Generalized entanglement entropy and the Ryu-Takayanagi proposal

Non-equilibrium systems with a long-term stationary state that possess as a spatio-temporally fluctuating quality $\beta$ can be described by a superposition of several statistics, ``superstatistics'' [1]. Recently [2,3] we have proposed entropy(ies) that depend only on the probability distribution $p_l$ and which expansion has as a first term the Shannon-entropy. We find the corresponding generalization of the von-Neumann entropy and calculate it for the model considered by Ryu and Takayangi. This results in $S=e^E (1-e^{-\frac{E}{e^E}}) \sim E-\frac{E^2}{2e^E} + ... (1)$, where $E=\frac{c}{3} \cdot {\rm log} \left( \frac{L}{\pi a} {\rm sin} \left( \frac{\pi l}{L} \right) \right),$ is the usual (2D CFT) entanglement entropy. In this set up the proposed ``area law'' $S_A=\frac{{\rm Area ~ of} ~ \gamma A}{4G^{(d+2)}_{N}}$ would need to be modified in order to have agreement with the entropy Eq.(1). It is beyond the scope of this abstract to suggest an expression for $S_{A-modified}$ and its implications for a modified theory of gravity. \\[4pt] [1] C. Beck and E.G.D. Cohen, Phys. A 322 (2003) 267.\\[0pt] [2] O. Obregon, Entropy 12(2010) 2067.\\[0pt] [3] O. Obregon and A. Gil-Villegas. Phys. Rev. E 88 (2013) 062146.