Formation of Power-law Electron Energy Spectra in Three-dimensional Low-$\beta$ Magnetic Reconnection

While observations have suggested that power-law electron energy spectra are a common outcome of strong energy release during magnetic reconnection, e.g., in solar flares, kinetic simulations have not been able to provide definite evidence of power-laws in energy spectra of non-relativistic reconnection. By means of 3D large-scale fully kinetic simulations, we study the formation of power-law electron energy spectra in non-relativistic low-$\beta$ reconnection. We find that both the global spectrum integrated over the entire domain and local spectra within individual regions of the reconnection layer have power-law tails with a spectral index $p \sim 4$ in the 3D simulation, which persist throughout the non-linear reconnection phase until saturation. In contrast, the spectrum in the 2D simulation rapidly evolves and quickly becomes soft. We show that 3D effects such as self-generated turbulence and chaotic magnetic field lines enable the transport of high-energy electrons across the reconnection layer and allow them to access several main acceleration regions. This leads to a sustained and nearly constant acceleration rate for electrons at different energies. We construct a model that explains the observed power-law spectral index in terms of the dynamical balance between particle acceleration and escape from main acceleration regions, which are defined based upon a threshold for the curvature drift acceleration term. This result could be important for explaining the formation of power-law energy spectrum in solar flares.