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DOE OSTI2020

Collective mode Brownian dynamics: A method for fast relaxation of statistical ensembles

Sampling equilibrium configurations of correlated systems of particles with long relaxation times (e.g., polymeric solutions) using conventional molecular dynamics and Monte Carlo methods can be challenging. This is especially true for systems with complicated, extended bond network topologies and other interactions that make the use and design of specialized relaxation protocols infeasible. We introduce a method based on Brownian dynamics simulations that can reduce the computational time it takes to reach equilibrium and draw decorrelated samples. Importantly, the method is completely agnostic to the particle configuration and the specifics of interparticle forces. In particular, we develop a mobility matrix that excites non-local, collective motion of N particles and can be computed efficiently in $\mathcal{O}(N)$ time. Particle motion in this scheme is computed by integrating the overdamped Langevin equation with an Euler–Maruyama scheme, in which Brownian displacements are drawn efficiently using a low-rank representation of the mobility matrix in position and wave space. We demonstrate the efficacy of the method with various examples from the realm of soft condensed matter and release a massively parallel implementation of the code as a plugin for the open-source package HOOMD-blue [J. A. Anderson et al., J. Comput. Phys. 227, 5342 (2008) and J. Glaser et al., Comput. Phys. Commun. 192, 97 (2015)] which runs on graphics processing units

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

A fast Monte Carlo cell-by-cell simulation for radiobiological effects in targeted radionuclide therapy using pre-calculated single-particle track standard DNA damage data

Introduction: We developed a new method that drastically speeds up radiobiological Monte Carlo radiation-track-structure (MC-RTS) calculations on a cell-by-cell basis. Methods: The technique is based on random sampling and superposition of single-particle track (SPT) standard DNA damage (SDD) files from a “pre-calculated” data library, constructed using the RTS code TOPAS-nBio, with “time stamps” manually added to incorporate dose-rate effects. This time-stamped SDD file can then be input into MEDRAS, a mechanistic kinetic model that calculates various radiation-induced biological endpoints, such as DNA double-strand breaks (DSBs), misrepairs and chromosomal aberrations, and cell death. As a benchmark validation of the approach, we calculated the predicted energy-dependent DSB yield and the ratio of direct-to-total DNA damage, both of which agreed with published in vitro experimental data. We subsequently applied the method to perform a superfast cell-by-cell simulation of an experimental in vitro system consisting of neuroendocrine tumor cells uniformly incubated with 177 Lu. Results and discussion: The results for residual DSBs, both at 24 and 48 h post-irradiation, are in line with the published literature values. Our work serves as a proof-of-concept demonstration of the feasibility of a cost-effective “in silico clonogenic cell survival assay” for the computational design and development of radiopharmaceuticals and novel radiotherapy treatments more generally.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2025

Low-Jitter Clock Receivers for Fast Timing Applications

Precision timing is a key requirement for emerging 4D particle tracking, Positron Emission Tomography (PET), beam and fusion plasma diagnostics, and other systems. Time-to-Digital Converters (TDCs) are commonly used to provide digital estimates of the relative timing between events, but the jitter performance of a TDC can be no better than the performance of the circuits that acquire the pulses and deliver them to the TDC. Several clock receiver and distribution circuits were evaluated, and a differential amplifier with resistive loads driving a pseudo-differential clock distribution network, developed using design guidelines for radiation tolerance and cryogenic compatibility, was fabricated as part of three prototypes: an analog front-end testbed chip for high-precision timing pixel readout, a dedicated TDC evaluation chip, and a Low-Gain Avalanche Detector (LGAD) readout circuit. Based on TDC measurements of the prototypes, we infer that the jitter added by the clock receiver and distribution circuits is less than 2.25 ps-rms. This performance meets the requirements of many future precision timing systems. The clock receiver and on-chip pseudo-differential driver were fabricated in commercial 28-nm CMOS technology and occupy 2288 µm 2 .

