Strange attractors induced by melting in systems with nonreciprocal effective interactions

Newton's third law—the action-reaction symmetry—can be violated for effective interbody forces in open and nonequilibrium systems that are ubiquitous in areas as diverse as complex plasmas, colloidal suspensions, active and living soft matter, and social behavior. While studying monolayer complex plasma (confined charged particles in an ionized gas) with nonreciprocal interactions mediated by plasma flows, in silico we found that an interplay between melting and thermal activation drastically transforms the collective dynamics: the order-disorder transition modifies the system's thermal steady state so that the crystal tends to melt, whereas the fluid tends to freeze, jumping chaotically between the two states. We identified this collective chaotic behavior as strange attractors formed in a monolayer complex plasma and link the strange attractor behavior to the specifics of interparticle interactions.

Figure 1

The schematic illustration of the model system under the study: (a) The side view of the dust particle monolayer confined in the vertical parabolic potential well. The direction of the plasma flow is indicated by the arrows, and the wakes are shown by the orange clouds. (b) Two negatively charged particles (colored in blue) in plasma flow interact nonreciprocally due to the positively charged wakes (orange clouds). ${\mathbf{F}}_{AB}^{\left(r\right)}=-{\mathbf{F}}_{BA}^{\left(r\right)}$ are the reciprocal (repulsive) forces; ${\mathbf{F}}_{AB}^{\left(w\right)}\ne -{\mathbf{F}}_{BA}^{\left(w\right)}$ denote attractive interactions between (negatively charged) particles and (positively charged) wakes. As result, the particle-wake interactions affect the reciprocity of action and reaction for effective forces, ${\mathbf{F}}_{BA}\ne -{\mathbf{F}}_{AB}$.

Figure 2

The hysteretic behavior of the average kinetic energy in the system at ${\mathrm{\Omega }}^2=11.25$ and $T_{\mathrm{th}}/T_m=0.85$: (a) The time evolution of the energy $K\left(t\right)$ from “cold” and “hot” initial states, shown by the blue and red solid lines, at different values of $\tau$. (b) The hysteretic dependence of the asymptotic energy $K_{\infty }$ on the damping time $\tau$. The orange symbols are MD data, the solid red lines are obtained with the balance method [48]. The gray area corresponds to the domain of thermally unstable states [there are no steady states here; the same domain is shown in Fig. 3]. (c) The calculated powers $P_{\mathrm{NR}}\left(K\right)$ (the orange thick line) and $P_D\left(K\right)$ (the blue lines at $\tau =330$, 670, and 1220) versus the kinetic energy $K$. The curves and states corresponding to the same conditions are labeled by the same letters $A–D$ (stable states), whereas the state $E$ is unstable.

Figure 3

Dissipative phase diagrams of the system at different confinement strength: Panels (a)–(c) correspond to the confinement frequencies ${\mathrm{\Omega }}^2=11.25$, 9.75, and 9.0, respectively. The symbols and solid lines are MD data and the results of the balance method, respectively, at different thermostat temperatures $T_{\mathrm{th}}$. The domains of thermally unstable states (there are no steady states here), obtained with the balance method [48], are gray-colored [the gray-colored domain (a) is exactly the same as shown in Fig. 2]. The dashed black lines correspond to the melting, $K_{\infty }/T_m=3/2$.

Figure 4

The energy release power and the excitation spectra: (a) Dependencies $P_{\mathrm{NR}}\left(K\right)$ of the energy release due to the interaction nonreciprocity at different ${\mathrm{\Omega }}^2$. The gap of the energy release power in crystal and fluid $\mathrm{\Delta }P$ at the melting line is shown in the inset. (b–g) Excitation spectra in crystal (b–d) and fluid (e–g) states. The color-coded velocity current spectra $C(\mathbf{q},\omega )$ were normalized to the maximum in $\mathrm{\Gamma }\mathrm{K}$ intervals in crystal and to the maximum at each $q$ in fluid. The first Brillouin zone in crystal is sketchily shown in (b). The blue symbols are results of single-mode analysis, and red ones are obtained with the three-oscillator model (see details in Sec. 2c). The parameters of simulations (for technical details see Sec. 2a) are (b) ${\mathrm{\Omega }}^2=11.25, T_{\mathrm{th}}/T_m=0.6, \tau =240$; (c) ${\mathrm{\Omega }}^2=9.75, T_{\mathrm{th}}/T_m=0.6, \tau =30$; (d) ${\mathrm{\Omega }}^2=9.0, T_{\mathrm{th}}/T_m=0.6, \tau =240$; (e) ${\mathrm{\Omega }}^2=11.25, T_{\mathrm{th}}/T_m=0.9, \tau =730$; (f) ${\mathrm{\Omega }}^2=9.75, T_{\mathrm{th}}/T_m=0.6, \tau =270$; (g) ${\mathrm{\Omega }}^2=9, T_{\mathrm{th}}/T_m=0.1, \tau =220$. The energy $K/T_m\simeq 2.4$ in the fluid states (e–g).

Figure 5

Strange dynamic attractor: (a) The dependence of kinetic energy $K_{\infty }\left(\tau \right)$ on the damping time $\tau$ at $T_{\mathrm{th}}/T_m=0.1$ and ${\mathrm{\Omega }}^2=9.0$. (b) The power of energy release $P_{\mathrm{NR}}\left(K\right)$ (thick orange line) and dissipation $P_d\left(K\right)$ at $\tau =30$, 97, and 246 (blue, red, and green lines), under the same conditions as in (a). At $\tau =97$, the steady state should have been in the gap shown by the dashed vertical line; as a result, the system is forced to oscillate around the point, as justified in panel (c) for dependencies $K\left(t\right)$ at $\tau =30$, 97, and 246. (d–f) Power spectrum densities of $K\left(t\right)$ at the same $\tau$ values as in (c).

Figure 6

Trajectory of the system in the phase space $\{K,\dot{K},P_{\mathrm{NR}}-P_d\}$: Blue and orange points correspond to (nonactivated) liquid and (activated) crystal, respectively. The red points depict the evolution over previous 300 dimensionless time units. See the supplemental movie [64] for detailed picture of the time evolution.

Figure 7

The bifurcation diagram of the discrete-time model (9): The color code represents the relative probabilities of the energy $K$ at fixed $\eta$, from blue to red. An example of the mapping function $f\left(K\right)$ given by Eq. (9) with $\eta =1.5$ and $f\left(K\right)=K$ are shown by the solid red and dashed gray lines in the inset. The jumps and chaotic behavior arise due to the irreversible (nonmonotonous) form of the mapping function $f\left(K\right)$.