STATISTICAL MECHANICS AND THE ROLE OF PROBABILITY IN NONLINEAR SYSTEMS

Abstract

This study investigates the connection between statistical mechanics and nonlinear system identification using an open-access benchmark dataset representing a cascaded-tanks system with soft and hard nonlinearities. The objective was to determine whether probabilistic principles—such as entropy, fluctuation, and ergodicity—can effectively characterise nonlinear dynamical behaviour beyond deterministic modelling. The methodology integrates statistical and probabilistic analyses, including correlation evaluation, probability density estimation, cumulative distribution analysis, entropy computation, and autocorrelation assessment. These techniques quantify structural coupling, memory effects, and information stability across estimation and validation phases. The results reveal that the system exhibits strong deterministic coupling between input and output signals, alongside measurable stochastic fluctuations. Probability distributions of system responses display near-Gaussian but bounded variability, while entropy values remain consistent between estimation and validation datasets. The observed finite correlation decay confirms that fluctuations are self-limiting, ensuring dynamic equilibrium. Collectively, these findings demonstrate that the system behaves as a quasi-ergodic ensemble governed by statistical equilibrium principles. This research contributes to nonlinear system identification by embedding statistical-mechanical reasoning into empirical analysis, offering a framework for understanding real-world nonlinear systems as probabilistically stable entities. The approach enhances interpretability, robustness, and predictive reliability for complex engineering systems operating under uncertainty.