47 OTHER INSTRUMENTATION↗

DOE OSTI2026

Unravelling fast-charging degradation in NMC/Gr pouch cells: Lithium plating and SEI properties

As fast-charging technology expands across the electric vehicle and emerging energy-storage applications, understanding its impact on battery performance and longevity is critical. In this study, 1.8 Ah LiNi 0.6 Mn 0.2 Co 0.2 O 2 /graphite pouch cells were charged at various charging rates (0.5C, 2C, 4C, and 6C) to investigate the degradation mechanisms. Our results showed that well-designed NMC/Gr pouch cells could reach over 1000 cycles with a 2C charging rate, while only reaching around 500 cycles with 4C and 6C charging rates. Fast-charging effects on NMC and graphite electrodes were obtained through a series of post-mortem characterizations, including electrochemical impedance spectroscopy (EIS), Raman spectroscopy, X-ray photoelectron spectroscopy (XPS), and scanning electron microscopy (SEM) with energy-dispersive X-ray spectroscopy (EDS). Although higher charging rates cause pulverization of NMC secondary particles, the dominant degradation mechanism driving the fading of fast-charging-related performance lies in the graphite anode, where lithium plating and LiF-rich solid electrolyte interphase (SEI) formation result in Li inventory loss and impedance growth. The postmortem results suggest that the formation of a LiF-rich SEI, which exacerbates anode impedance and some irreversible Li + ion loss, is likely driven by the substantial decomposition of PF 6 − during fast charging, an effect often overlooked in smaller laboratory-scale studies.

Luo, Mei [Argonne National Laboratory (ANL), Argon↗

DOE OSTI2024

Formation of collisionless shocks driven by strongly magnetized relativistic electrons in the laboratory

In experiments performed with the OMEGA EP laser system, proton deflectometry captured magnetic field dynamics consistent with collisionless shock formation driven by strongly magnetized relativistic electrons. During laser-foil interactions, shocks can form as relativistic electrons and strong surface magnetic fields generated by a short-pulse laser impinge on a cooler plasma produced by a longer-pulse laser. Three-dimensional particle-in-cell simulations reproduce the magnetic draping and fast formation speeds measured in the experiment and reveal that this relativistic-electron-driven shock forms at an interface that is unstable to shear and streaming instabilities. The simulation results provide insight into the microphysics that may influence high-energy shocks observed in extreme astrophysical environments. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1994

Particle acceleration in the dynamic magnetotail: Orbits in self-consistent three-dimensional MHD fields

The acceleration of protons in a dynamically evolving magnetotail is investigated by tracing particles in the fields obtained from a three-dimensional resistive magnetohydrodynamic (MHD) simulation. The MHD simulation, representing plasmoid formation and ejection through a near-Earth reconnection process, leads to cross-tail electric fields of up to approximately 4 mV/m with integrated voltages across the tail of up to approximately 200 kV. Energization of particles takes place over a wide range along the tail, due to the large spatial extent of the increased electric field together with the finite cross-tail extent of the electric field region. Such accelerated particles appear earthward of the neutral line over a significant portion of the closed field line region inside of the separatrix, not just in the vicinity of the separatrix. Two different acceleration processes are identified: a 'quasi-potential' acceleration, due to particle motion in the direction of the cross-tail electric field, and a 'quasi-betatron' effect, which consists of multiple energy gains from repeated crossings of the acceleration region, mostly on Speiser-type orbits, in the spatially varying induced electric field. The major source region for accelerated particles in the hundreds of keV range is the central plasma sheet at the dawn flank outside the reconnection site. Since this source plasma is already hot and dense, its moderate energization by a factor of approximately 2 may be sufficient to explain the observed increases in the energetic particle fluxes. Particles from the tail are the source of beams at the plasma sheet/lobe boundary. The temporal increase in the energetic particle fluxes, estimated from the increase in energy gain, occurs on a fast timescale of a few minutes, coincident with a strong increase in B(sub z), despite the fact that the inner boundary ('injection boundary') of the distribution of energized particles is fairly smooth.

