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 $\epsilon -\mu \ll T$, where $\epsilon$ is the particle energy, $\mu$ is the chemical potential, $T$ is the temperature. The break energies are the rest energy and $-\mu$. For $\epsilon \ll -\mu \ll T$, the energy spectral index $\alpha$ is $\delta /2+1$ and $\delta +1$ for non- and ultra-relativistic particles, respectively, with $\delta$ an effective fractal dimension of background magnetic field lines. The spectral index for $-\mu \ll \epsilon \ll T$ is $\alpha +1$. 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 $\delta \approx 1.4$.

arXiv commentAccepted for publication in Astroparticle Physics

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