctuations in active nematics and in other systems is that even local density fluctuations relax only logarithmically. Figure 2 shows the autocorrelation of the number fluctuations at three densities. The logarithmic relaxation is seen only at the highest of these densities, when giant number fluctuations are also seen. This is interpreted [SOM of (1)] as the outcome of local fluctuations being relaxed by modes at all length scales [see SOM text for (1)]. The theory of (11) predicts logarithmic relaxation for similar reasons. We are not aware of similar observations or predictions in inelastic gases of spheres.

Fig. 2.

Density autocorrelations showing clear logarithmic behavior in the giant number fluctuation regime (N = 2820) but not at lower particle density (N = 800, 1500).

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