Birn, Joachim↗

NASA NTRSDate not supplied

Towards an Optimal Estimation Retrieval of Cirrus Cloud Optical and Microphysical Properties Using Hyperspectral Shortwave Instruments and A Fast Radiative Transfer Algorithm

Cirrus cloud retrieval products (here, cloud optical depth, effective particle size, and cloud top height) are important inputs into numerical weather and climate models. Uncertainties in such retrieval products, as a matter of course, propagate downstream, impacting model calculations. Improvements in high-quality global cirrus cloud optical and microphysical data products from satellite observations are needed to understand and reduce retrieval uncertainties. Hyperspectral instruments produce high-resolution and information-dense radiance spectra, thus offering the opportunity to reduce uncertainties in retrieval products. An optimal estimation-based cirrus cloud optical and microphysical product retrieval is under development for the NASA Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) and the Earth Surface Mineral Dust Source Investigation (EMIT) instruments, and the forthcoming Climate Absolute Radiance and Refractivity Observatory Pathfinder (CLARREO-Pathfinder). The spectral coverage of all three instruments includes the ultraviolet, visible, and near-infrared. In this study a very fast radiative transfer model, the Principal Component-based Radiative Transfer Model in the solar spectral region (PCRTM-Solar), is used for forward modeling computations. This will reduce the computational burden in the forward radiance and Jacobian calculations as the cost function is minimized, enabling the entire AVIRIS, EMIT, and CLARREO-Pathfinder spectral range to be used in the optimal estimation framework. Using the entire spectrum will maximize the information content, resulting in a more robust and more accurate retrieval. As a first step, the retrieval is being designed for single layer ice clouds over open ocean water. Preliminary results will be shown.

Jeffrey Mast↗

NASA NTRS1989

Computer simulations of cosmic-ray diffusion near supernova remnant shock waves

A plasma simulation model was used to study the resonant interactions between streaming cosmic-ray ions and a self-consistent spectrum of Alfven waves, such as might exist in the interstellar medium upstream of a supernova remnant shock wave. The computational model is a hybrid one, in which the background interstellar medium is an MHD fluid and the cosmic-rays are discrete kinetic particles. The particle sources for the electromagnetic fields are obtained by averaging over the fast cyclotron motions. When the perturbed magnetic field is larger than 10 percent of the background field, the macro- and microphysics are no longer correctly predicted by quasi-linear theory. The particles are trapped by the waves and show sharp jumps in their pitch-angles relative to the background magnetic field, and the effective ninety-degree scattering time for diffusion parallel to the background magnetic field is reduced to between 5 and 30 cyclotron periods. Simulation results suggest that Type 1 supernova remnants may be the principal sites of cosmic ray acceleration.

Max, C. E.↗

DOE OSTI2024

Laser-driven ion and electron acceleration from near-critical density gas targets: Towards high-repetition rate operation in the 1 PW, sub-100 fs laser interaction regime

Ion acceleration from gaseous targets driven by relativistic-intensity lasers was demonstrated as early as the late 1990s, yet most of the experiments conducted to date have involved picosecond-duration, Nd:glass lasers operating at low repetition rate. Here, we present measurements on the interaction of ultraintense ( ≈ 10 20 W cm − 2 , 1 PW ) , ultrashort ( ≈ 70 fs ) Ti:Sa laser pulses with near-critical ( ≈ 10 20 cm − 3 ) helium gas jets, a debris-free targetry with the potential for future compatibility with high ( ≈ 1 Hz ) repetition rate operation. We provide evidence of α particles being forward accelerated up to ≈ 2.7 − MeV energy with a total flux of ≈ 10 11 sr − 1 as integrated over > 0.1 − MeV energies and detected within a 0.5 − mrad solid angle. We also report on on-axis emission of relativistic electrons with an exponentially decaying spectrum characterized by a ≈ 10 − MeV slope, i.e., five times larger than the standard ponderomotive scaling. The total charge of these electrons with energy above 2 MeV is estimated to be of ≈ 1 nC , corresponding to ≈ 0.1 % of the laser drive energy. In addition, we observe the formation of a plasma channel, extending longitudinally across the gas density maximum and expanding radially with time. These results are well captured by large-scale particle-in-cell simulations, which reveal that the detected fast ions most likely originate from reflection off the rapidly expanding channel walls. The latter process is predicted to yield ion energies in the MeV range, which compare well with the measurements. Finally, direct laser acceleration is shown to be the dominant mechanism behind the observed electron energization. Published by the American Physical Society 2024

