Quantum coarse-grained entropy and thermodynamics

We extend classical coarse-grained entropy, commonly used in many branches of physics, to the quantum realm. We find two coarse grainings, one using measurements of local particle numbers and then total energy, and the second using local energy measurements, which lead to an entropy that is defined outside of equilibrium, is in accord with the thermodynamic entropy for equilibrium systems, and reaches the thermodynamic entropy in the long-time limit, even in genuinely isolated quantum systems. This answers the long-standing conceptual problem, as to which entropy is relevant for the formulation of the second thermodynamic law in closed quantum systems. This entropy could be in principle measured, especially now that experiments on such systems are becoming feasible.

Figure 1

Time evolution of $S_{xE}$ (line) and the factorized observational entropy FOE (dashed line) starting in a pure thermal state. After $t=30$, the right wall is expanded to double the system size and the system continues to evolve. The straight lines represent thermodynamic entropies of the canonical ensemble. This graph shows that for a typical state, $S_{xE}$ and FOE models the dynamical process of equilibriation between the two regions.

Figure 2

The top curves show observational entropies $S_{xE}$ (light red) and FOE (dark blue) for a microcanonical state (line), a random superposition of neighboring energy eigenstates (crosses), and energy eigenstates (dots), from top to bottom. The lowest curve is the microcanonical entropy given by logarithm of the density of states. Both $S_{xE}$ and FOE of random superposition of neighboring energy eigenstates approximate the thermodynamic entropy. Since these states model typical states of an isolated quantum system in far future, this graph provides an extensive numerical evidence that for most initial states, $S_{xE}$ and FOE converge to thermodynamic entropy, even for genuinely closed quantum systems.