Maximum Entropy for Reduced Moment Problems

Abstract

The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of R d is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.

Convex duality and entropy-based moment closures: Characterizing degenerate densities

“…Whether they hold in general is an interesting and (to our knowledge) open question in analysis and algebraic geometry. In the process of investigating D m , we also recover and extend many previous results from both [20,21] and [31].…”

“…Another reason to study D m is that the equilibrium densities, i.e., those densities which are moments of a Maxwellian distribution (7), lie on its boundary [20,21,24,31]. Because the kinetic entropy drives solutions of (3) toward local thermal equilibrium, we expect that trajectories defined by solutions to (16) will, at times, come very close to D m .…”

“…Previous studies of the set D m can be found in [20,21,31]. In [20], Junk provides a geometric description for D m in a one-dimensional setting (d = 1) with…”

“…Later, in [21], Junk considers a more general case in which m consists of a radial component |v| N , for some even integer N ≥ 2, plus polynomial components of lower degree. For such cases, he provides an integrability condition to determine whether D m is nonempty.…”

“…In cases where the moments are continuous with respect to the relevant topology, there is always an entropy minimizer [6,21]. Unfortunately, in classical gas dynamics, this is not usually the case.…”