Kinetic Theory with Casimir Invariants-Toward Understanding of Self-Organization by Topological Constraints

A topological constraint, characterized by the Casimir invariant, imparts non-trivial structures in a complex system. We construct a kinetic theory in a constrained phase space (infinite-dimensional function space of macroscopic fields), and characterize a self-organized structure as a thermal equilibrium on a leaf of foliated phase space. By introducing a model of a grand canonical ensemble, the Casimir invariant is interpreted as the number of topological particles.