Noise Activated Granular Dynamics

We study the behavior of two particles moving in a bistable potential, colliding inelastically with each other and driven by a stochastic heat bath. The system has the tendency to clusterize, placing the particles in the same well at low drivings, and to fill all of the available space at high temperatures. We show that the hopping over the potential barrier occurs following the Arrhenius rate, where the heat bath temperature is replaced by the granular temperature. Moreover, within the clusterized “phase” one encounters two different scenarios: For moderate inelasticity, the jumps from one well to the other involve one particle at a time, whereas for strong inelasticity the two particles hop simultaneously.

Figure 1

Relative distance $x_2-x_1$ as a function of time for a system with $r=0.9$ and $T_b=4.0$. The solid line indicates the diameter of the rods; the dashed line marks the well separation $L\simeq 10.95$.

Figure 2

Arrhenius plot of mean escape times $\tau$. Open symbols refer to elastic case: The escape time is ${\tau }_1$ (circles) when a well is singly occupied, and ${\tau }_2$ (squares) when a well is doubly occupied. Full symbols, instead, correspond to the inelastic system ($r=0.9$). Linear behavior indicates the validity of Kramers theory with renormalized parameters and the slopes agree with values obtained from Eq. (6). Upper inset: enlargement of the crossover region. Lower inset: data collapse of distribution of escape times for different temperatures. The arguments of the exponentials (dashed lines) in the same figure have been obtained by applying formula (6). The overall agreement between the above prediction and the values of ${\tau }_1$ and ${\tau }_2$ obtained by simulation is rather good.

Figure 3

Plot of the temperature $T_2$ vs $T_b$ for various choices of the elasticity parameter $r$. From top to bottom, the curves indicate $r=0.98$, 0.9, 0.8, 0.7, 0.6, 0.5, and 0.4. As $r$ grows toward the value 1, the lines approach the diagonal (solid line) where $T_2=T_b$. The inset shows the ratio $T_2/T_b$ obtained by the slopes of $T_2$ vs $T_b$ for the same values of $r$. The continuous line indicates the theoretical estimate given by formula (7).

Figure 4

Arrhenius plot of the mean escape times ${\tau }_2$ versus $1/T_b$ for several values of $r$. Thick solid line indicates the escape time ${\tau }_m$ of single particle of mass $2M$ corresponding to simultaneous jumps. At about $T_b=10$ and $r=0.5$, we observe a crossover between ${\tau }_2$ and ${\tau }_m$ signalling the formation of a stable molecule. The inset shows the difference ${\tau }_m-{\tau }_2$ vs $T_b$ to pinpoint the crossover where ${\tau }_m\simeq {\tau }_2$.