On the double-adiabatic equations in the relativistic regime

arXiv:2603.25594 · physics.plasm-ph, astro-ph.HE · Submitted 2026-03-26 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Today's paper: "On the double-adiabatic equations in the relativistic regime".

Jocelyn: The gist: This work analytically solves the drift kinetic equation for a homogeneous, magnetized, collisionless plasma and obtains general solutions for time-dependent, gyrotropic distribution functions in nonrelativistic, relativistic,

Vera: First, who's behind it and why it matters.

Title and authors: Vera: To summarize what they are doing here in "On the double-adiabatic equations in the relativistic regime," they analytically solve the drift kinetic equation to get a time-dependent distribution function f alpha t p⊥ p∥, and then they calculate the moments of this solution to get double-adiabatic evolution equations for P⊥ and P∥.

Jocelyn: They are focusing on solving that drift kinetic equation in three different regimes: nonrelativistic, relativistic, and ultrarelativistic initial distributions. This shows how the physics handles changes from normal speeds up to speeds much closer to the speed of light.

Subrahmanyan: The summary highlights that for a nonrelativistic Maxwellian initial distribution, they recover the classic CGL double-adiabatic equations for T⊥ t and T∥ t. This is just checking if their general method works when things are slow.

Vera: Then they move to the relativistic extension, where they derive new evolution equations for P⊥ and P∥ that depend only on density n(t) and field strength B(t), which is a big step because it decouples the pressure evolution.

Jocelyn: And when they get to the ultrarelativistic case, they obtain analytical expressions for the perpendicular pressure P⊥ (two point one eight) and parallel pressure P∥ (two point two one) that are tractable when kBT over mc squared is greater than one times the speed of light squared, which is what makes those high-speed calculations possible <ref:2603.25594#pg1,obtain analytical expressions for the>.

Subrahmanyan: The paper shows that these analytical expressions for the ultrarelativistic case are also consistent with the general state equations for a relativistic anisotropic plasma from Gedalin in one thousand nine hundred ninety-one showing they hold up across different theoretical frameworks <ref:2603.25594#pg2>.

Vera: So, essentially, they are taking a distribution function evolution and turning it into simple equations for pressure components based on how fast the density and field are changing.

Jocelyn: And this is important because it gives us a way to track how anisotropy develops in these extreme environments without having to solve the full kinetic equation every time.

The paper's summary: Vera: One of the main improvements they propose is that they can generate new, relativistic expressions for the evolution of both perpendicular and parallel pressures by numerically integrating those analytical solutions derived from the drift kinetic equation, like equations two point three three and two point three five.

Jocelyn: They’re essentially using those analytical results to create a new way to get double-adiabatic evolution for the relativistic and ultrarelativistic cases that builds on the classical CGL model by giving it relativistic versions.

Subrahmanyan: This means they've managed to extend the CGL equations, which are useful for constructing fluid models with anisotropic closures to MHD equations, into a regime where particle speeds are significant fractions of light speed.

Vera: They also show that their derived distribution function f t p p is the analytical form of the distribution function that Particle-in-Cell simulations evolve when initialized with a Maxwell-Jüttner distribution, until some instability kicks in.

Jocelyn: So, they’ve connected the abstract kinetic solutions directly to what we see in simulations, which gives us a concrete way to predict how the plasma actually behaves before it breaks down.

Subrahmanyan: This connection is powerful because it suggests that their analytical form for P⊥ (two point three three) and P∥ (two point three five) correctly describes the evolution of P⊥ i and P∥ i in shearing simulations for mildly relativistic temperatures, which is a very specific test case.

Vera: And they also mentioned that this general solution works even if you start with other initial distributions, like power-law or kappa distributions, which are relevant for things like cosmic rays and solar wind studies.

The paper's improvements: Jocelyn: So to wrap up the paper "On the double-adiabatic equations in the relativistic regime," they show that their derived distribution function is a general solution for the time-dependent drift kinetic equation, which allows them to predict pressure evolution based on density and field variations.

Vera: They provide new, relativistic expressions for P⊥ and P∥ when starting with an initial relativistic Maxwellian distribution, and these results are validated by Particle-in-Cell simulations of both shearing and compressing boxes in both mildly relativistic and ultrarelativistic initial temperatures.

Subrahmanyan: The implication is that the relativistic double-adiabatic equations now provide a new closure for extended magnetohydrodynamic MHD theories in the context of collisionless plasmas where heat fluxes are negligible and spatial gradients are large, which is relevant for understanding pulsar wind nebulae or black hole accretion flows.

Jocelyn: They also confirm that their analytical expressions for P⊥ in the ultrarelativistic case are consistent with those from the relativistic moment equations, showing a robust connection across different theoretical approaches.

