Energy Diffusion and Advection Coefficients in Kinetic Simulations of Relativistic Plasma Turbulence

TL;DR

This work tests the applicability of the Fokker-Planck diffusion–advection framework to nonthermal particle acceleration in turbulent, relativistic pair plasmas using fully kinetic PIC simulations. By tracking large ensembles of particles and extracting energy-space FP coefficients, the study demonstrates that diffusion dominates with a universal $D(\gamma) = D_0(t) \gamma^{2}$ scaling in the nonthermal tail, and that the diffusion prefactor scales with instantaneous magnetisation roughly as $D_0 \propto \sigma^{3/2}$ at low $\sigma$ (approaching a flatter $\sigma$ dependence at higher $\sigma$). An analytic advection form $A(\gamma,t) = A_0(t) \gamma \log(\gamma/\gamma_A^*(t))$ is derived and confirmed, linking FP coefficients to the time evolution of the power-law index $\alpha(t)$ which decays exponentially toward a convergent value. The results challenge the traditional $D_0 \propto \sigma$ expectation from second-order Fermi acceleration in this regime and illustrate the FP framework as a robust diagnostic for kinetic processes in astrophysical plasmas, with implications for modeling NTPA in shocks and magnetic reconnection.

Abstract

Turbulent, relativistic nonthermal plasmas are ubiquitous in high-energy astrophysical systems, as inferred from broadband nonthermal emission spectra. The underlying turbulent nonthermal particle acceleration (NTPA) processes have traditionally been modelled with a Fokker-Planck (FP) diffusion-advection equation for the particle energy distribution. We test FP-type NTPA theories by performing and analysing particle-in-cell (PIC) simulations of turbulence in collisionless relativistic pair plasma. By tracking large numbers of particles in simulations with different initial magnetisation and system size, we first test and confirm the applicability of the FP framework. We then measure the FP energy diffusion ($D$) and advection ($A$) coefficients as functions of particle energy $γm c^2$, and compare their dependence to theoretical predictions. At high energies, we robustly find $D \sim γ^2$ for all cases. Hence, we fit $D = D_0 γ^2$ and find a scaling consistent with $D_0 \sim σ^{3/2}$ at low instantaneous magnetisation $σ(t)$, flattening to $D_0 \sim σ$ at higher $σ\sim 1$. We also find that the power-law index $α(t)$ of the particle energy distribution converges exponentially in time. We build and test an analytic model connecting the FP coefficients and $α(t)$, predicting $A(γ) \sim γ\log γ$. We confirm this functional form in our measurements of $A(γ,t)$, which allows us to predict $α(t)$ through the model relations. Our results suggest that the basic second-order Fermi acceleration model, which predicts $D_0 \sim σ$, may not be a complete description of NTPA in turbulent plasmas. These findings encourage further application of tracked particles and FP coefficients as a diagnostic in kinetic simulations of various astrophysically relevant plasma processes like collisionless shocks and magnetic reconnection.

Figures (24)

• Figure 1: Evolution of particle energy distribution for simulations with $N = 768$ and different $\sigma_0 \in~\{3/2, 3/8, 3/32, 3/128\}$. Five times (separated by equal durations) are shown for each simulation, including the initial time, three intermediate times, and the final time (in order of decreasing color fade).
• Figure 2: Moving average of rate of change of average particle energy $\langle d\gamma_{\rm avg}/dt \rangle$ (normalized by $v_{A0}/L$) versus time for simulations with $N = 768$ and $\sigma_0 \in \{3/2, 3/4,..., 3/128\}$. The width of the moving average is $L/{v_{\mkern-1mu {A}}}_0$, and this filtering is denoted by angled brackets. Black dashed lines correspond to $\dot{\gamma}_{\rm avg} \propto \sigma_0$.
• Figure 3: Particle trajectory in the vicinity of an acceleration event, overlaid with magnetic field (blue vectors) at the time when the particle was located at the red marker. The latest time in the trajectory is marked by the black dot.
• Figure 4: Particle energy distribution $f(\gamma)$ (a) and the local power-law index $-\alpha_{\rm loc} = \partial \log{f}/\partial \log{\gamma}$ (b) at several different times for a representative simulation, having $\sigma_0 = 3/8$ and $L/{\rho_e}_0=512$. In (a), dashed lines show the power-law fits. In (b), dashed lines are the fitted cubics, crosses mark the geometric mean of $\gamma_\textnormal{avg}$ and $\gamma_\textnormal{max}$, while circles mark the first turning point of the cubic or the inflection point if there are no turning points. See the main text for more details.
• Figure 5: Time evolution of power-law indices $\alpha$ for simulations with different system size $L/{\rho_e}_0$ but the same $\sigma_0 = 3/8$, with exponential fits (dashed lines). Data points for each simulation are slightly time-shifted for visibility.