Rigorous and General Definition of Thermodynamic Entropy

In this paper, a general definition of entropy is presented, based on operative definitions of all the concepts employed in the treatment, designed to provide a clarifying and useful, complete and coherent, minimal but general, rigorous logical framework suitable for unambiguous fundamental discussions on Second Law implications. Operative definitions of system, state, isolated system, environment of a system, process, separable system, system uncorrelated from its environment and parameters of a system are stated, which are valid also in the presence of internal semipermeable walls and reaction mechanisms. The concepts of heat and of quasistatic process are never mentioned, so that the treatment holds also for nonequilibrium states, both for macroscopic and few particles systems. The role of correlations on the domain of definition and on the additivity of energy and entropy is discussed: it is proved that energy is defined for any separable system, even if correlated with its environment, and is additive for separable subsystems even if correlated with each other; entropy is defined only for a separable system uncorrelated from its environment and is additive only for separable subsystems uncorrelated from each other; the concept of decorrelation entropy is defined. A definition of thermal reservoir less restrictive than in previous treatments is adopted: it is fulfilled, with an excellent approximation, by any single-constituent simple system contained in a fixed region of space, provided that the energy values are restricted to a suitable finite range. The proof that entropy is a property of the systemis completed by a new explicit proof that the entropy difference between two states of a system is independent of the initial state of the auxiliary thermal reservoir chosen to measure it. The definition of a reversible process is given with reference to a given scenario, i.e., the largest isolated system whose subsystems are available for interaction; thus, the operativity of the definition is improved and the treatment becomes compatible also with recent interpretations of irreversibility in the quantum mechanical framework. Rigorous extensions of the definitions of energy and entropy to open systems are stated. The existence of a fundamental relation for the stable equilibrium states of an open system with reactive constituents is proved rigorously; it is shown that the amounts of constituents which correspond to given fixed values of the reaction coordinates should appear in this equation.