Dynamical collision network in granular gases

We address the problem of recollisions in cooling granular gases. To this aim, we dynamically construct the interaction network in a granular gas, using the sequence of collisions collected in an event driven simulation of inelastic hard disks from time $0$ until time $t$. The network is decomposed into its $k$-core structure: particles in a core of index $k$ have collided at least $k$ times with other particles in the same core. The difference between cores $k+1$ and $k$ is the so-called $k$-shell, and the set of all shells is a complete and nonoverlapping decomposition of the system. Because of energy dissipation, the gas cools down: its initial spatially homogeneous dynamics, characterized by the Haff law, i.e., a $t^{-2}$ energy decay, is unstable toward a strongly inhomogeneous phase with clusters and vortices, where energy decays as $t^{-1}$. The clear transition between those two phases appears in the evolution of the $k$-shells structure in the collision network. In the homogeneous state the $k$-shell structure evolves as in a growing network with a fixed number of vertices and randomly added links: the shell distribution is strongly peaked around the most populated shell, which has an index $k_{\max }\sim 0.9⟨d⟩$ with $⟨d⟩$ the average number of collisions experienced by a particle. During the final nonhomogeneous state a growing fraction of collisions is concentrated in small, almost closed, communities of particles: $k_{\max }$ is no more linear in $⟨d⟩$ and the distribution of shells becomes extremely large developing a power-law tail $\sim k^{-3}$ for high shell indexes. We conclude proposing a simple algorithm to build a correlated random network that reproduces, with few essential ingredients, the whole observed phenomenology, including the $t^{-1}$ energy decay. It consists of two kinds of collisions (links): single random collisions with any other particle and long chains of recollisions with only previously encountered particles. The algorithm disregards the exact spatial arrangement of clusters, suggesting that the observed stringlike structures are not essential to determine the statistics of recollisions and the energy decay.

Figure 1

Evolution of the network of collisions for a system of six particles. (a) The ordered sequence of collisions. (b) Collision network at each time step, where $nc$ is the number of realized collisions. All particles are in the core of index 0. Particles not appearing in higher cores are also in the shell of index 0. In general a $k$-core has all the particles with shell index $⩾k$.

Figure 2

(Color online) (a) Decay of granular temperature $T_g$ vs time $t$. The dashed red line marks the Haff law, $T_g=1∕{(1+t∕t_e)}^2$ with $t_e\approx 52$, the dotted green line is a $t^{-1}$ decay. (b) Cumulated number of collisions per particle $\tau$ vs time $t$, the dashed red line shows the logarithmic growth of collisions in the Haff phase, $\tau =a\ln (1+t∕t_e)$ with $a\approx 10$. In the inset a linear plot of $\tau$ vs $t$ is shown, focusing on the behavior at large times $(t⪢{10}^4)$. The system is a gas of $N={10}^6$ inelastic hard disks with restitution coefficient $\alpha =0.9$ in a box of size $3163\times 3163$ (units of a diameter) with periodic boundary conditions. Averages over ten different initial conditions have been performed. Units in the figures are arbitrary, based on simulation parameters.

Figure 3

Three snapshots of a small part of the gas of ${10}^6$ inelastic hard disks with restitution coefficient $\alpha =0.9$. In the snapshots a square of surface $200\times 200$ is portrayed, in units of a diameter, where the whole system is $3163\times 3163$; therefore the pictures portray a fraction 0.004 of the total surface. (a) Just after thermalization [0 collisions per particle (cpp)]; (b) after 100 cpp; and (c) after 200 cpp.

Figure 4

This figure shows a graph and its decomposition in $k$-cores. The shell index is the maximum core that a node belongs to, then the 2-core has nodes with shell indexes 2 and 3.

