On the Structural Distinction Between Entropy and Time in Dynamical Theories

The relation between entropy and time is central to debates on thermodynamic irreversibility and the arrow of time. This paper clarifies that relation by distinguishing several roles often associated with entropy in such debates: temporal ordering, temporal orientation, temporal flow and measurement, and thermodynamic asymmetry. The paper does not deny that entropy increase, together with a low-entropy past and suitable coarse-graining, may explain the thermodynamic arrow or help orient an already ordered sequence of states. It also does not deny that thermodynamic or statistical structure may contribute to the selection or measurement of physically meaningful temporal flow in special frameworks. It addresses a narrower question: whether standard entropy notions can themselves supply temporal ordering or serve as general temporal parameters. Using thermodynamic, Boltzmann, Gibbs, and coarse-grained entropy within a minimal dynamical-systems framework, we show that they do not satisfy this role in general. Entropy functionals may be non-injective along trajectories; fine-grained Gibbs entropy is invariant under Hamiltonian flow; coarse-grained entropy depends on descriptive partitions; and entropy monotonicity depends on boundary conditions rather than an intrinsic temporal orientation. An open-system example is included only to illustrate that subsystem entropy may decrease while the dynamical time parameter continues to order the evolution. The novelty is therefore not in the bare claim that entropy and time are non-identical, nor in the attribution of a crude entropy-equals-time thesis to the literature, but in the explicit role-separation argument showing why entropy can characterize asymmetry, help orient an already ordered history, or contribute to temporal-flow selection only after suitable dynamical, statistical, or ordering structure is already given. Entropy remains central to statistical-mechanical accounts of irreversibility, but under standard definitions, it cannot itself supply temporal ordering.