Interaction, coalescence, and collapse of localized patterns in a quasi-one-dimensional system of interacting particles

We study the path toward equilibrium of pairs of solitary wave envelopes (bubbles) that modulate a regular zigzag pattern in an annular channel. We evidence that bubble pairs are metastable states, which spontaneously evolve toward a stable single bubble. We exhibit the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive, whereas it is repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: It is attractive for NF systems and repulsive for F systems and decreases exponentially with the bubbles distance. Moreover, for NF systems, the bubbles come closer and eventually merge as a single bubble, in a coalescence process. We also evidence a collapse process, in which one bubble shrinks in favor of the other one, overcoming an energetic barrier in phase space. This process is relevant for both NF systems and F systems. In NF systems, the coalescence prevails at low temperature, whereas thermally activated jumps make the collapse prevail at high temperature. In F systems, the path toward equilibrium involves a collapse process regardless of the temperature.

Figure 1

Schematic representation of the particles configurations for a NF system in (a) and for a F system in (b). In both plots the blue $+$ signs (respectively, red $\times$) represent the virtual position of the particles in the presence only of the left bubble (respectively, right bubble), the open black circles correspond to the actual position of the particles in presence of both bubbles, and the cyan dashed lines correspond to the analytic transverse displacement field presented Eq. (8). In (b) the black box indicates the location of the topological defect.

Figure 2

Plot of the modulated interaction potential energy $E_P$ [in ${10}^{-5}$ nJ; see Eq. (18)] as a function of the distance $D$ between bubbles (in mm). The inset is a zoom on the large distance behavior that evidences the local minima of energy.

Figure 3

Snapshots of instantaneous configurations of the particles (both axis in mm) inside a NF system (left plots) and a F system (right plots). For the NF system the configurations are displayed at time 0 s (a), 90 s (c), and 108 s (e). For the F system the configurations are displayed at time 0 s (b), 50 s (d), and 100 s (f). The cyan solid line is the ansatz (8). For both systems there are $N=128$ particles and the confinement is $\epsilon =0.05$ (see also the Supplemental Material of this paper [34]).

Figure 4

Positions of the two bubbles (in mm) as a function of the time (in s) for a NF system of $N=128$ particles, a confinement $\epsilon =0.05$, and with two bubbles initially separated by a distance of $D_0=39.7$ mm.

Figure 5

Plot of the distance $D\left(t\right)$ (in mm) between a bubble pair as a function of time $t$ (in s) for a NF system of $N=128$ particles. For four different initial distances: 35.4 mm (a), 39.7 mm (b), 42.8 mm (c), and 45.6 mm (d). Solid black line: For a confinement ${\epsilon }_1=0.085$ and $\mathrm{\Delta }E\left({\epsilon }_1\right)=7.9{10}^{-10}$ nJ. Solid cyan (light gray) line: For a confinement ${\epsilon }_2=0.093$ and $\mathrm{\Delta }E\left({\epsilon }_2\right)=1.0{10}^{-10}$ nJ. Dashed black line: Analytical solution, Eq. (16), with a periodic potential energy amplitude $\mathrm{\Delta }E\left(\epsilon \right)=0$.

Figure 6

Acceleration $\ddot{D}$ [in mm ${\mathrm{s}}^{-2}$; (a) linear scale, (b) logarithmic scale] extracted from the trajectories obtained by simulation, such as in Figs. 4 and 3, as function of the distance $D$ between bubbles [in mm; (a) and (b) linear scales]. For both plots the cyan (light gray) curve corresponds to a NF system and the blue (dark gray) curve corresponds to a F system. In (a) the thin black curves correspond to the theoretical expression of the acceleration calculated from Eq. (15). For a system of $N=128$ particles, $\epsilon =0.05$, with initial distances of $D_0=42.8$ mm for the NF system and of $D_0=34.6$ mm for the F system.

Figure 7

Plot of the configurational energy (in ${10}^{-5}$ nJ) as a function of the distance between bubbles (in mm), with the zero energy taken when the bubbles are at infinite distance. The thick cyan (light gray) solid line curve corresponds to the configurational energy of the NF system in Fig. 6, and the thick magenta (darker) solid line corresponds to the F system in Fig. 6. The thin black solid line is the opposite of the F-system energy and is shown to be identical to the NF-system energy.

Figure 8

Positions of the two bubbles (in mm) as a function of the time (in s) for a F system of $N=128$ particles, a confinement $\epsilon =0.05$, and with two bubbles initially separated by a distance of $D_0=34.6$ mm.

Figure 9

Probability to observe the reorganization of the bubble pair system via the coalescence process (cyan dots) or via the collapse process (magenta squares) as a function of the temperature (${10}^9$ K). For NF systems of $N=64$ particles and confinement $\epsilon =0.13$, with bubbles initially separated by $D_0=44.1$ mm.

Figure 10

Positions $x$ of the two bubbles (in mm) as a function of the time (in s) for a NF system of $N=64$ particles, a confinement $\epsilon =0.13$, and with two bubbles initially separated by a distance of $D_0=44.1$ mm. The temperature is $T=3.{10}^9$ K and the damping is $\gamma =1{\mathrm{s}}^{-1}$. (a) Coalescence process. (b) Collapse process. At this temperature both processes are equally likely, see Fig. 9.

Figure 11

Two instantaneous configurations (both axis in mm) of the particles inside a NF system taken during the final steps of the coalescence, following the series presented Fig. 3. $N=128$ particles and $\epsilon =0.05$. (a) At time 110 s; the cyan solid line is the ansatz (8). (b) At time 340 s; the cyan solid line is the analytic solution (5).

Figure 12

Transverse positions (in mm) of two successive particles located just at the middle between the two bubbles as a function of the time (in s). For a system of $N=128$ particles, $\epsilon =0.05$, at zero temperature and without dissipation.

Figure 13

Instantaneous configurational energy (in nJ) as a function of the time (in s). The configurational energy is averaged over many simulations (see the text for details). In both plots $N=64, \epsilon =0.13$, the temperature is $T={10}^9$ K and the damping constant is $\gamma =1{\mathrm{s}}^{-1}$. (a) For a NF system, with the time origin taken at the coalescence event. (b) For an F system, with the time origin taken at the collapse event.

Figure 14

Plot of the mean time (in s) taken by the system to reorganize toward one bubble via the collapse process as a function of the temperature (in ${10}^9$ K). The red dots correspond to NF systems and the blue squares correspond to F systems. In both cases $N=64$ particles, $\epsilon =0.13$, and the damping constant is $\gamma =1{\mathrm{s}}^{-1}$.

Figure 15

Left plots: Maximum bubble amplitude $h_{\max }$ (in mm) for a metastable bubble pair and then for the stable single bubble, as a function of time (in s). Right plots: Nearest-neighbors distance $d^{*}$ (in mm) for particles outside the bubbles, for a metastable bubble pair, and then for the stable single bubble, as a function of time (in s). Top plots for a coalescence process and bottom plots for a collapse process. In each plots the dashed red line indicates the time at which coalescence or collapse happen. For a system of $N=64$ particles, $\epsilon =0.15$, at a temperature $T=5.\times {10}^9$ K and with a damping constant $\gamma =1{\mathrm{s}}^{-1}$.