Figure 5

Distribution of number of links per node (collisions done by each particle) at different times for the four homogeneous cases taken into account: a small inelastic system of hard disks (solid line), a gas of elastic disks (dashed line), a gas of inelastic disks simulated with the DSMC algorithm (dot-dashed line), and a randomly growing network where each link (collision) is done choosing with uniform probability over the set of all possible pairs of nodes (dotted line).

Figure 6

(Color online) Distribution of $k$-shells for the four homogeneous systems against time $\tau$ measured in collisions per particle (cpp). (a)–(d) Time is on the $x$ axis, while shell index $k$ is on the $y$ axis. Brightness of color indicates population (number of particles in the shell). (a) The small inelastic molecular dynamics (MD), (b) the elastic MD, (c) the DSMC, and (d) the random network. (e) Sections of the left graph at four different times: the solid line is the small system of inelastic hard disks, the dashed line is the elastic system of hard disks, the (green) dot-dashed represents the system of inelastic hard disks simulated with the DSMC algorithm, and the dotted line is the random network. The bold dashed (red) line in the two bottom graphs marks an exponential growth $\sim \exp \left(0.24k\right)$.

Figure 7

Histograms of degree distribution $f(d,\tau )$ for different values of $\tau$, i.e., of number of collisions per particle: comparison between the large inelastic system (solid line) and the elastic system (dashed line).

Figure 8

(Color online) Distribution of $k$-shells in the large inelastic system against time $\tau$ measured in collisions per particle (cpp). (a) Time is on the $x$ axis, while shell index $k$ is on the $y$ axis. Brightness of color indicates population (number of particles in the shell). The solid white line marks the $1.8\tau$ law observed in the homogeneous case. (b) Sections of the left graph at six different times (solid line) compared with the elastic system of the same size (light dashed lines). The bold dashed line marks a power law decay with a $-3$ exponent.

Figure 9

(Color online) Plot of the number of particles in disconnected components in each core, as a function of the core index, at time $\tau =200$ cpp. At small core indexes, the number of observed disconnected components is 1, then it grows up to $\sim 6$.

Figure 10

(Color online) Left: statistics of physical observables in different shells at a given time ($\tau =200$ cpp). Units are arbitrary, depending on simulation parameters. (a) Total number of collisions performed by particles in a given shell. (b) Dissipated energy per collision, together with an exponential fit of the first decay $\sim \exp (-0.03k)$. (c) Dissipated energy summed over all collisions done by particles in a given shell. (d) Average energy competing to each particle in a given shell and total energy contained in a given shell, rescaled in order to appear on the same scale. Right: a snapshot of the system at $\tau =200$ cpp, showing in different gray tones (colors) the shell each grain belongs to.

Figure 11

(Color online) Visualization of the collision network using LaNet-vi [23]. (a)–(c) A homogeneous case, and (d)–(f) A nonhomogeneous case. Both visualizations have been done using a uniformly sampled network because of computer memory constraints (only 20% of the collisions have been taken). Each sampled network has the complete visualization [(a) and (d)], a zoom of a central part [(b) and (e)], and half of a longitudinal cut [(c) and (f)].

Figure 12

(Color online) Simulations of the recollision model. (a) The number of nodes per shell, with $p_{\mathit{hom}}=1$ (black dashed curve with squares) after that $40N$ links have been added, $p_{\mathit{hom}}=0.9$ (red solid curve with circles) after that $40N$ links have been added, $p_{\mathit{hom}}=0.8$ (blue solid curve with diamonds) after that $25N$ links have been added, and a case (red dashed line) with $p_{\mathit{hom}}=0.9$ and $nr$ chosen randomly uniformly in $[1,d_i]$, after $40N$ links. In all cases $N={10}^5$. The thick dot-dashed line is a $k^{-3}$ algebraic decay. (b) The decay of the energy in the recollision model, for the same cases. The two dot-dashed lines denote the two algebraic decays $t^{-2}$ and $t^{-1}$. Here $N={10}^6$ and $\alpha =0.9$. Units are arbitrary, depending on simulation parameters.