Leaking outside the box: probing the nonthermal particle acceleration in a steady-state, kinetic turbulence modelling

We present a new framework for Particle-in-Cell (PIC) simulations of kinetic turbulence in extreme astrophysical environments. By introducing a novel "leaky-box" setup, with continuous particle injection from an external heat bath and diffusive escape, we achieve true steady-state conditions that allow direct investigation of nonthermal particle acceleration in turbulent plasmas. This approach yields stable, nonthermal particle energy distributions. We find that magnetic and kinetic pressures equilibrate naturally, placing an upper limit on the efficiency of stochastic acceleration. As the plasma magnetization (σ) increases, the particle energy spectrum hardens and acceleration becomes more efficient. Notably, for σ> 1, escaping particles carry over 50% of the dissipated energy, highlighting the potential of turbulence as an efficient cosmic-ray accelerator. At high magnetization (σ~100), this energy-carrying fraction approaches 70%. Our results demonstrate that steady-state kinetic turbulence can robustly produce high-energy particles, providing a controlled platform for probing acceleration physics beyond transient or "closed-box" simulations. This framework has direct relevance to high-energy astrophysical systems where turbulence and relativistic plasmas coexist, such as relativistic jets, pulsar wind nebulae, and galaxy clusters.