Stochastic Agent-Based Active Matter

• Stochastic agent-based models describe systems of self-driven agents, where individual energy uptake, fluctuation, and coupling produce emergent oscillatory behavior.
• The model uses coupled Lotka–Volterra equations to simulate tradeable performance and private robustness metrics, highlighting the impact of probabilistic state transitions.
• Nonlinear feedback and noise suppression via private goods stabilize synchronized groups, ensuring robust collective dynamics even under stochastic shocks.

A stochastic agent-based model of active matter describes systems composed of many interacting self-driven agents whose energy uptake, internal fluctuations, and cooperative or competitive interactions generate complex, emergent collective behavior far from equilibrium. These models formalize how individual-level stochastic processes—such as energy depot dynamics, probabilistic state transitions, and random fluctuations—combine with nonlinear coupling and energy-driven feedback to produce macroscopically organized states, including synchronization, robust limit cycles, and spontaneous stabilization phenomena.

1. Structure of the Stochastic Agent-Based Model

The system consists of $N$ agents. Each agent $i$ ($i=1, \ldots, N$) features:

• An internal energy depot $e^i(t)$, which is filled at a constant or stochastic uptake rate and dissipated through production activities and baseline losses.
• Two coupled “goods”: $x^i(t)$ is a tradeable or performance-oriented metric; $y^i(t)$ is a private, robustness-enhancing variable (“good”).
• A binary decision variable $\theta^i$ encoding the agent’s investment preference between $x$ and $y$.
• Stochastic dynamics in both the depot and production activities, driven by internal (e.g., metabolic noise) and external sources.

Energy depot evolution is governed by

$$\frac{d e^i}{dt} = -\gamma_e e^i + \left[\alpha_0^i - \alpha_1^i e^i\right] + S^i \xi^i(t)$$

where $S^i$ is a noise amplitude modulated by $y^i(t)$, typically $S^i[y^i(t)] = S \exp(-\beta y^i(t))$, and $\xi^i(t)$ is white noise.

The “goods” evolve according to generalized, coupled Lotka–Volterra equations: $$\begin{aligned} \frac{dx^i}{dt} &= x^i \left[ a_x^i + \frac{1}{N^{(ij)}} \sum_j b_x^{(ij)} x^j + c_x^i y^i \right] \ \frac{dy^i}{dt} &= y^i \left[ a_y^i + b_y^i y^i + c_y^i x^i \right] \end{aligned}$$ with $a_x^i = a_{x0} + \bar{\chi}^i\theta^i$, $a_y^i = a_{y0} + \bar{\chi}^i(1-\theta^i)$, where $a_{x0}$, $a_{y0}$ are negative (decay in absence of energy input), $\bar{\chi}^i$ encodes individual differences, and the $b$, $c$ coefficients control coupling.

The interaction structure for $x^i$ is controlled by the sign and structure of $b_x^{(ij)}$—competition for $b_x^{(ij)} < 0$, cooperation for $b_x^{(ij)} > 0$. The mean-field form replaces the sum by the average $X(t) = (1/N)\sum_i x^i$.

2. Stochastic Dynamics, Nonlinearities, and Fluctuation Control

Each dynamical variable is subject to direct or indirect stochastic fluctuations:

• The energy depot noise $S^i \xi^i(t)$ is multiplicative, with effective suppression by production of $y$, reflecting the stabilizing role of $y$ in reducing variability in agent activity.
• The equation for $x^i$ is nonlinearly coupled to both the mean field (aggregated agent activity) and to the agent’s own $y^i$, and is thus highly sensitive to both collective oscillations and individual robustness investments.

Because all production and consumption terms are nonlinear—e.g., $x^i$ multiplies the bracketed growth term, and energy depot conversion is itself nonlinear—the model supports complex bifurcation scenarios, multistability, and emergent oscillations.

Stochasticity not only drives agent-level variability but also enters the macroscopic dynamics through synchronization phenomena and the formation of robust collective oscillations.

3. Synchronization, Cooperation, and Synergetics

A central phenomenon is the emergence of dynamical order via the combination of critical energy supply and cooperative agent interactions, in line with the principles of synergetics. When $b_x > 0$ (cooperation), coupling via the mean field $X(t)$ enables underactive agents to be “pulled” into oscillations by more active ones, resulting in the formation of synchronized groups of agents—each group sustaining its own limit cycle.

The degree of phase coherence is quantified by the Kuramoto order parameter: $$R(t) e^{i\Phi(t)} = \frac{1}{N}\sum_j e^{i\phi^j(t)}$$ where the oscillatory cycles of $(x^i, y^i)$ are mapped into complex phase trajectories via

$$x^i(t) - x^i_{stat} = r^i(t)\cos\phi^i(t), \quad y^i(t) - y^i_{stat} = r^i(t)\sin\phi^i(t)$$

A high $R(t)$ within a group confirms internal synchronization, while smaller $R$ across groups demonstrates coexistence of multiple, weakly coupled limit cycles (analogous to chimera states).

Production of $y^i$ is essential for sustained synchronization: by suppressing fluctuations through its exponential effect on energy depot noise, $y^i$ enables $x^i$ to remain in the oscillatory dynamical regime despite persistent stochastic perturbations.

4. Stability and Robustness in Response to Shocks

The synchronized collective state persists under “shocks” simulating temporary, agent-level switches between cooperation and competition (modeled via stochastic sign changes in $b_x^{(ij)}$). Simulations confirm that while local switching induces transient distortions or secondary cycles in the affected agent’s $x^i$ and $y^i$, the overall structure of the synchronized state—with grouping and group-level phase coherence—remains stable, especially when the majority of the population maintains cooperative coupling. The feedback loop provided by $y^i$ stabilizes the stochastic depot and production dynamics, further enhancing resilience.

5. Mathematical Properties and Parameter Dependence

The existence and nature of oscillatory collective states depend on critical control parameters:

• Sufficient energy uptake rate $q_0$ relative to depot dissipation ($\gamma_e$) and competition/cooperation balance ($b_x$).
• Magnitude of private investment in $y$ (high $\beta$ promotes robustness by reducing $S^i[y^i]$).
• Distribution and heterogeneity of individual agent parameters ($\bar{\chi}^i$, $a_{x0}$, $a_{y0}$, etc.).

In the regime of cooperative mean-field coupling ($b_x > 0$), the system generically supports multistable, robust synchronized oscillations; competition ($b_x < 0$) suppresses collective synchronization, favoring fixed points or dominance by a single agent.

6. Conceptual Significance and Implications

This modeling framework illustrates the principles of synergetics within active matter: macroscopic order (synchronized oscillations, group formation) arises spontaneously from the interplay of microscopic stochasticity, critical energy uptake, and nonlinear cooperative feedback. The emergence of distinct, coherent groups with internally high Kuramoto order parameter and robust resilience to individual-level shocks demonstrates that stochastic agent-based models are not only effective for capturing the statistical mechanics of active, energy-consuming systems but also provide a quantitative lens on the self-organization of real-world collectives—be they biological, economic, or technological.

The detailed structure—energy depot noise, coupled non-linear production, mean-field feedback, and fluctuation-suppressing private goods—provides insight into mechanisms stabilizing collective dynamics in noisy, out-of-equilibrium systems and offers a basis for designing artificial active materials and socio-economic collectives with built-in resilience and adaptability (Schweitzer et al., 17 Oct 2025).