Collisionless relaxation as the origin of the anisotropic, non-thermal, and multi-temperature momentum distributions observed in space plasmas

Anisotropic, non-thermal, and multi-temperature distributed particle momenta are commonly observed in collisionless space plasmas, such as the solar wind. Using Liouville’s theorem, we argue that anisotropic compression or expansion of the plasma, followed by a relaxation of the resulting anisotropic stress must lead to non-equilibrium states that are either anisotropic, non-thermal distribution functions, different electron and ion temperatures, or a combination of these effects. We present arguments showing that a plasma in thermal equilibrium undergoing anisotropic compression or expansion cannot return to thermal equilibrium in the absence of particle collisions. Since most astrophysical plasmas are practically collisionless and experience significant anisotropic compression or expansion, we expect anisotropic, non-thermal, and multi-temperature particle distributions to be ubiquitous, in agreement with solar wind measurements.

💡 Research Summary

The paper by Enßlin and Pfrommer tackles a long‑standing puzzle in space plasma physics: why collisionless plasmas such as the solar wind frequently exhibit anisotropic, non‑thermal, and multi‑temperature particle momentum distributions. The authors argue that these features are a natural consequence of collisionless relaxation following anisotropic compression or expansion, and they substantiate this claim using fundamental principles of Hamiltonian dynamics, specifically Liouville’s theorem.

The argument begins with an idealized, initially isotropic and homogeneous plasma described by a monotonically decreasing distribution function g₀(p). When the plasma is compressed (or expanded) along the magnetic‑field direction by a factor r, the phase‑space coordinates transform as (x, y, z) → (x, y, z/r) and (pₓ, p_y, p_z) → (pₓ, p_y, r p_z). This operation creates an anisotropic distribution f′₀ that retains the same total number of particles but now possesses a directional bias in momentum space.

The core of the analysis is a proof by contradiction. Assuming that collisionless relaxation (i.e., Vlasov‑Maxwell dynamics without binary collisions) can completely isotropize the distribution, the final state would be f₁(p)=g₁(p), a purely isotropic function of momentum magnitude. However, the relaxation must respect four conserved quantities: particle number, total momentum, total energy, and, crucially, the phase‑space density (the Casimir invariants). By writing down the integral constraints for particle number and energy before and after relaxation, the authors show that no monotonic g₁(p) can simultaneously satisfy both constraints unless r=1 (no compression). Hence, some imprint of the original anisotropy must survive the relaxation.

The surviving imprint can manifest in three ways: (i) a residual anisotropy in the pitch‑angle distribution, (ii) a non‑thermal high‑energy tail (often modeled as a kappa distribution), and (iii) a temperature difference between species (e.g., electrons and ions). The paper connects these outcomes to observed solar‑wind features: the frequent detection of kappa‑like suprathermal tails, the presence of fire‑hose, mirror, and gyrothermal instabilities that are driven by anisotropy, and the measured electron‑ion temperature ratios.

The authors also critically examine related theoretical frameworks. They discuss Lynden‑Bell’s violent relaxation theory, which maximizes an entropy while conserving Casimirs, and point out that in a strictly collisionless system the Casimirs prevent any entropy change, making the maximization procedure internally inconsistent. They further critique quasi‑linear diffusion theory, noting that its prediction of momentum‑space diffusion conflicts with Liouville’s theorem because diffusion would imply mixing of phase‑space densities, which is prohibited in a Hamiltonian flow.

A notable insight is the role of free energy distribution across momentum space. High‑energy particles carry relatively little free energy to excite waves, so wave activity is concentrated at lower momenta. Consequently, the relaxation process efficiently isotropizes the core of the distribution while leaving the high‑energy tail relatively untouched, naturally producing a kappa‑type spectrum.

In the concluding section, the authors argue that because most astrophysical plasmas are essentially collisionless and undergo significant anisotropic compressions (e.g., due to expansion of the solar wind, shock passage, or large‑scale flows), the mechanisms described should be ubiquitous. They suggest that the same reasoning can be extended to other environments such as galaxy‑cluster outskirts, accretion flows, and supernova remnants. Future work is proposed to involve kinetic simulations that track the detailed evolution of the phase‑space density, and to compare the predicted residual anisotropies and temperature ratios with high‑resolution spacecraft measurements.

Overall, the paper provides a compelling, physics‑first explanation for the prevalence of anisotropic, non‑thermal, and multi‑temperature particle distributions in space plasmas, grounding the argument in the immutable conservation laws of collisionless dynamics.