Microscopic Dynamical Entropy I: Quantifying Hamiltonian Irreversibility in Large and Small Systems

We introduce a Microscopic Dynamical Entropy (MDE) for Hamiltonian systems, defined with respect to a chosen partition of degrees of freedom into a system X and its environment Y. The construction is based on the conditional phase-space volume (CPV), or conditional Boltzmann entropy, associated with the unmonitored degrees of freedom Y. The MDE is a microscopically defined entropy functional of the marginal distribution $\rho_X(t)$, obtained by discarding conditional microscopic information associated with Y from the Gibbs entropy of the joint XY system, while retaining exact Hamiltonian dynamics. This construction clarifies the microscopic origin of thermal entropy. The dependence of MDE solely on $\rho_X(t)$ is consistent with the thermodynamic assumption that the entropy increment of a heat bath Y depends on its heat content and temperature, not on details of its probability distribution. Indeed, the MDE recovers dS = dQ/T connecting entropy increments to heat flow between system and environment. More generally, it provides a consistent description of irreversible relaxation under exact Hamiltonian dynamics, while permitting transient entropy decreases in small systems and in spin-echo type protocols. Under time-scale separation between X and Y, the MDE becomes strictly monotonic in time, recovering the familiar structure of irreversible thermodynamics. The MDE can foreshadow thermodynamics even in a small isolated Hamiltonian system, if X is a well chosen subset of its degrees of freedom. An example is the centre of mass of interacting particles confined to a box. Even for as few as N=10 particles, the MDE increases during relaxation towards a maximum at equilibrium, with increasing monotonicity at larger N. Taken together, our results show that the MDE offers a microscopic interpretation of nonequilibrium thermal entropy and its time dependence within exact Hamiltonian dynamics.