Microscopic diagonal entropy and its connection to basic thermodynamic relations

We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as $S_d=-\sum_n \rho_{nn}\ln \rho_{nn}$ with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy $S_n=-{\rm Tr}\, \rho\ln\rho$. However, in contrast to $S_n$, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy distribution function and thus it is measurable, at least in principle. Under very generic assumptions of the locality of the Hamiltonian and non-integrability the d-entropy becomes a unique function of the average energy in large systems and automatically satisfies the fundamental thermodynamic relation. This relation reduces to the first law of thermodynamics for quasi-static processes. The d-entropy is also automatically conserved for adiabatic processes. We illustrate our results with explicit examples and show that $S_d$ behaves consistently with expectations from thermodynamics.