Entropy defect: Algebra and thermodynamics

We investigate the way the entropy of a system can be partitioned into the entropies of its constituents in consistency with thermodynamics. This partitioning is described through the concept of an entropy defect, which measures the missing entropy between the sum of entropies of a system's constituents and the entropy of the combined system; this decrease of entropy corresponds to the order induced by the additional long-range correlations developed among the constituents of the combined system. We conclude that the most generalized addition rule is the one characterizing the kappa entropy; when the system resides in stationary states, the kappa entropy becomes the one associated with kappa distributions, while, in general, this entropy applies more broadly, in stationary or nonstationary states. Moreover, we develop the specific algebra of the addition rule with entropy defect. The addition rule forms a mathematical group on the set of any measurable physical-quantity (e.g., entropy). Finally, we use these algebraic properties to restate the generalized zeroth law of thermodynamics so that it is applicable for nonstationary as well as stationary states: If a body C measures the entropies of two other bodies, A and B, then, their combined entropy is measured as the connected A and B entropy, where the entropy defect is involved in all measurements.