From Binary Randomness to Macroscopic Emergence Microscopic Derivation and Numerical Verification of Nonuniformity Emergence

Abstract:The macroscopic phenomenological equations for open systems—the income-and-expenditure master equation, the free-energy gradient flow, the quasi-potential landscape—were previously proposed as phenomenological postulates, and their rigorous connection to the microscopic stochastic dynamics had not been established. This paper supplies that step: starting from a minimal random axiom (A0), and passing through a triple limit (thermodynamic limit, temporal coarse-graining, slow-manifold reduction), we rigorously derive the macroscopic master equation, and we give a numerical verification—by exact stochastic simulation (the Gillespie algorithm, not tau-leaping)—for every link in the derivation chain. The main results are as follows. First, the ceiling constraint on the active fraction (the (1−Φ) factor) is not imposed externally but is forced to emerge by the law of large numbers; its absence makes the order parameter diverge without bound. Second, the macroscopic drift term equals the difference of the microscopic birth and death rates, while the fluctuation term is proportional to the reciprocal of the system size, so in the thermodynamic limit the macroscopic equation is a deterministic ordinary differential equation with fluctuations of order N to the minus one-half—we confirm this numerically by the convergence of stochastic trajectories to the deterministic solution. Third, the stationary distribution of a single unit's size is the negative binomial distribution, whose shape parameter equals the ratio of the additive to the multiplicative birth rate, giving the microscopic identity of the power-law tail exponent (heavy-tailed when this ratio is below one). Fourth, the bistable structure of the macroscopic potential (the coexistence of a dead state and an active state) requires the presence of a positive-feedback term, whose necessary and sufficient condition is stated explicitly and confirmed by stability analysis and a hysteresis loop. Fifth, as the completion of the distributional classification, we map the four basic combinations of a random sequence (summation, multiplication, taking the extreme, no combination) to five aggregation channels, underwritten jointly by three limit theorems (Lévy–Khintchine, Kesten, Fisher–Tippett–Gnedenko), the extreme-value operation giving the three generalized-extreme-value types. The quantitative indicators of all numerical verifications (KS distance, relative error, goodness of fit) are reported throughout.