01 Optimal-Order-Multi-Agent-Many-Body-Systems 2026
The paper proposes a unified theoretical framework for analyzing complex adaptive systems composed of heterogeneous agents that interact through feedback loops between their actions and collective observations. Two fundamental agent‑level variables are introduced: power (Ai), measuring an agent’s influence on system outcomes, and response function (Bi), governing how agents react to observed changes. A third coefficient (Ci) captures response to exogenous forces. From these microscopic parameters the author derives macroscopic system properties such as total power (W), useful power (D = Σ Ai Bi), entropy (S = (W‑D)/W), order/synchronization, fragility, and mobility/viscosity (V = Σ|Bi|, M = 1/V). A utility function u(W)=W^g, parameterized by a risk‑appetite coefficient g (linked to CRRA risk aversion), is used to formalize the trade‑off between growth and risk. By maximizing expected utility (or a linearized objective G that combines growth, variance, and mobility), the paper identifies conditions for an “optimal order” – a balance where agents are sufficiently synchronized to generate productivity but not so concentrated that fragility and loss of mobility become excessive. The analysis connects to classic literature (mean‑field games, stochastic games, dynamical systems, ABM) and introduces concentration measures (Shannon, Gini, Simpson) to quantify power distribution. The author argues that order, entropy, and information are task‑dependent, system‑relative concepts, and demonstrates the framework’s relevance across domains: large language model architectures, biological neural networks, autonomous robotic swarms, public‑opinion dynamics, financial markets, and macro‑economic policy. The paper concludes with a call for empirical measurement of response functions, formalization of mobility, and practical methods to detect optimal order in real‑world systems.
Topics
Feedback‑Loop Model of Heterogeneous Agents
Introduces the core mathematical structure: agents i have power A_i, response coefficients B_i (to observations) and C_i (to exogenous forces), with dynamics dS_i = C_i dE + B_i dO and aggregate outcome dO driven by weighted actions.
Deriving Macroscopic System Quantities from Agent Parameters
Shows how total power W, useful energy D = Σ A_i B_i, entropy S = (W‑D)/W, and order/synchronization emerge analytically from the microscopic variables, linking micro‑heterogeneity to macro‑behaviour.
Utility Function and Risk‑Appetite Parameterization
Defines system‑level utility u(W)=W^g, explains the mapping between g and CRRA risk aversion (γ = 1‑g), and derives expected utility approximations that balance growth (R) against variance (σ_W^2).
Optimal Order: Balancing Growth, Stability, and Mobility
Formulates the optimal‑order problem as maximizing an objective G that combines expected wealth growth, risk (variance), and mobility (inverse viscosity), revealing trade‑offs between concentration, synchronization, and adaptability.
Concentration, Synchronization, and Mobility Metrics
Introduces concentration indices (Shannon, Gini, Simpson) to quantify power distribution, defines viscosity V = Σ|B_i| and mobility M = 1/V, and links these to fragility and systemic risk.
Task‑Dependent Definitions of Order, Entropy, and Information
Argues that order, entropy, and information are relative to system goals and receivers; order is a configuration enabling a task, entropy measures unusable power, and information is a control signal whose value depends on the acting system.
Application to Large Language Model Architectures
Maps neural components (neurons, attention heads, expert modules) onto agents with power A_i and response B_i, suggesting that optimal AI performance requires a balance between synchronization (consensus) and diversity (exploration).
Implications for Biological Neural Systems
Interprets neuronal heterogeneity, synaptic strength, and feedback loops through the same A_i‑B_i framework, linking learning, memory, and pathological feedback (e.g., addiction) to changes in useful energy and order.