  A paradigm based on the absolute equilibrium of Galerkin-truncated inviscid
systems to aid in understanding turbulence [T.-D. Lee, "On some statistical
properties of hydrodynamical and magnetohydrodynamical fields," Q. Appl. Math.
10, 69 (1952)] is taken to study gyrokinetic plasma turbulence: A finite set of
Fourier modes of the collisionless gyrokinetic equations are kept and the
statistical equilibria are calculated; possible implications for plasma
turbulence in various situations are discussed. For the case of two spatial and
one velocity dimension, in the calculation with discretization also of velocity
$v$ with $N$ grid points (where $N+1$ quantities are conserved, corresponding
to an energy invariant and $N$ entropy-related invariants), the negative
temperature states, corresponding to the condensation of the generalized energy
into the lowest modes, are found. This indicates a generic feature of inverse
energy cascade. Comparisons are made with some classical results, such as those
of Charney-Hasegawa-Mima in the cold-ion limit. There is a universal shape for
statistical equilibrium of gyrokinetics in three spatial and two velocity
dimensions with just one conserved quantity. Possible physical relevance to
turbulence, such as ITG zonal flows, and to a critical balance hypothesis are
also discussed.
