Abstract
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.
Highlights
The process of self-organization of a structure may appear to be an antithesis of the maximum entropy ansatz, yet various nonlinear systems display what may be viewed as the simultaneous existence of order and disorder
An appropriate scale hierarchy, encompassing large-scale order and small-sale disorder, is established by exploiting phase-space foliation provided by the adiabatic invariants of the system; the corresponding invariant measure is specified
A leaf of the foliated phase space is identified as a grand canonical ensemble of macroscopic quasi-particles representing coarse grained orbits
Summary
The process of self-organization of a structure may appear to be an antithesis of the maximum entropy ansatz, yet various nonlinear systems display what may be viewed as the simultaneous existence of order and disorder. Magnetospheric plasmas (the naturally occurring ones such as the planetary magnetospheres [2,3,4], as well as their laboratory simulations [5,6,7,8]) are selforganized around the dipole magnetic fields in which charged particles cause a variety of interesting phenomena: the often observed inward diffusion (or up-hill diffusion) of particles injected from the outer region is of particular interest This process is driven by some spontaneous fluctuations (symmetry breaking) that violate the constancy of angular momentum. Instead, a clear-cut description of equilibria that maximize entropy simultaneously with bearing steep density gradients Such an equilibrium will be formulated as a grand canonical distribution on a leaf of foliated phase space that represents a macro-hierarchy. In a strongly inhomogeneous magnetic field (typically a dipole magnetic field), the phase-space metric of magnetized particles is distorted; the projection of the equipartition distribution onto the flat space of the laboratory frame yields peaked profile because of the connecting inhomogeneous Jacobian weight
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