Page Entropy of Proton System in Deep-Inelastic-Scattering at Small-$x$

The partons model reveals the dynamical structure of nucleons (protons and neutrons). Studies related to thermodynamic quantities of nucleons are interesting and topical questions. In this work, for the first time we apply Page's theory of the studies of black hole to investigate the entropy of proton system. Inspired by the quantum entanglement entropy in black hole information theories, we establish the proton entanglement with the similar way. Based our calculations, the proton entanglement entropy has the approximate form $S = \ln m -1/2$, where $m$ represents the partons density of proton (mainly gluon and sea quark contributions) at small Bjorken $x$. Our calculations using Page's theory are well in agreement with the recent DIS measurements from H1 collaboration.