Entropy Balance Equation of Spacetime Thermodynamics in f(R) Gravity

We study spacetime thermodynamics for non-equilibrium processes. We first generalize the formulation of spacetime thermodynamics by using an observer outside the horizon. Then we construct the entropy balance equation of spacetime thermodynamics for non-equilibrium processes in f® gravity. The coefficients of the expansion and shear terms are equal to the viscosities of the black hole membrane paradigm, and a new entropy production term appears.