Power-law spectrum of energetic particles in classical thermal equilibrium by pitch-angle scattering process

Abstract

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.