Operationalizing the Arrow of Time in mesoscopic: A Unified Framework for Non-equilibrium Matter

Abstract: What sustains a non-equilibrium system against fluctuations from within - as witnessed in non-equilibrium steady states, glassy relaxation, and even living organisms? Here we show that the arrow of time itself can be operationalized into a measurable physical quantity on mesoscopic particles - the eigen-phase displacement. This displacement gives rise to a non-local generalized force, the thermodynamic inertia force, which emerges from the integrated contribution of local constraints rather than as a conventional local force. It actively counteracts fluctuations and its algebraic structure is a semi-group, fundamentally distinct from the Lie group of Newton inertia, thereby encoding the irreversibility of time's arrow. Building on this foundation, we construct a unified Microstate-Sequence-Mode-Coupling (MSS-MCT) theory. Its thermodynamic limit is defined by Microstate Sequence (MSS) theory, and its dynamical action is captured by a consequent mode-coupling theory (MCT). From this single first-principles framework, we simultaneously resolve two long-standing puzzles: it predicts the giant non-Gaussian parameter(1~10), closing the order-of-magnitude gap with experiments that standard mode-coupling theory could not explain; and it delivers a first-principles, non-fitting derivation of the universal polymer constant $C_{1} \approx 16.7$ with merely 1 percent error - the most accurate theoretical prediction to date, dramatically surpassing the Adam-Gibbs and others. Our work establishes thermodynamic inertia as a foundational principle for non-equilibrium matter, bridging thermodynamic and dynamic descriptions from glassy relaxation to the maintenance of life.