Fractional calculus, completely monotonic functions, a generalized Mittag-Leffler function and phase-space consistency of separable augmented densities

TL;DR

This paper establishes necessary and sufficient conditions for the non-negativity of distribution functions in spherical systems with separable augmented densities, using fractional calculus and properties of generalized Mittag-Leffler functions.

Contribution

It generalizes previous conditions for phase-space consistency to arbitrary separable systems using fractional calculus and monotonicity of generalized Mittag-Leffler functions.

Findings

Necessary conditions involve complete monotonicity of certain functions.

Sufficient conditions extend previous results to more general systems.

Criteria are applied to systems with monotonic anisotropy profiles.

Abstract

Under the separability assumption on the augmented density, a distribution function can be always constructed for a spherical population with the specified density and anisotropy profile. Then, a question arises, under what conditions the distribution constructed as such is non-negative everywhere in the entire accessible subvolume of the phase-space. We rediscover necessary conditions on the augmented density expressed with fractional calculus. The condition on the radius part R(r^2) -- whose logarithmic derivative is the anisotropy parameter -- is equivalent to R(1/w)/w being a completely monotonic function whereas the condition on the potential part is stated as its derivative up to the order not greater than 3/2-b being non-negative (where b is the central limiting value for the anisotropy parameter). We also derive the set of sufficient conditions on the separable augmented density for the non-negativity of the distribution, which generalizes the condition derived for the generalized Cuddeford system by Ciotti & Morganti to arbitrary separable systems. This is applied for the case when the anisotropy is parameterized by a monotonic function of the radius of Baes & Van Hese. The resulting criteria are found based on the complete monotonicity of generalized Mittag-Leffler functions.