Maxwell's Demon for Emergent Page Curve and Split Property

Abstract

Seeking for the proper situation of Emergent Gravity, we recent reveal that the entropic mechanism only happens when extremal surfaces are varied, which is similar to the non-gravitational-bath-coupled setup of the island development. In this paper, we consider perturbing thin shell state outside horizon during equilibrating, to find the evolution of these Euclidean states requires an external force to be $ F_{ex}\propto T_{H}\delta A(\mu_a)$, proportional to area variation of the apparent horizon which could transform into the actual event horizon as extremal surface. We analogize the falling tendency to the 2nd law tendency, and then explore the potential tendency violation of Hawking radiation. This analogy recalls the role of Maxwell's Demon to interpret the mechanism of the bath for the Page curve to emerge in a close universe. It could reconcile the Split Problem concerning quantum gravity raised by Raju.