  Adiabatic invariants foliate phase space, and impart a macro-scale hierarchy
by separating microscopic variables. On a macroscopic leaf, long-scale ordered
structures are created while maximizing entropy. A plasma confined in a
magnetosphere is invoked for unveiling the organizing principle ---in the
vicinity of a magnetic dipole, the plasma self-organizes to a state with a
steep density gradient. The resulting nontrivial structure has maximum entropy
in an appropriate, constrained phase space. One could view such a phase space
as a leaf foliated in terms of Casimir invariants ---adiabatic invariants
measuring the number of quasi-particles (macroscopic representation of periodic
motions) are identified as the relevant Casimir invariants. The density clump
is created in response to the inhomogeneity of the energy level (frequency) of
quasi-particles.