Vera: So, if you’re looking at how plasma pressure anisotropy evolves in these extreme settings, this paper provides a new tool—the double-adiabatic equations in the relativistic regime—that can be used to model things like jets and radiation belts.

Subrahmanyan: I just think the work confirms that the specific form of these double-adiabatic equations depends on the initial distribution function used, which is a key point for anyone trying to apply this theory to real cosmic ray physics.

Jocelyn: It’s a solid piece of work connecting kinetic theory, simulations, and general fluid models for anisotropic plasmas.

Vera: We're all ready for whatever comes next on the arXiv.

Conclusion: Vera: So we’ve been looking at "On the double-adiabatic equations in the relativistic regime," and essentially, this paper solves the drift kinetic equation to get a general solution for a time-dependent distribution function, which then lets them derive new evolution equations for perpendicular and parallel pressures.

Jocelyn: Right. They’re taking that kinetic equation—the drift kinetic equation—and showing how you can get those double-adiabatic equations for P⊥ and P∥ in the relativistic and ultrarelativistic regimes based only on the density and magnetic field changes.

Subrahmanyan: The big picture here is that this gives us a new way to close extended magnetohydrodynamic theories when you have collisionless plasmas where heat fluxes are negligible but spatial gradients are large, which is super relevant for modeling astrophysical phenomena.

Vera: And they do connect this to simulations, you know? They use Particle-in-Cell simulations of shearing and compressing boxes to confirm that their analytical expressions for P⊥ and P∥ actually describe the pressure evolution seen in those runs across different initial temperatures.

Jocelyn: That’s important because it means we can take those empirical histograms from the simulations, like the faMJ(t, p⊥) distribution, and compare them directly to the analytical marginal distributions they get from their solution.

Subrahmanyan: If this holds up as robustly as it does in shearing simulations for mildly relativistic temperatures, then this method might be a way to predict pressure anisotropies in things like pulsar wind nebulae or jets that we can’t quite reach with standard MHD codes.

Vera: They showed the solution works for several initial conditions, from nonrelativistic Maxwellians all the way up to ultrarelativistic Maxwell-Jüttner distributions, which is pretty comprehensive.

Jocelyn: It really shows how versatile this kinetic approach is; it’s not just a niche thing for one temperature but covers a wide range of particle speeds and initial states.

Subrahmanyan: The paper also suggests using these equations as a new closure relation for those MHD systems, which means you keep the pressure anisotropies in your stress-energy tensor even when you're using fluid approximations.

Vera: So the main implication is that we now have a more rigorous framework for how pressure anisotropy develops in collisionless magnetized plasmas across different relativistic scales.

Jocelyn: It’s a solid piece of work connecting kinetic theory, simulations, and general fluid models for anisotropic plasmas.

Subrahmanyan: We should keep an eye on how this method extends to other initial distributions like power-law or kappa distributions when we look at cosmic ray or solar wind studies.

Departamento de Física, Facultad de Ciencias Físicas y Matemıcas, Universidad de Chile · Department of Astronomy, University of Wisconsin--Madison · Department of Physics, University of Wisconsin--Madison

physics.plasm-ph, astro-ph.HE

Submitted: 2026-03-26

Updated: 2026-07-28

Comments: Accepted at Journal of Plasma Physics, typos corrected; 30 pages, 8 figures

DOI: 10.1017/S0022377826102220

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 80/100

The gist: The gist: This work analytically solves the drift kinetic equation for a homogeneous, magnetized, collisionless plasma and obtains general solutions for time-dependent, gyrotropic distribution

Key concepts

Drift Kinetic Equation: This is the fundamental equation describing how the distribution of particles changes over time due to their motion (drift) within a magnetized plasma. It tracks the evolution of the particle distribution function f(t, p⊥, p∥), which depends on momentum components perpendicular and parallel to the magnetic field.
Double-Adiabatic Equations: These are equations that describe how two key thermodynamic quantities—perpendicular pressure (P⊥) and parallel pressure (P∥)—evolve independently over time. The paper extends these classic equations to account for relativistic effects, providing a more accurate description of plasma behavior.
Maxwell-Jüttner Distribution: This is a specific type of distribution function used to describe the momentum states of particles in a relativistic plasma. It is an extension of the nonrelativistic Maxwellian distribution and is crucial for modeling ultrarelativistic conditions where particle speeds approach the speed of light.

Terminology

Anisotropy — 36× in this paper · explained in 4 episodes
Adiabatic Evolution — 12× in this paper · explained in 2 episodes
Accretion Flows — 6× in this paper · explained in 1 episode
Benchmark — 1× in this paper · explained in 2 episodes
Boltzmann Equation — 1× in this paper · explained in 2 episodes
Decoupling — 1× in this paper · explained in 2 episodes
Dynamical Instability — 1× in this paper · explained in 2 episodes
International Space Science Institute — 1× in this paper
Robustness — 1× in this paper · explained in 5 episodes

Summary

The gist: This work analytically solves the drift kinetic equation for a homogeneous, magnetized, collisionless plasma and obtains general solutions for time-dependent, gyrotropic distribution functions in nonrelativistic, relativistic, and ultrarelativistic regimes.

