General Theory of Large D Membranes Consistent with Second Law of Thermodynamics

We write down the most general membrane equations dual to black holes for a general class of gravity theories, up to sub-leading order in $1/D$ in large $D$ limit. We derive a minimal entropy current which satisfies a local form of second law from these membrane equations. We find that consistency with second law requires the membrane equations to satisfy certain constraints. We find additional constraints on the membrane equations from the existence of membrane solutions dual to stationary black holes. Finally we observe a tension between second law and matching with Wald entropy for dual stationary black hole configurations, for the minimal entropy current. We propose a simple modification of the membrane entropy current so that it satisfies second law and also the stationary membrane entropy matches the Wald entropy.