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Power-law Spectrum of Energetic Particles in Classical Thermal Equilibrium by Pitch-angle Scattering Process

The Boltzmann-Gibbs thermodynamic equilibrium state of charged particles pitch-angle scattered by weak plasma waves is discussed. Degrees of freedom of these waves play a fundamental role in constructing the grand canonical ensemble. Via the gyro-resonance condition, fast particles have an inverse break power-law spectrum for $ \varepsilon -μ\ll T $, where $ \varepsilon $ is the particle energy, $ μ$ is the chemical potential, $ T $ is the temperature. The break energies are the rest energy and $ -μ$. For $ \varepsilon \ll -μ\ll T $, the energy spectral index $ α$ is $ δ/2+1 $ and $ δ+1 $ for non- and ultra-relativistic particles, respectively, with $ δ$ an effective fractal dimension of background magnetic field lines. The spectral index for $ -μ\ll \varepsilon \ll T $ is $ α+1 $. This thermal equilibrium scenario, combined with the leaky-box model and cosmic-ray observations, seems to suggest that the Galactic magnetic field is super-diffusive with $ δ\approx 1.4 $.