Active matter synchronization and synergetics

TL;DR

This work investigates how energy intake and cooperative interactions in an agent-based active-matter model generate collective synchronization. Using a mean-field-coupled generalized Lotka-Volterra framework with two goods, $x$ (performance) and $y$ (robustness), the authors show that cooperation can yield persistent, multi-group synchronization, while competition alone destabilizes the system. The model demonstrates two coexisting synchronized domains with high intra-group coherence and robustness against shocks that switch agents between cooperation and competition, aligning with synergetics principles of energy-driven self-organization. The results offer a conceptual bridge between active matter, non-equilibrium phase transitions, and economic production dynamics, illustrating how energy, cooperation, and feedback shape emergent order and resilience.

Abstract

We study the collective behavior in a stochastic agent-based model of active matter. Provided a critical take-up of energy, agents produce two types of goods $x$, $y$ that follow a generalized Lotka-Volterra dynamics. For isolated agents, production would either reach a fixed point or diverge. Coupling agents' production via a mean field of $x$, however, can lead to synchronized oscillations if agents cooperate in the production of $x$. The production of $y$ supports the emergence of the synchronized dynamics by suppressing fluctuations and mitigating competition between agents, this way stabilizing the production of $x$. We find that in the synchronized state different groups of agents coexist, each following their own limit cycle. The Kuramoto order parameter is large within groups, and small across groups. The collective state is stable against shocks from agents temporarily switching between cooperation and competition. The model dynamics illustrates the principles of synergetics, i.e., the spontaneous emergence of order given a critical energy supply and cooperative interactions.

Figures (6)

• Figure 1: Isolated dynamics:$x(t)$ (blue), $y(t)$ (red). Dashed lines: Stationary solutions: (a)$\theta$=1, $x^{\mathrm{stat}}$=4, (b)$\theta$=0, $x^{\mathrm{stat}}$=$y^{\mathrm{stat}}$=2, Parameters:$a_{x0}$=$-0.3$, $a_{y0}$=$-0.1$, $Q$=0.5, $b_{x}$=$-0.05$, $b_{y}$=$-0.1$, $c_{x}$=0.2, $c_{y}$=$-0.1$, $Q$=0.5, $\beta$=0.1. The simulations were performed over $10^2$ time-steps with a Runge-Kutta 4 integration with constant $dt$=$0.005$.
• Figure 2: Competition dynamics of two agents:$x^{(1)}(t)$ (blue), $x^{(2)}(t)$ (black). $\theta$=1, $X_{0}$=0.3, $X_{s}$=4, (a) early growth, (b) long-term competition. Parameters:$a_{x0}$=$-0.3$, $Q$=0.5, $b_{x}$=$-0.05$, $c_{x}$=0, $s^{(1)}$=0.04, $s^{(2)}$=0.06. The simulations were performed over $2.5\times 10^3$ time-steps with a Runge-Kutta 4 integration with constant $dt$=$0.05$.
• Figure 3: Competition dynamics.(a)$\Delta x^{12}=x^{(1)}(t)-x^{(2)}(t)$ (blue), $\Delta y^{12}=y^{(1)}(t)-y^{(2)}(t)$ (red) in a 2-agent system with strong noise ($s$ = $0.25$, $\beta$=$0.1$). The horizontal dashed lines represent the stationary solutions. (b) Evolution of $x_{i}$, $y_{i}$, and the Kuramoto order parameter over time in a 20-agent system with weak noise ($s$ = $0.01$, $\beta$=$0.1$). The vertical dashed lines indicate a jump to the last $50\,000$ timesteps. Parameters:$a_{x0}$=$-0.3$, $a_{y0}$=$-0.1$, $Q$=$0.5$, $\theta$=0. The following parameters are linearly spaced for the agents in the ranges: $b_{x}$$\in$$-[0.029, 0.056]$, $b_{y}$$\in$$-[0.009, 0.063]$, $c_{x}$$\in$$[0.09, 0.27]$, $c_{y}$$\in$$-[0.09, 0.27]$. The simulations were performed over $10^6$ time-steps with a Runge-Kutta 4 integration with constant $dt$=$0.005$.
• Figure 4: Cooperation dynamics in a multi-agent system.(a) Evolution of the median $x$ and $y$ over time, with error bars indicating the interquartile range (25th to 75th percentiles) and Kuramoto order parameter. The vertical dashed lines indicate a jump to the last $50\,000$ timesteps. (b) x(t) vs y(t) for the last $50\,000$ timesteps for 6 agents. The points in the center of the cycles indicate the fixed points for the individual agents. Parameters:$a_{x0}$=$-0.3$, $a_{y0}$=$-0.1$, $Q$=$0.5$, $s$=0, $\theta$=0. The following parameters are linearly spaced for the 20 agents in the ranges: $b_{x}$$\in$$[0.029, 0.056]$, $b_{y}$$\in$$-[0.009, 0.063]$, $c_{x}$$\in$$[0.09, 0.27]$, $c_{y}$$\in$$-[0.09, 0.27]$. The simulations were performed over $10^6$ time-steps with a Runge-Kutta 4 integration with constant $dt$=$0.005$.
• Figure 5: Synchronization of two groups in a multi-agent system.(a) Evolution of the Kuramoto order parameter over time for the first 10 (top) and the second 10 (bottom) agents. The vertical dashed lines indicate a jump to the last $50\,000$ timesteps. (b) Kuramoto order parameter calculated for all agent pairs and averaged over the last $50\,000$ timesteps visualized as a heatmap. Parameters:$a_{x0}$=$-.3$, $a_{y0}$=$-.1$, $Q$=$0.5$, $s$=0, $\theta$=0. The following parameters are linearly spaced for the 20 agents in the ranges: $b_{x}$$\in$$[0.029, 0.056]$, $b_{y}$$\in$$-[0.009, 0.063]$, $c_{x}$$\in$$[0.09, 0.27]$, $c_{y}$$\in$$-[0.09, 0.27]$. The simulations were performed over $10^6$ time-steps with a Runge-Kutta 4 integration with constant $dt$=$0.005$.