Theory, observations, and simulations of kinetic entropy in a magnetotail electron diffusion region

Abstract

We examine velocity-space kinetic entropy, a spatially local measure of entropy for systems out of thermal equilibrium, during an encounter of an electron diffusion region at a magnetic reconnection site in Earth's magnetotail by the Magnetospheric Multiscale (MMS) mission. We start by generalizing the theory of kinetic entropy to the case of non-uniform velocity space grids and transforming the equations into spherical energy coordinates useful to experimental plasma detectors. The theory is then applied to MMS data and compared to particle-in-cell simulations of reconnection. We demonstrate that the entropy-based non-Maxwellianity measure from the MMS data is of sufficiently high precision to reliably identify non-Maxwellian distributions and therefore the measurements when kinetic effects are most significant. By comparing two different non-Maxwellian measures, we show that total entropy density suffers from “information loss” because it lacks a dependence on the velocity space grid, and so has lost information about how well a distribution function is resolved. Local velocity-space kinetic entropy density recovers this information. We quantify information loss and argue that the considerations needed to minimize it are crucial for instruments designed to measure distribution functions in situ.

The need for accurate measurements of thermal velocity distribution functions in the solar wind

“…Further, if our measurements effectively "smear out" the ion VDF, we cannot discriminate whether the ion VDF changes shape across the shock or how it evolves. This leaves us with crude parameters like the total ion temperature, which can be extremely misleading in such kinetic processes [e.g., (Wilson et al, 2014a;Wilson et al, 2014b;Goldman et al, 2020;Wilson et al, 2020;Goldman et al, 2021;Argall et al, 2022)].…”

Temporal, Spatial, and Velocity‐Space Variations of Electron Phase Space Density Measurements at the Magnetopause

“…Because the magnetopause environment is believed to be collisionless to an excellent approximation, we do not believe this deviation from zero to be physical; rather, we suspect the instrumentation on MMS, while groundbreaking, was not specifically designed to measure the entire velocity space with the exactness that satisfying the Vlasov equation would require, consistent with Argall et al. (2022). Nevertheless, we emphasize the remarkable qualitative and visual agreement between v ⋅ ∇ f e and ( F / m e ) ⋅ ∇ v f e ( i.e.…”

“…Due to the limitations of the MMS temporal, spatial, and velocity-space resolutions, we note that the Vlasov equation terms we computed (e.g., row F in Figure 5) do not sum to zero exactly throughout velocity-space. Because the magnetopause environment is believed to be collisionless to an excellent approximation, we do not believe this deviation from zero to be physical; rather, we suspect the instrumentation on MMS, while groundbreaking, was not specifically designed to measure the entire velocity space with the exactness that satisfying the Vlasov equation would require, consistent with Argall et al (2022). Nevertheless, we emphasize the remarkable qualitative and visual agreement between v ⋅ ∇f e and (F/m e ) ⋅ ∇ v f e (i.e., comparing rows D and E in Figure 5) that MMS successfully captures.…”

Relating the Phases of Magnetic Reconnection Growth to the Temporal Evolution of X-line Structures in a Collisionless Plasma

“…Another aspect worth exploring in the future is how temporal evolution of the x-line and the direction of energy flow relate to entropy, disorder, and the arrow of time. Studies using PIC simulations 27,28 and MMS data 29 suggest that even in collisionless reconnection, the velocity-space kinetic entropy does increase and the reconnection process is irreversible.…”

“…Within the central EDR, electromagnetic fields do work to lower the local plasma entropy, making the local electron VDFs more ordered and less Maxwellian before those electrons become thermalized in the exhaust and increase in entropy again 29 . The VDF structures associated with the remagnetization of the electron jet in the outer EDR 24 also appear to form near the time of the maximum reconnection rate 14,15 .…”