Theoretical Foundation

The evolution of the particle distribution function is governed by the drift kinetic equation (2.1) which describes the evolution of fα(t, p⊥, p∥) where p⊥ and p∥ are the perpendicular and parallel momenta with respect to the direction of the mean magnetic field.
The conservation of particle adiabatic invariants M = p2⊥/2B and J = Hp∥dl provides a natural decoupling, allowing the evolution of P⊥α and P∥α to be treated independently.
The general time-dependent, homogeneous distribution function fα(t, p, µ) is obtained by solving equation (2.4) using the method of characteristics for an initial distribution f0(p, µ).

Regime-Dependent Solutions

For a nonrelativistic Maxwellian initial distribution, the solution recovers the classic CGL double-adiabatic equations for T⊥(t) and T∥(t).
The relativistic extension to the double-adiabatic equations, derived from solving the drift kinetic equation, yields new evolution equations for P⊥ and P∥ dependent only on density n(t) and field strength B(t).
For an ultrarelativistic Maxwell-Jüttner distribution, analytical expressions for the perpendicular pressure P⊥ (2.18) and parallel pressure P∥ (2.21) are obtained, which are tractable for kBT /mc2 ≳ 1.
The general state equations proposed in Gedalin (1991) are satisfied by the derived double-adiabatic equations in both relativistic and ultrarelativistic regimes.

Simulation Validation

Particle-in-cell (PIC) simulations of shearing and compressing boxes were used to confirm the analytical results.
In shearing simulations, the magnetic field amplifies as B(t) = B0√1 + s2t2, while density remains constant, leading to a continuous drive for pressure anisotropy P⊥,i > P∥,i.
In compressing simulations, both density and magnetic field amplify as n(t) = n0(1 + qt)2 and B(t) = B0(1 + qt)2, driving continuous ion and electron pressure anisotropy.
The analytical expressions for P⊥ (2.33) and P∥ (2.35), derived from the relativistic moment equations, correctly describe the evolution of P⊥,i and P∥,i in shearing simulations for mildly relativistic temperatures.
Similarly, these equations accurately describe the evolution of electron pressures in both shearing and compressing simulations across various initial temperature cases.

Conclusions and Applications

The derived distribution function (2.30) is shown to be the analytical form of the distribution function that PIC simulations evolve when initialized with a Maxwell-Jüttner distribution, until an instability is excited.
The relativistic double-adiabatic equations provide a new closure for extended magnetohydrodynamic (MHD) theories in the context of collisionless plasmas where heat fluxes are negligible and spatial gradients are large.
The results can be applied to astrophysical phenomena such as pulsar wind nebulae, relativistic jets, black hole accretion flows, and Van Allen radiation belts.
The work confirms that the specific form of the double-adiabatic equations depends on the initial distribution function used.
The general solution admits other types of initial distributions, including power-law and kappa distributions, relevant for cosmic-ray and solar wind studies.

How it works

The process involves analytically solving the drift kinetic equation (2.1) to find a time-dependent distribution function fα(t, p, µ). This solution is then used to calculate the moments of this distribution to obtain evolution equations for the perpendicular and parallel pressures P⊥ and P∥.

The paper systematically examines three initial conditions: nonrelativistic Maxwellian, relativistic Maxwell-Jüttner, and ultrarelativistic Maxwell-Jüttner distributions.

For the ultrarelativistic case, the integral moments are tractable, leading to analytical evolution equations for P⊥ and P∥ that are applicable for kBT /mc2 ≳ 1.

The derived distribution function f(t, p⊥, p∥) is a novel anisotropic extension of the Maxwell-Jüttner distribution.

The general solution (2.5) allows one to obtain evolution equations for P⊥ and P∥ as a function of B(t) and n(t).

The final results demonstrate that the analytical expressions for P⊥ in the ultrarelativistic case are consistent with the general formula (2.33), which is derived from relativistic moment equations.

The comparison between simulation histograms and the analytical marginal distributions faMJ(t, p⊥) and faMJ(t, p∥) validates the solution across shearing and compressing simulations.

The analysis of pressure anisotropy evolution in the Braginskii limit shows that 1/P d∆P/dt = A B˙B + C n˙n.

