Microscopic diagonal entropy and its connection to basic thermodynamic relations

Understand

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.