Maximaldichte

Maximaldichte is a German term used in different fields to denote the greatest possible density that can be achieved under a given set of constraints. It is not tied to a single, universal definition, but rather describes an extreme density in the specific context being considered.

In physics and materials science, maximal density often refers to the densest possible packing of matter within a fixed volume. A classical problem in this area asks how much of space can be occupied by non-overlapping particles of a given shape. For equal spheres in three dimensions, the densest known arrangements are the face-centered cubic and hexagonal close packing, which achieve a density of π/(3√2) ≈ 0.74048. In two dimensions, the densest packing of equal circles is the hexagonal lattice, with density π/(2√3) ≈ 0.9069. Real materials typically fall short of these theoretical maxima due to defects, porosity, and irregular particle shapes. Processing methods such as sintering, hot pressing, or hot isostatic pressing aim to reduce porosity and approach maximal density.

In mathematical and statistical contexts, density has related but distinct meanings. A probability density function assigns likelihood over outcomes, and its maximal density is the peak value of that function. In set theory and combinatorics, notions of density (natural density, upper density) describe how large a subset is relative to a larger set. Thus, Maximaldichte surfaces as a descriptive term across disciplines, indicating an extreme or upper bound on how densely something can be arranged, filled, or distributed under specified rules.

See also: packing density, porosity, density of states, probability density function, natural density.