Summary of Findings

The research successfully extends the double-adiabatic equations to the relativistic regime and presents a novel anisotropic version of the Maxwell-Jüttner distribution. The analytical solution for the drift kinetic equation provides a general, time-dependent solution for the distribution function in terms of density and magnetic field variations. The paper obtains new, relativistic expressions for the evolution of both perpendicular and parallel pressures in terms of integral moments that can be solved numerically for any temperature θ = kBT /mc2. The analytical expressions are shown to be applicable for kBT /mc2 ⩾ 1. The derived double-adiabatic equations agree with the general state equations for a relativistic anisotropic plasma from Gedalin (1991) and are validated using PIC simulations of shearing and compressing boxes. The work demonstrates that the distribution function correctly describes the momentum distributions of both shearing and compressing PIC simulations in both initial isotropic and final anisotropic states.

Key Results

** Nonrelativistic Case:**

The nonrelativistic case recovers the classic CGL double-adiabatic equations when considering a Maxwellian distribution for f0(p, µ).

** Relativistic Case (Ultrarelativistic):**

For ultrarelativistic initial conditions, the evolution of P⊥ and P∥ is described by analytical expressions (2.18) and (2.21).

The general equations (2.33) and (2.35), derived from relativistic moment equations, are shown to be consistent with the ultrarelativistic analytical results.

The time-dependent distribution function f(t, p, µ) is a new anisotropic extension for a relativistic Maxwellian.

** Simulation Agreement:**

PIC simulations of shearing and compressing boxes confirm that the derived analytical expressions for P⊥ and P∥ correctly describe their evolution in both mildly relativistic and ultrarelativistic initial temperatures.

The marginal distributions faMJ(t, p⊥) and faMJ(t, p∥) obtained from the distribution function f(t, p⊥, p∥) show very good agreement with simulation results at both t · s = 0 and t · s = 1.

** Robustness:**

The results show a robust independence of both mass ratio (mi/me) and initial plasma beta (βiniti) in the shearing and compressing simulations.

The relativistic extension of the double-adiabatic equation is applicable to a wide variety of astrophysical scenarios where one or more species are relativistically hot.

** Future Directions:**

The work suggests using these equations as a new closure for extended MHD models and studying the linear stability of the time-dependent anisotropic Maxwell-Jüttner distribution.

Improvements for AI systems

The AI system can analytically solve time-dependent drift kinetic equations for homogeneous, magnetized collisionless plasmas by obtaining a general solution for a time-dependent, gyrotropic distribution function f(t, p⊥, p∥), for a given initial, well-behaved distribution f0(p⊥, p∥) (Eq. 2.5). This allows the system to predict the evolution of perpendicular and parallel pressures in relativistic and ultrarelativistic regimes based only on density and magnetic field variations.
The system can generate new, relativistic expressions for the evolution of both perpendicular and parallel pressures by numerically integrating analytical solutions derived from the drift kinetic equation (e.g., equations 2.33, 2.35, 2.18, 2.21). This capability extends classical models like CGL by providing new double-adiabatic evolution for the relativistic and ultrarelativistic cases (Section 6).
The AI can validate its physical models against complex astrophysical scenarios by performing simulations using fully kinetic particle-in-cell (PIC) codes, confirming that our analytical results agree with PIC simulations of shearing and compressing boxes in both mildly relativistic and ultrarelativistic initial temperatures.
The system can characterize the momentum distribution of plasma species directly from simulation data by comparing empirical histograms, such as the marginal distribution faMJ(t, p⊥) (solid red line) shows very good agreement with the simulation at this stage (Figure 5), against its analytically derived solutions like equation (4.12).
The AI can implement closure relations for extended magnetohydrodynamics (MHD) theories by using the derived distribution function and double-adiabatic equations to maintain pressure anisotropies in the stress-energy tensor, providing a closure to the MHD fluid system while retaining pressure anisotropies (Section 5.1).

Abstract

We revisit the double adiabatic evolution equations and extend them to the relativistic and ultrarelativistic regimes. We analytically solve the relativistic, time-dependent drift kinetic equation for a homogeneous, magnetized, collisionless plasma and obtain a solution explicitly dependent on the magnetic field and density variations. In the case of an initial relativistic Maxwellian distribution, a natural extension to an anisotropic Maxwell-Jüttner is obtained. We calculate the moments of this time-dependent solution and obtain analytical expressions for the evolution of the perpendicular and parallel pressures in the ultrarelativistic case. We numerically solve the moment equations in the relativistic case and obtain general expressions for the double-adiabatic equations in this regime. We confirm our results using fully kinetic particle-in-cell simulations of shearing and compressing boxes. Our findings can be readily applied to relativistic species including cosmic-rays and electron-positron pairs, present in astrophysical plasmas like pulsar wind nebulae, astrophysical jets, black hole accretion flows, and Van Allen radiation belts.

Sources

Double-Adiabatic Equations of State for Relativistic Plasmas