Quantifying Energy Conversion in Higher-Order Phase Space Density Moments in Plasmas

Weakly collisional and collisionless plasmas are typically far from local thermodynamic equilibrium (LTE), and understanding energy conversion in such systems is a forefront research problem. The standard approach is to investigate changes in internal (thermal) energy and density, but this omits energy conversion that changes any higher-order moments of the phase space density. In this Letter, we calculate from first principles the energy conversion associated with all higher moments of the phase space density for systems not in LTE. Particle-in-cell simulations of collisionless magnetic reconnection reveal that energy conversion associated with higher-order moments can be locally significant. The results may be useful in numerous plasma settings, such as reconnection, turbulence, shocks, and wave-particle interactions in heliospheric, planetary, and astrophysical plasmas.

Figure 1

Schematic showing energy conversion channels according to their impact on the phase space density $f_{\sigma }$. The initial $f_{\sigma }$ is depicted as Maxwellian for illustrative purposes on the left. The final $f_{\sigma }$ is to their right. The descriptions of the changes in $f_{\sigma }$ are to their right.

Figure 2

Electron energy conversion in a PIC simulation of magnetic reconnection. (a) Out-of-plane current density $J_z$, with projections of magnetic field lines and segments of electron velocity streamlines overplotted in black and orange, respectively. (b) Electron entropy-based non-Maxwellianity ${\overline{M}}_{\mathrm{KP},e}$. Time rates of change per particle of (c) work $d{\mathcal{W}}_e/dt$, (d) internal energy $d{\mathcal{E}}_{e,\mathrm{int}}/dt$, and (e) relative energy $d{\mathcal{E}}_{e,\mathrm{rel}}/dt$. (f) ${\log }_{10}[|(d{\mathcal{E}}_{e,\mathrm{rel}}/dt)/(d{\mathcal{E}}_{e,\mathrm{int}}/dt)|]$. 1D cuts of the terms in (c)–(e) in the (g) $x$ and (h) $y$ directions. (i)–(l) Reduced electron phase space density $f_e(v_x,v_z)$ at locations denoted by the colored $\mathsf{x}$’s at the top left of the plots corresponding to the $\mathsf{x}$’s in (b) along a streamline.

Figure 3

Sketch illustrating energy conversion from Eq. (10). Arrows show conversion channels between work (blue), heat (pink), energy (orange), and collisions (green), with standard channels in black and relative channels in red. The light blue dashed arrow signifies how the relative terms couple to thermodynamic terms.