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