Bridging Boltzmann and Prigogine: A Dissipative Extension of Statistical Entropy

This working paper develops a theoretical bridge between Boltzmann’s statistical interpretation of entropy and Prigogine’s theory of dissipative structures. The aim is not to modify Boltzmann’s entropy relation, but to extend its conceptual reach toward open, non-equilibrium systems in which local order can emerge, persist, and decay under continuous energy and matter throughput. The proposed framework introduces an effective accessible-state perspective linked to a time-windowed, exergy-aware order-selection index. This index combines available work, usable gradients, structuring coupling, environmental sink capacity, microscopic state-transition compatibility, and disruptive pressure. In this way, the framework connects statistical state accessibility at the micro- and mesoscopic level with macroscopic thermodynamic constraints. The second law of thermodynamics remains fully preserved as the global admissibility condition. The proposed extension is explicitly phenomenological, system-class dependent, and falsifiable. Its variables, observation windows, thresholds, and comparison models must be defined in advance, and the framework must be rejected or revised if it fails to outperform simpler alternatives or contradicts established physical control parameters. Rayleigh–Bénard convection is used as a worked reference case to demonstrate how the general framework must reduce to known non-equilibrium stability criteria where these are already available. The accompanying materials include a Python implementation and a practical experimental application for testing the proposed Boltzmann–Prigogine bridge in a controlled dissipative system.