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 ɛ − μ ≪ T , where ɛ is the particle energy, μ is the chemical potential, T is the temperature. The break energies are the rest energy and − μ. For ɛ ≪ − μ ≪ 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 ≪ ɛ ≪ 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 ≈ 1. 4. [ABSTRACT FROM AUTHOR]