Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2006-11-27
Nonlinear Sciences
Exactly Solvable and Integrable Systems
40 pages, presented at IUTAM Moscow August 2006 Plenary Talk, Collective behaviour in Macroscopic Systems, Lake Como, Italy, D
Scientific paper
Exact solutions of a family of Heisenberg-Ising spin-lattice models for a coupled barotropic flow - massive rotating sphere system under microcanonical constraint on relative enstrophy is obtained by the method of spherical constraint. Phase transitions representative of Bose-Einstein condensation in which highly ordered super and sub-rotating states self-organize from random initial vorticity states are calculated exactly and related to three key parameters - spin of sphere, kinetic energy of the barotropic flow which is specified by the inverse temperature and amount of relative enstrophy which is held fixed. Angular momentum of the barotropic fluid relative to the rotating frame of the infinitely massive sphere is the main order parameter in this statistical mechanics problem $-$ it is not constrained either canonically nor microcanonically as coupling between the fluid and the rotating sphere by a complex torque is responsible for its change. This coupling and exchange of angular momentum is a necessary condition for condensation in this spin-lattice system. There is no low temperature defects in this model - the partition function is calculated in closed form for all positive and negative temperatures. Also note-worthy is the fact that this statistical equilibrium model is not a mean field model and can be extended to treat fluctuations if required in more complex coupled flows.
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