Exact Evolution Law for Action-Weighted Path Ensembles and the Dynamics of 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 process has not been established. This paper derives an exact kinematic law for evolving action-weighted canonical path ensembles, and decomposes the dynamics of the ensemble-average action. In the precision-driven regime, the evolution closes: the rate of change is governed by the action variance, so increasing selectivity concentrates probability weight toward lower-action trajectories, causing the average action to decrease monotonically and behave as a Lyapunov-type quantity. Constant and decreasing selectivity lead to stationary and broadening behavior. Endogenous reduction of the average action defines self-organization, while externally prescribed modulation defines controlled ensemble evolution. The framework further admits system-dependent realizations of selectivity dynamics, including reconstruction from observable statistics and feedback-driven evolution, without modifying the underlying kinematic law. By using stochastic action as the organizing trajectory-level quantity, the approach connects path-ensemble organization to stochastic least-action ordering while yielding measurable diagnostics, finite-time constraints, and falsifiable signatures in open stochastic systems.