Diffusion coefficient and DC conductivity of anisotropic static black hole

Abstract: In this study we apply two different methods in the context of $AdS/CFT$ correspondence and calculate the diffusion coefficient and $DC$ conductivity of a four-dimensional spatially anisotropic static black hole. First, the \emph{modified} transport coefficients is obtained by stretched horizon method and Fick's law in the context of the membrane paradigm. In order to do such calculation, we use the Maxwell equations with electromagnetic gauge field propagating in two dimensions. Two dimensional propagating gauge field leads to the complex transport coefficients which is proved by present paper. In second step, we explain electro-thermal method and employ an effective vector field and extract retarded Green's function on the classical boundary. Then, $DC$ conductivity and diffusion coefficient are obtained by using Kubo formula. Our calculation can be applied on two well-known examples of anisotropic black holes as the Einstein-Maxwell-dilaton-axion model and AdS-Einstein-Maxwell-dilaton-axion in massive gravity.