tokens: since once the magnetic field twists into an overwhelmingly toroidal configuration, as measured in the comoving jet frame, it is very difficult for it to share additional energy with the particle flow (Begelman Li 1994 ). The above results apply to a pair-dominated jet; the effect of an ion component on radiatively cooled turbulence is outside the scope of this paper. The ions would effectively not cool at all, but would probably receive the majority of dissipated energy (Zhdankin et al. 2019 ), suggesting that the true value of σ could be much lower due to a hot ion component. A self-consistency condition that needs to be met in order for external IC to dominate over synchrotron cooling, , could be violated if δ ph or Γ j is too small. A cooled turbulence model is also attractive for explaining flares from BL Lac-type blazars, which lack an intense ambient radiation field. Here it is plausible that the jet emission is self-contained and likely due to the SSC process, and even larger values of σ (at least the σ associated with the electron or pair content) would be required. 6 CONCLUSION In this work, we conducted and studied the results from a series of PIC simulations of driven turbulence in relativistic collision-less pair plasma with external IC radiation. We focused on understanding the statistical properties of turbulent fluctuations and particle distributions during the statistical steady state, where injected energy from external driving is balanced by radiative energy losses. We performed a parameter scan over the steady-state magnetization 〈σ〉 and system size L /2〈ρ e 〉, to the extent that computational resources allowed. We now summarize our main results: Regardless of initial conditions, turbulence eventually acquires a rigorous statistical steady state (by heating or cooling) with an equilibrium temperature that is close to analytic predictions (equation 7 ) and can be controlled by adjusting the energy density of the external photon bath, U ph . Radiative turbulence thus provides an ideal scenario for studying the fundamental properties of kinetic turbulence. The properties of turbulence change from low 〈σ〉 to high 〈σ〉. The former looks like standard non-relativistic MHD turbulence, apart from the tendency to have rotational magnetic fluctuations. The latter has fluctuations that are highly relativistic, compressible, and electromagnetically dominated. The steady-state particle energy distributions are always close to the thermal (Maxwell–Jüttner) distribution, with only a modest non-thermal population (broadening the tail by a factor of ∼2 in energy). The non-thermal population is more temporally variable and energetically significant at high 〈σ〉. These steady-state particle energy distributions are well fit by analytic solutions to the Fokker–Planck equation describing stochastic particle acceleration, for simple forms of the momentum advection and diffusion coefficients (linear and quadratic in energy, respectively). The
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