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

Gyrokinetic statistical absolute equilibrium and turbulence