Quantum thermalization and average entropy of a subsystem

Page's seminal result on the average von Neumann (VN) entropy does not immediately apply to realistic many-body systems which are restricted to physically relevant smaller subspaces. We investigate here the VN entropy averaged over the pure states in the subspace $\mathcal{H}_E$ corresponding to a narrow energy shell centered at energy $E$. We find that the average entropy is $\overline{S}_{1} \simeq \ln d_1$, where $d_1$ represents first subsystem's effective number of states relevant to the energy scale $E$. If $d_E = \dim{(\mathcal{H}_E)}$ and $D$ ($D_1$) is the Hilbert space dimension of the full system (first subsystem), we estimate that $d_1 \simeq D_1^\gamma$, where $\gamma = \ln (d_E) / \ln (D)$ for nonintegrable (chaotic) systems and $\gamma < \ln (d_E) / \ln (D)$ for integrable systems. This result can be reinterpreted as a volume-law of entropy, where the volume-law coefficient depends on the density-of-states for nonintegrable systems, and remains below the maximal possible value for integrable systems. We numerically analyze a spin model to substantiate our main results.