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Intercomparison of Aerosol Instrumental Responses Using AEROTRAK, SMPS, and APS in a Simulated Building

In this work, three aerosol instruments were intercompared by using data collected from field campaigns in a simulated building environment during the last quarter of 2020 from October to December at Oak Ridge National Laboratory. The three aerosol instruments—AeroTrak Model 9110, Scanning Mobility Particle Spectrometer (SMPS) 3080L + 3025A, and Aerodynamic Particle Sizer (APS) 3320—were manufactured by TSI Inc. to perform in situ measurements of particle number concentration in the air. These instruments were co-located in one room in which an aerosol generator was used to produce a defined aerosol population for the instruments to sample and detect in each campaign. The SMPS measured aerosol particles with diameters from 0.01 to 0.7 µm, APS measured aerosol particles with diameters from 0.5 to 20 µm, and AeroTrak measured aerosol particles with diameters from 0.1 to 10 µm. The results suggest that although AeroTrak is fast and easy to use, it does not provide measurements comparable with those obtained by SMPS and APS. The differences in particle counts were orders of magnitude in size (i.e., the AeroTrak data were unreasonably small in the tested scenarios). The results clearly indicate that AeroTrak should not be used in ambient applications in which the level of particle concentration is expected to be high. This would cause difficulty to a campaign in which the dilution factor is unknow a priori. The high particle concentration could also lead to an erroneous aerosol size distribution (e.g., a bimodal size distribution was measured as opposed to a single-mode one). The number concentration limit that AeroTrak is expected to run reliably (i.e., with a counting coincidence error less than ±10%) is approximately three to four per cubic centimeter. This suggests that the instrument is best used in a clean room environment, not in an ambient or indoor application, such as the Flexible Research Platform campaigns.

47 OTHER INSTRUMENTATION↗

NASA NTRS2011

Consequences of the Breakout Model for Particle Acceleration in CMEs and Flares

The largest and most efficient particle accelerators in the solar system are the giant events consisting of a fast coronal mass ejection (CME) and an intense X-class solar flare. Both flares and CMEs can produce l0(exp 32) ergs or more in nonthermal particles. Two general processes are believed to be responsible: particle acceleration at the strong shock ahead of the CME, and reconnection-driven acceleration in the flare current sheet. Although shock acceleration is relatively well understood, the mechanism by which flare reconnection produces nonthermal particles is still an issue of great debate. We address the question of CME/flare particle acceleration in the context of the breakout model using 2.5D MHD simulations with adaptive mesh refinement (AMR). The AMR capability allows us to achieve ultra-high numerical resolution and, thereby, determine the detailed structure and dynamics of the flare reconnection region. Furthermore, we employ newly developed numerical analysis tools for identifying and characterizing magnetic nulls, so that we can quantify accurately the number and location of magnetic islands during reconnection. Our calculations show that flare reconnection is dominated by the formation of magnetic islands. In agreement with many other studies, we find that the number of islands scales with the effective Lundquist number. This result supports the recent work by Drake and co-workers that postulates particle acceleration by magnetic islands. On the other hand, our calculations also show that the flare reconnection region is populated by numerous shocks and other indicators of strong turbulence, which can also accelerate particles. We discuss the implications of our calculations for the flare particle acceleration mechanism and for observational tests of the models.

