Dynamics of a first-order transition to an absorbing state

A granular system confined in a quasi-two-dimensional box that is vertically vibrated can transit to an absorbing state in which all particles bounce vertically in phase with the box, with no horizontal motion. In principle, this state can be reached for any density lower than the one corresponding to one complete monolayer, which is then the critical density. Below this critical value, the transition to the absorbing state is of first order, with long metastable periods, followed by rapid transitions driven by homogeneous nucleation. Molecular dynamics simulations and experiments show that there is a dramatic increase on the metastable times far below the critical density; in practice, it is impossible to observe spontaneous transitions close to the critical density. This peculiar feature is a consequence of the nonequilibrium nature of this first-order transition to the absorbing state. A Ginzburg-Landau model, with multiplicative noise, describes qualitatively the observed phenomena and explains the macroscopic size of the critical nuclei. The nuclei become of small size only close to a second critical point where the active phase becomes unstable via a saddle node bifurcation. It is only close to this second critical point that experiments and simulations can evidence spontaneous transitions to the absorbing state while the metastable times grow dramatically moving away from it.

Figure 1

Normalized horizontal kinetic energy per particle as a function of time. The three simulations shown in black, blue (dark gray), and green (light gray) have the same number of particles, $N=5160$, and differ only in the grains' initial random conditions.

Figure 2

Average transition time $\langle \tau \rangle$ to the absorbing state as a function of the number of particles $N$ (normalized to the $N_{\max }$ in the top axis). The average and error bars are computed from $N_r$ independent realizations. For the latter we use the standard error definition ${\sigma }_{\tau }/\sqrt{N_r}$, where ${\sigma }_{\tau }$ is the standard deviation of the $N_r$ measurements of the fluidization time $\tau$. The solid line corresponds to the double exponential fit (1). The inset shows the standard deviation ${\sigma }_{\tau }$ versus $\langle \tau \rangle$. The straight line represents the prediction for a Poissonian process, ${\sigma }_{\tau }=\langle \tau \rangle$.

Figure 3

Sequence of snapshots of the system as it nucleates to the absorbing state. The gray intensity is proportional to the horizontal kinetic energy of each grain, with black grains having vanishing horizontal energy. The time difference between snapshots is $\Delta t=73.4T$. The number of particles is $N=5128$. For this number of particles, the average transition time is $\langle \tau \rangle \approx 1151T$. The full video can be found in the Supplemental Material [29].

Figure 4

Transition probability to the absorbing state starting with a nucleus seed of radius $R$. Symbols correspond to the results of the molecular dynamics simulations for different numbers of particles in the box. From left to right $N=5170$, 5300, 6000, and 8000. Continuous lines are the result of fits for each case using expression (2).

Figure 5

Radius (error bars, left axis) and relative area (solid circles, right axis) of the critical nuclei as a function of the number of particles. The error bars correspond to the width $\Delta$ of the transition probability fit according to Eq. (2). The connecting solid lines are placed as a guide. The horizontal solid line indicates the maximum radius allowed by the box size, $R_{\max }=50\sqrt{2}\sigma$, while the horizontal dashed line corresponds to $90\%$ of the total box area.

Figure 6

Details of the experimental setup. (a) Schematic illustration of the experimental cell, showing the horizontal dimensions and the frame that holds the bottom and top glass plates. (b) Side view pictures of the cell and the base with the illumination LED array off (above) and on (below). (c) Typical image used for particle position detection. From two consecutive images particle velocities are measured as well.

Figure 7

Average transition time $\langle \tau \rangle$ to the absorbing state as a function of the dimensionless acceleration $\Gamma$, obtained in simulations. The average and error bars are computed from $N_r$ independent realizations. The solid line corresponds to an exponential fit $\langle \tau \rangle /T=a+exp[c({\Gamma }_0-\Gamma )]$ with $a=660\pm 43$, $c=279\pm 7$, ${\Gamma }_0=1.94\pm 0.01$. The number of particles is fixed to $N=5160$.

Figure 8

Normalized horizontal kinetic energy per particle $K$ versus time for three driving accelerations, $\Gamma =2.05\pm 0.01$ (a), $\Gamma =2.22\pm 0.03$ (b), and $\Gamma =2.24\pm 0.04$ (c), corresponding to a pure fluidized state (a) and alternations between fluidized and absorbing states (b),(c). The normalization is done by the characteristic energy scale of one particle of mass $m$ with speed $A\omega$.

Figure 9

Time averaged normalized horizontal kinetic energy per particle $K$ versus $\Gamma$. Data from four different $\Gamma$ ramps are presented, each one with a different symbol. Error bars are computed from the standard deviation of each $K$ time series. In this case, the time series were of 300 s duration.

