Small-number statistics near the clustering transition in a compartmentalized granular gas

Statistical fluctuations are observed to profoundly influence the clustering behavior of granular material in a vibrated system consisting of two connected compartments. When the number of particles $N$ is sufficiently large ($N\approx 300$ is sufficient), the clustering follows the lines of a standard second-order phase transition and a mean-field description works. For smaller $N$, however, the enhanced influence of statistical fluctuations breaks the mean-field behavior. We quantitatively describe the competition between fluctuations and mean-field behavior (as a function of $N$) using a dynamical flux model and molecular dynamics simulations.

Figure 1

Experimental snapshots from a clustering experiment in a compartmentalized container with $N=300$ glass beads. At strong shaking (amplitude $a=1.0\mathrm{mm}$, frequency $f=70\mathrm{Hz}$) the particles are distributed evenly over the two compartments (a). When the shaking strength is reduced ($a=1.0\mathrm{mm}$, $f=50\mathrm{Hz}$) an asymmetric clustered state is seen to develop (b)–(d). The pictures are taken 5 (b), 10 (c), and 25 s (d) after reducing the shaking strength.

Figure 2

Molecular dynamics simulations for $N=300$ particles. Left: Time evolution of the number of particles in the left compartment $\left[N_1\left(t\right)\right]$ starting from the symmetric distribution $N_1\left(0\right)=N_2\left(0\right)=N∕2=150$. Right: Probability distribution function showing the statistical distribution of the particles over the two compartments. Three different regimes are distinguished, depending on the shaking strength. (I) At mild shaking (top, $f=50\mathrm{Hz}$), the particles cluster in one of the two compartments. (II) At intermediate shaking strength (middle, $f=55\mathrm{Hz}$), they still tend to cluster, but the system is intermittently driven out of this state by the statistical fluctuations. (III) For strong shaking (bottom, $f=70\mathrm{Hz}$), the system is fluctuating around the symmetric distribution.

Figure 3

(Color online) The clustering transition for three different particle numbers: $N=300$ (top left), 100 (top right), and 50 particles [bottom left (starting with both compartments equally filled) and right (starting from a clustered state)]. The contours give the probability. These diagrams are constructed from the PDFs (cf. Fig. 2) for a range of shaking frequencies $f$ around the critical value $f_c$; the PDFs of Fig. 2 are vertical cuts through this figure. The circles along the backbone curve (red) represent the maximum of the PDF at each measured frequency. For $N=300$ the diagram has all the characteristics of a standard second-order phase transition: a pitchfork bifurcation with its branches opening as ${(f_c-f)}^{\beta }$, with the mean-field critical exponent $\beta =1∕2$. For $N=100$ the branches do not follow this power law anymore due to the increased influence of statistical fluctuations, and for $N=50$ even the branches themselves have deteriorated.

Figure 4

(Color online) Contour plots showing the probability density of having a given outflow of particles from a compartment as a function of the number of particles $N_1$ in the compartment, for $f=50$ (upper), 55 (middle), and 70 Hz (lower), all at driving amplitude $a=1\mathrm{mm}$. The probability density has been normalized to 1 for each value of $N_1$ (integrating along a vertical line in the figure) and its value can be read off from the gray-scale (color) bar. The (blue) backbone line corresponds to the averaged particle flux $F\left(N_1\right)$, which can slightly differ from the most probable flux. The plots are based on MD simulations during 20 000 driving strokes.

Figure 5

(Color online) The averaged fluxes $F\left(N_1\right)$ (black) and $F\left(N_2\right)$ [gray (red)] in a two-compartment system, obtained from MD simulations for $f=50$ (top), 55 (middle), and 70 Hz (bottom) with $a=1\mathrm{mm}$. Where the two curves intersect, the flux of particles out of compartment 1 (black, starting out from zero at $N_1=0$) is balanced by the particles it receives from compartment 2 [gray (red), starting out from zero at $N_1=300$, i.e., $N_2=0$].

Figure 6

(Color online) The net flux $G\left(N\right)=-F\left(N_1\right)+F\left(N_2\right)$ (left column) and its average $⟨G\left(N\right)⟩=⟨-F\left(N_1\right)+F\left(N_2\right)⟩$ (middle column) obtained from MD simulations for $f=50$ (top), 55 (middle), and 70 Hz (bottom). The amplitude of the driving is $a=1\mathrm{mm}$. The right column shows the potential $V\left(N_i\right)$ corresponding to the average net fluxes (see Sec. 4). For mild shaking ($f=50\mathrm{Hz}$, top) the potential consists of two wells (representing the clustered states) separated by a barrier. At the critical shaking frequency (close to $f=55\mathrm{Hz}$, middle) the barrier disappears, and for higher shaking strengths the potential has just one single minimum at the symmetric state $N_i=N∕2$ ($f=70\mathrm{Hz}$, bottom). The data have been fitted to a quartic potential as in Eq. (10), with the coefficients $\{V_0,a,b\}$ taking on the values $\{-36.34,-0.0848,2.77\times {10}^{-6}\}$ for $f=50\mathrm{Hz}$, $\{-20.68,-0.0248,1.16\times {10}^{-6}\}$ for $f=55\mathrm{Hz}$, and {43.48, 0.1396, $0.87\times {10}^{-6}$} for $f=70\mathrm{Hz}$. Precisely at the critical point, the coefficient $a$ (associated with the quadratic term of the potential) goes through zero.

