Phase-space mixing and the merging of cusps

Collisionless stellar systems are driven towards equilibrium by mixing of phase-space elements. I show that the excess-mass function D(f) = integral((F) over bar (x, v) > f) [(F) over bar (x, v) - f] d(3)xd(3)v [where (F) over bar (x, v) is the coarse-grained distribution function] always decreases on mixing. D(f) gives the excess mass from values of F(x, v) > f. This novel form of the mixing theorem extends the maximum phase-space density argument to all values of f. The excess-mass function can be computed from N-body simulations and is additive: the excess mass of a combination of non-overlapping systems is the sum of their individual D(f). I propose a novel interpretation for the coarse-grained distribution function, which avoids conceptual problems with the mixing theorem. As an example application, I show that for self-gravitating cusps (rho proportional to r(-gamma) as r -> 0) the excess mass D proportional to f(-2(3-gamma)/(6-gamma)) as f -> infinity, i. e. steeper cusps are less mixed than shallower ones, independent of the shape of surfaces of constant density or details of the distribution function ( e. g. anisotropy). This property, together with the additivity of D( f) and the mixing theorem, implies that a merger remnant cannot have a cusp steeper than the steepest of its progenitors. Furthermore, I argue that the cusp of the remnant should not be shallower either, implying that the steepest cusp always survives.