Figure 10

Example of fluidization time measurement from horizontal kinetic energy time series, $\Gamma =2.24\pm 0.04$. The original signal (blue, thin line) is smoothed (red, thick line). The horizontal dashed and continuous lines show two possible threshold values, $K_c=K_{\mathrm{int}}$ and $K_c=2K_{\mathrm{int}}$, respectively, where the transition kinetic energy $K_{\mathrm{int}}$ is defined in Appendix pp2. Up and down triangles show times where a fluidization stage starts and ends, which define $\tau$ as the time lapse between theses events, represented by the gray areas below the filtered signal.

Figure 11

Average metastable fluidization time $\langle \tau \rangle$ versus $\Gamma$ measured in experiments. The solid line is a double exponential fit. The inset shows the standard deviation ${\sigma }_{\tau }$ versus average fluidization time $\langle \tau \rangle$. The continuous line shows the Poissonian process result ${\sigma }_{\tau }=\langle \tau \rangle$.

Figure 12

Landau free energy as a function of the horizontal kinetic energy. (Top) Case with $\epsilon <0$ ($\epsilon =-0.1$). Only the state with $K\sim 1$ is a stable equilibrium. (Middle) Case with $\epsilon >0$ ($\epsilon =0.18$). Two stable equilibrium states are found with $K=0$ and $K=K_1$. The barrier is at $K\sim \epsilon$. (Bottom) Case with $\epsilon =1/4$. The equilibrium at $K=K_1$ becomes unstable via a saddle point.

Figure 13

Bifurcation diagram of the continuous model (6). Solid and dashed lines represent stable and unstable equilibria, respectively. Note that only positive values for the order parameter are possible. The critical points are located at $\epsilon =0$ and $\epsilon =1/4$.

Figure 14

Characterization of the final state in $\epsilon -D$ parameter space of numerical solution of Eq. (8). At high $D$ and small $\epsilon$ values the system remains in the fluid state (solid circles), at low $D$ and large $\epsilon$ values there is a transition to the absorbing state (solid squares), and in the intermediate region, the final state depends on the realization of the noise (stars). For $\epsilon \le 0.14$ no transition to the absorbing state is observed.

Figure 15

Radius (solid circles, left axis) and relative area (open squares, right axis) of the critical nuclei as a function of the bifurcation parameter $\epsilon$ obtained in the continuous model (8) on a square box of unit size. The maximum nucleus radius is $R_{\max }=\sqrt{2}/2\approx 0.71$; beyond that value the nucleus covers the complete system. The diffusion coefficient is $D=0.05$ and the noise intensity is ${\Gamma }_{\mathrm{n}}=0$.

Figure 16

Acceleration and normalized horizontal kinetic energy without (a) and with (b) PID control. The measured accelerations are $\Gamma =2.24\pm 0.04$ and $\Gamma =2.29\pm 0.02$, respectively. With no PID control, $\Gamma$ is clearly correlated with $K$: a transition to the absorbing state, indicated by a strong decrease in $K$, is concomitant with a small decrease in $\Gamma$, of about $5\%$.

Figure 17

Driving voltage signal $V_{\mathrm{s}}$ (top) and horizontal kinetic energy (bottom) with PID control, $\Gamma =2.29\pm 0.02$. $V_{\mathrm{s}}$ increases when the granular system transits from the fluidized to the quasiabsorbing state. The bottom panel shows the normalized horizontal kinetic energy $K/\left(m{A}^2{\omega }^2\right)$ obtained from particle tracking (continuous line) and the synthetic horizontal kinetic energy obtained from the voltage signal, $K_V=a_{\mathrm{v}}[max\left(V_{\mathrm{s}}\right)-V_{\mathrm{s}}]$ with $a_{\mathrm{v}}=5$ V${}^{-1}$ (dashed line).

Figure 18

(a) Representative histograms of normalized horizontal kinetic energy, for $\Gamma =2.18\pm 0.02$ ($\circ$), $2.24\pm 0.04$ ($\square$), and $2.45\pm 0.03$ ($▵$). Solid lines correspond to either the fit of $F_1$ alone or the fit of the addition of $F_1$ and $F_2$. The dashed and dot-dashed lines correspond to the two fits $F_1$ and $F_2$. Solid circles ($•$) correspond to the intersection values for the two largest $\Gamma$, $K_{\mathrm{int}}/\left(m{A}^2{\omega }^2\right)=0.089$ and $0.069$, for which there are indeed transitions between the fluidized and quasiabsorbing states. (b) $K_{\mathrm{int}}/\left(m{A}^2{\omega }^2\right)$ versus $\Gamma$ for the same four realizations of Fig. 9. The continuous line is a phenomenological linear fit $a\Gamma +b$, discarding the two largest $K_{\mathrm{int}}$ values.

Figure 19

$\tau$ versus $\Gamma$ obtained from both the horizontal kinetic energy and voltage signals (dots and crosses, respectively). Over 2000 fluidization times are shown. Average values are also shown ($\circ$). Error bars correspond to the standard error definition.