Figure 7

The standard deviation $\sigma \left(N_1\right)$ [or equivalently, the fluctuation term $\xi \left(N_1\right)$ in Eq. (4)] of the roughly Gaussian profiles that one gets by vertically cutting through the contour plots of the net flux in Fig. 6; the measurements are given by the symbols, which have been connected to guide the eye. The magnitude of the fluctuations is seen to be highest around the symmetric distribution $(N_i=N∕2)$ and grows mildly with increasing driving frequency $f=50$, 55, 70 Hz. The thin fluctuating lines represent the quantity $2.8{[F\left(N_1\right)+F\left(N_2\right)]}^{1∕2}$ [with $F\left(N_1\right)$ and $F\left(N_2\right)$ taken from Fig. 5], which follow the curves of $\sigma \left(N_1\right)$ reasonably well, in agreement with Eggers’ prediction Eq. (7).

Figure 8

(Color online) The amplitude of the fluctuations [represented by the maximal standard deviation $\sigma (N∕2)$, i.e., $\xi (N∕2)$, see Fig. 7; (red) squares connected by dashed line] and the height of the potential barrier $h_b(N∕2)$ [cf. Fig. 6; (blue) dots connected by solid line] as functions of the driving frequency $f$, for $N=300$, 100, and 50 particles (top to bottom). There is a region below the critical frequency $f_c$ (at which the barrier height becomes nonzero) where the fluctuations are still larger than the barrier height, which means that the system will switch intermittently from one potential well to the other, thus frustrating the mean-field behavior. The size of this region (and hence the overall influence of the fluctuations) grows for decreasing particle number $N$. For $N=50$ particles the fluctuations are seen to dominate at all frequencies.

Figure 9

(a) The bifurcation diagram for $N=300$ particles, driven at an amplitude of $a=1\mathrm{mm}$. Just below the critical frequency $f_c\approx 57\mathrm{Hz}$, the mean-field behavior (indicated by the dashed line) is slightly thwarted by the statistical fluctuations in the system. The inset shows the same diagram with the quantity along the vertical axis squared: the mean-field behavior now is represented by a straight line. (b) The same for $N=100$ particles. It is seen (also in the inset) that the mean-field behavior is much more disturbed than in (a).

Figure 10

The potential $V\left(N_i\right)$ for $N=100$ particles at mild shaking $(f=25\mathrm{Hz})$ and strong shaking $(f=36\mathrm{Hz})$, corresponding to the symmetric and clustered regimes, respectively (cf. Fig. 3, middle plot). The raw data from the average net flux have been fitted to a quartic polynomial as in Eq. (10); the coefficients $\{V_0,a,b\}$ are {1.90, $-0.0231,5.81\times {10}^{-6}$} for $f=25\mathrm{Hz}$, and {1.00, 0.0442, $2.14\times {10}^{-6}$} for $f=36\mathrm{Hz}$. At the critical frequency $(f_c\approx 32\mathrm{Hz})$, the coefficient $a$ goes through zero and the form of the potential changes from double well to single well.

Figure 11

The normalized autocorrelation function $C\left(\tau \right)$ for $N=300$ particles, for various driving frequencies $f$ (indicated in the plot). The curves all decrease from the initial value 1, but at different rates depending on $f$. Close to the critical frequency $f_c\approx \mathrm{56–57}\mathrm{Hz}$ the decay is slowest. The curves corresponding to $f<f_c$ are dashed, to distinguish them more easily from those for $f>f_c$ (solid lines).

Figure 12

(Color online) The correlation decay rates $1∕{\tau }_0$ as function of the shaking frequency for $N=300$, 100, and 50 particles, respectively. The standard behavior of $1∕{\tau }_0$ going to zero at the critical frequency $f_c$ is still recognizable for $N=300$ and 100, but deteriorates for $N=50$. The plots for $N=300$ and 100 show that $1∕{\tau }_0$ approaches zero linearly, i.e., as ${\mid f_c-f\mid }^{\gamma }$ with the mean-field critical exponent $\gamma =1$. The solid (blue) stars are based on the raw data; the open (red) ones have been corrected for the intermittency that occurs just below the critical point, according to the recipe given in the text [see Eqs. (13, 14)].