Statistical properties of the final state in one-dimensional ballistic aggregation

We investigate the long time behavior of the one-dimensional ballistic aggregation model that represents a sticky gas of $N$ particles with random initial positions and velocities, moving deterministically, and forming aggregates when they collide. We obtain a closed formula for the stationary measure of the system which allows us to analyze some remarkable features of the final “fan” state. In particular, we identify universal properties which are independent of the initial position and velocity distributions of the particles. We study cluster distributions and derive exact results for extreme value statistics (because of correlations these distributions do not belong to the Gumbel-Fréchet-Weibull universality classes). We also derive the energy distribution in the final state. This model generates dynamically many different scales and can be viewed as one of the simplest exactly solvable model of $N$-body dissipative dynamics.

Figure 1

An example of a configuration of six particles with identical initial mass $m_i=1$ and varying initial velocities, evolving in the long time limit into the fan state: the velocities are now increasing from left to right and there are no more colllisions.

Figure 2

The initial configuration is represented by a graph (thin line) joining $P_0$ to $P_N$ (we have taken $m_j=1\forall j$). This graph can be interpreted as a random walk in the cumulative momentum and cumulative mass space. After the first collision the particles $i$ and $i+1$ aggregate into one cluster: in the associated graph, the angle $\left(P_{i-1}{P}_i{P}_{i+1}\right)$ is replaced by the segment $\left(P_{i-1}{P}_{i+1}\right)$.

Figure 3

We draw the convex minorant of the graph representing the same initial configuration as in Fig. 2. This convex minorant represents the final “fan” state in which no more collisions can occur. Each straight segment of the convex minorant corresponds to a cluster of $n_l$ particles with momentum $n_l{u}_l$.

Figure 4

The limiting cumulative velocity distribution $F_{\infty }\left(v\right)=F(z=v∕\sigma )$ plotted against $z$, for the Gaussian distribution $\varphi \left(v\right)=\exp (-v^2∕2{\sigma }^2)∕\sqrt{2\pi {\sigma }^2}$.

Figure 5

(Color online) Main: Distribution of the total energy for $N=2000$. The solid (red) line plots the saddle-point approximation (87), and the dashed (blue) line plots the exact form of Eq. (76), evaluated using MATHEMATICA. Inset: The solid (red) line plots the large deviation function (88), and the dashed (blue) line plots $-{\left(\ln N\right)}^{-1}\ln Q_N\left(\frac{z}{2}\ln N\right)-{\left(\ln N\right)}^{-1}\ln \left[{\left(\frac{3\pi }{2}\ln N\right)}^{1∕2}{z}^{5∕6}\Gamma \left(z^{1∕3}\right)\right]$ as a function of $z$, using the exact form of $Q_N\left(E\right)$ given in Eq. (76).