Self-organization by topological constraints: hierarchy of foliated phase space

Abstract

Topological constraints are the key to an understanding of how a macrosystem can be different from the simple sum of microelements. The emergence of a macrostructure is a reflection of reduced degrees of freedom, because the realization of all degrees of freedom, on an equal footing, maximizes the entropy and eliminates any inhomogeneity. Here, we formulate topological constraints as foliation of the phase space. A macrohierarchy is, then, a leaf (submanifold) embedded in the total phase space. A plasma confined in a magnetic field is invoked for explaining the organizing principle. In a magnetosphere, the plasma self-organizes to a state with a steep density gradient. The resulting nontrivial structure has maximum entropy in an effective phase space that is reduced by adiabatic invariants and the corresponding coarse-grained angle variables. Formally, the adiabatic invariants may be viewed as Casimir invariants, and the effective phase space is the Casimir leaf. Conversely, we may deem any Casimir invariant as an adiabatic invariant derived by separating a ‘micro’ conjugate variable; the topological constraint of the Casimir invariant can be unfrozen when the conjugate variable is recovered. We put this interpretation into the test in the context of ideal magnetohydrodynamics which is an infinite-dimensional Hamiltonian system obeying an infinite number of topological constraints. Diverse structures realized in plasmas are described as creations on different hierarchy of foliation.