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We develop the framework of classical Observational entropy, which is amathematically rigorous and precise framework for non-equilibriumthermodynamics, explicitly defined in terms of a set of observables.Observational entropy can be seen as a generalization of Boltzmann entropy tosystems with indeterminate initial conditions, and describes the knowledgeachievable about the system by a macroscopic observer with limited measurementcapabilities; it becomes Gibbs entropy in the limit of perfectly fine-grainedmeasurements. This quantity, while previously mentioned in the literature, hasbeen investigated in detail only in the quantum case. We describe thisframework reasonably pedagogically, then show that in this framework, certainchoices of coarse-graining lead to an entropy that is well-defined out ofequilibrium, additive on independent systems, and that grows towardsthermodynamic entropy as the system reaches equilibrium, even for systems thatare genuinely isolated. Choosing certain macroscopic regions, this dynamicalthermodynamic entropy measures how close these regions are to thermalequilibrium. We also show that in the given formalism, the correspondencebetween classical entropy (defined on classical phase space) and quantumentropy (defined on Hilbert space) becomes surprisingly direct and transparent,while manifesting differences stemming from non-commutativity ofcoarse-grainings and from non-existence of a direct classical analogue ofquantum energy eigenstates.

Classical dynamical coarse-grained entropy and comparison with the quantum version