Antiochos, S. K.↗

NASA NTRS1990

A study of the scintillation light induced in liquid xenon by electrons and alpha particles

The time dependence and the intensity of the primary scintillation light in liquid Xe excited by Am-241 alpha particles and Bi-207 internal-conversion electrons were measured at different electric-field strengths. High-purity liquid Xe was used to fill a parallel-plate ionization chamber equipped with a CaF2 window coupled to a UV-sensitive photomultiplier tube. The effect of the specific ionization density on the scintillation light and the time correlation between the light signal and the charge signal is reported. It is demonstrated that the fast scintillation signal produced in liquid Xe by an ionizing particle provides an ideal trigger in a detector aiming at a complete three-dimensional event reconstruction with an excellent background rejection capability.

Aprile, Elena↗

DOE OSTI2022

Regularized inversion of aerosol hygroscopic growth factor probability density function: application to humidity-controlled fast integrated mobility spectrometer measurements

Abstract. Aerosol hygroscopic growth plays an important role in atmospheric particle chemistry and the effects of aerosol on radiation and hence climate. The hygroscopic growth is often characterized by a growth factor probability density function (GF-PDF), where the growth factor is defined as the ratio of the particle size at a specified relative humidity to its dry size. Parametric, least-squares methods are the most widely used algorithms for inverting the GF-PDF from measurements of the humidified tandem differential mobility analyzer (HTDMA) and have been recently applied to the GF-PDF inversion from measurements of the humidity-controlled fast integrated mobility spectrometer (HFIMS). However, these least-squares methods suffer from noise amplification due to the lack of regularization in solving the ill-posed problem, resulting in significant fluctuations in the retrieved GF-PDF and even occasional failures of convergence. In this study, we introduce nonparametric, regularized methods to invert the aerosol GF-PDF and apply them to HFIMS measurements. Based on the HFIMS kernel function, the forward convolution is transformed into a matrix-based form, which facilitates the application of the nonparametric inversion methods with regularizations, including Tikhonov regularization and Twomey's iterative regularization. Inversions of the GF-PDF using the nonparameteric methods with regularization are demonstrated using HFIMS measurements simulated from representative GF-PDFs of ambient aerosols. The characteristics of reconstructed GF-PDFs resulting from different inversion methods, including previously developed least-squares methods, are quantitatively compared. The result shows that Twomey's method generally outperforms other inversion methods. The capabilities of Twomey's method in reconstructing the pre-defined GF-PDFs and recovering the mode parameters are validated.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1992

Anomalous ion mixing within a Kelvin-Helmholtz vortex in a collisionless plasma

Anomalously fast ion mixing is observed in a hybrid code simulation (particle ions and a massless electron fluid) of the Kelvin-Helmholtz instability in a collisionless plasma. While the traditional view predicts that the ion mixing occurs on a time scale longer than the roll-up time T(v) of the Kelvin-Helmholtz vortex by a factor of (velocity shear scale length)/(ion Larmor radius), the simulation results show that the mixing occurs within a time scale comparable to T(v). It is shown that this fast mixing is due to the scattering of ions by unsteady electromagnetic fields.

Terasawa, T.↗

DOE OSTI2022

Effect of anisotropic fast ions on internal kink stability in DIII-D negative and positive triangularity plasmas

