When do generalized entropies apply? How phase space volume determines entropy

We show how the dependence of phase space volume Ω(N) on system size N uniquely determines the extensive entropy of a classical system. We give a concise criterion when this entropy is not of Boltzmann-Gibbs type but has to assume a generalized (non-additive) form. We show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen and statistically inactive. Systems governed by generalized entropies are therefore systems whose phase space volume effectively collapses to a lower-dimensional "surface". We illustrate these results in three concrete examples: accelerating random walks, a microcanonical spin system on networks and constrained binomial processes. These examples suggest that a wide class of systems with "surface-dominant" statistics might in fact require generalized entropies, including self-organized critical systems such as sandpiles, anomalous diffusion, and systems with topological defects such as vortices, domains, or instantons.