Exact Evolution Law for Canonical Path Ensembles with a Variance Controlled Precision Regime for Self-Organization

Abstract

Self-organizing open systems sustained by source--sink fluxes transform stochastic motion into ordered behavior, yet a general dynamical criterion governing this transformation has not been established. Building on a stochastic--dissipative path-ensemble formulation, this work derives an exact kinematic law for evolving canonical path ensembles under time-dependent parameter variation, decomposing the dynamics of the average stochastic action into contributions from selectivity modulation and structural deformation. In the precision-driven regime, the evolution closes: the rate of change is governed by the variance, yielding strict monotonicity under increasing selectivity, plateaus under stationary selectivity, and broadening under decreasing selectivity. Self-organization corresponds to endogenous, feedback-driven parameter evolution that reduces the average action, with the precision-driven regime providing a Lyapunov realization, while more general structural evolution yields system-dependent behavior; externally prescribed variation corresponds to controlled evolution. These results establish an exact ensemble-level law for organization in open stochastic systems with directly testable signatures in measurable statistics.