Recent DIII-D experiments show that sawtooth stability is strongly affected by anisotropic fast ions from neutral beam injection (NBI) in both negative and positive triangularity plasmas. Fast ions from co-current NBI are stabilizing for the sawtooth stability, resulting in longer sawtooth periods. On the other hand, fast ions from counter-current NBI are destabilizing, leading to small and frequent sawteeth. The relative change of sawtooth period and amplitude is more than a factor of two. These observations appear to hold in both plasma shapes. Non-perturbative toroidal modeling, utilizing the magnetohydrodynamic-kinetic hybrid stability code MARS-K (Liu et al 2008 Phys. Plasmas 15 112503), reveals an asymmetric dependence of the stability of the n = 1 (n is the toroidal mode number) internal kink mode on the injection direction of NBI, being qualitatively consistent with the experimentally observed sawtooth behavior. The MARS-K modeling results suggest that anisotropic fast ions affect the mode growth rate and frequency through both adiabatic and non-adiabatic contributions. Here, the asymmetry of the internal kink mode instability relative to the NBI direction is mainly due to the non-adiabatic contribution of passing fast ions, which stabilize (destabilize) the internal kink with the co-(counter-) current NBI as compared to the fluid counterpart. However, finite orbit width (FOW) correction to passing particles partially cancels the asymmetry. Trapped particles are always stabilizing due to precessional drift resonance. Modeling also shows that fast ions affect the internal kink in a similar manner in both negative and positive triangularity plasmas, although being slightly more unstable in the negative triangularity configuration already in the fluid limit. The similarity is mainly attributed to the fact that the mode is localized in the plasma core region, with very similar eigenmode structures in both negative and positive configurations. Furthermore, MARS-K modeling indicates that other factors, such as the plasma rotation and the drift kinetic effects of thermal plasmas, weakly modify the mode stability as compared to the drift kinetic resonance effects and FOW correction of fast ions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Kinetic Simulations of Instabilities and Particle Acceleration in Cylindrical Magnetized Relativistic Jets

Relativistic magnetized jets, such as those from AGN, GRBs, and XRBs, are susceptible to current- and pressure-driven MHD instabilities that can lead to particle acceleration and nonthermal radiation. Here, we investigate the development of these instabilities through 3D kinetic simulations of cylindrically symmetric equilibria involving toroidal magnetic fields with electron–positron pair plasma. Generalizing recent treatments by Alves et al. and Davelaar et al., we consider a range of initial structures in which the force due to toroidal magnetic field is balanced by a combination of forces due to axial magnetic field and gas pressure. We argue that the particle energy limit identified by Alves et al. is due to the finite duration of the fast magnetic dissipation phase. We find a rather minor role of electric fields parallel to the local magnetic fields in particle acceleration. In all investigated cases, a kink mode arises in the central core region with a growth timescale consistent with the predictions of linearized MHD models. In the case of a gas-pressure-balanced (Z-pinch) profile, we identify a weak local pinch mode well outside the jet core. We argue that pressure-driven modes are important for relativistic jets, in regions where sufficient gas pressure is produced by other dissipation mechanisms.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

The Origin of Power-law Spectra in Relativistic Magnetic Reconnection

Abstract Magnetic reconnection is often invoked as a source of high-energy particles, and in relativistic astrophysical systems it is regarded as a prime candidate for powering fast and bright flares. We present a novel analytical model—supported and benchmarked with large-scale three-dimensional kinetic particle-in-cell simulations in electron–positron plasmas—that elucidates the physics governing the generation of power-law energy spectra in relativistic reconnection. Particles with Lorentz factorγ≳ 3σ(here,σis the magnetization) gain most of their energy in the inflow region, while meandering between the two sides of the reconnection layer. Their acceleration time is t acc ∼ γ η rec − 1 ω c − 1 ≃ 20 γ ω c − 1 , whereη rec ≃ 0.06 is the inflow speed in units of the speed of light andω c =eB 0 /mcis the gyrofrequency in the upstream magnetic field. They leave the region of active energization aftert esc , when they get captured by one of the outflowing flux ropes of reconnected plasma. We directly measuret esc in our simulations and find thatt esc ∼t acc forσ≳ few. This leads to a universal (i.e.,σ-independent) power-law spectrum dN free / d γ ∝ γ − 1 for the particles undergoing active acceleration, and dN / d γ ∝ γ − 2 for the overall particle population. Our results help to shed light on the ubiquitous presence of power-law particle and photon spectra in astrophysical nonthermal sources.

Astronomy & Astrophysics↗

DOE OSTI2024

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Boltzmann equation↗