Non-thermal Particle Distributions Research Articles

Advanced fuel fusion, phase space engineering, and structure-preserving geometric algorithms

Non-thermal advanced fuel fusion trades the requirement of a large amount of recirculating tritium in the system for that of large recirculating power. Phase space engineering technologies utilizing externally injected electromagnetic fields can be applied to meet the challenge of maintaining non-thermal particle distributions at a reasonable cost. The physical processes of the phase space engineering are studied from a theoretical and algorithmic perspective. It is emphasized that the operational space of phase space engineering is limited by the underpinning symplectic dynamics of charged particles. The phase space incompressibility according to the Liouville theorem is just one of many constraints, and Gromov's non-squeezing theorem determines the minimum footprint of the charged particles on every conjugate phase space plane. In this sense and level of sophistication, the mathematical abstraction of phase space engineering is symplectic topology. To simulate the processes of phase space engineering, such as the Maxwell demon and electromagnetic energy extraction, and to accurately calculate the minimum footprints of charged particles, recently developed structure-preserving geometric algorithms can be used. The family of algorithms conserves exactly, on discretized spacetime, symplecticity and thus incompressibility, non-squeezability, and symplectic capacities. The algorithms apply to the dynamics of charged particles under the influence of external electromagnetic fields as well as the charged particle–electromagnetic field system governed by the Vlasov–Maxwell equations.

Nonlinear diffusive shock acceleration of cosmic rays: quasi-thermal and non-thermal particle distributions

Nonlinear diffusive shock acceleration of cosmic rays: quasi-thermal and non-thermal particle distributions

Acceleration of cosmic rays in presence of magnetohydrodynamic fluctuations at small scales

ABSTRACT This work investigates the evolution of the distribution of charged particles (cosmic rays) due to the mechanism of stochastic turbulent acceleration (STA) in presence of small-scale turbulence with a mean magnetic field. STA is usually modelled as a biased random walk process in the momentum space of the non-thermal particles. This results in an advection-diffusion type transport equation for the non-thermal particle distribution function. Under quasi-linear approximation, and by assuming turbulent spectra with power being available only in the sub-gyroscale range, we find that the Fokker–Planck diffusion coefficients Dγγ and Dμμ scale with the Lorentz factor γ as Dγγ ∝ γ−2/3 and Dμμ ∝ γ−8/3. We consider Alfvèn and fast waves in our calculations, and find a universal trend for the momentum diffusion coefficient irrespective of the properties of the small-scale turbulence. Such universality has already been reported regarding the spatial diffusion of the cosmic rays, and, here too, we observe a universality in the momentum diffusion coefficient. Furthermore, with the calculated transport coefficients, we numerically solve the advection-diffusion-type transport equation for the non-thermal particles. We demonstrate the interplay of various mircophysical processes such as STA, synchrotron loss, and particle escape on the particle distribution by systematically varying the parameters of the problem. We observe that the effect of the small-scale turbulence is more impactful for the high-energy protons as compared to the electrons and such turbulence is capable of sustaining the energy of the protons from catastrophic radiative loss processes. Such a finding is novel and helps us to enhance our understanding about the hadronic emission processes that are typically considered as a competitor for the leptonic emission for certain astrophysical systems.

Particle acceleration with magnetic reconnection in large-scale RMHD simulations – I. Current sheet identification and characterization

ABSTRACT We present a new algorithm for the identification and physical characterization of current sheets and reconnection sites in 2D and 3D large-scale relativistic magnetohydrodynamic numerical simulations. This has been implemented in the pluto code and tested in the cases of a single current sheet, a 2D jet, and a 3D unstable plasma column. Its main features are (i) a computational cost that allows its use in large-scale simulations and (ii) the capability to deal with complex 2D and 3D structures of the reconnection sites. In the performed simulations, we identify the computational cells that are part of a current sheet by a measure of the gradient of the magnetic field along different directions. Lagrangian particles, which follow the fluid, are used to sample plasma parameters before entering the reconnection sites that form during the evolution of the different configurations considered. Specifically, we track the distributions of the magnetization parameter σ and the thermal to magnetic pressure ratio β that – according to particle-in-cell simulation results – control the properties of particle acceleration in magnetic reconnection regions. Despite the fact that initial conditions of the simulations were not chosen ‘ad hoc’, the 3D simulation returns results suitable for efficient particle acceleration and realistic non-thermal particle distributions.

Prospects for Ray-tracing Light Intensity and Polarization in Models of Accreting Compact Objects Using a GPU

The Event Horizon Telescope (EHT) has recently released high-resolution images of accretion flows onto two supermassive black holes. Our physical understanding of these images depends on the accuracy and precision of numerical models of plasma and radiation around compact objects. The goal of this work is to speed up radiative-transfer simulations used to create mock images of black holes for comparison with the EHT observations. A ray-tracing code for general relativistic and fully polarized radiative transfer through plasma in strong gravity is ported onto a graphics processing unit (GPU). We describe our GPU implementation and carry out speedup tests using models of optically thin advection-dominated accretion flow onto a black hole realized semianalytically and in 3D general relativistic magnetohydrodynamic simulations, low and very high image pixel resolutions, and two different sets of CPU+GPUs. We show that a GPU with high double precision computing capability can significantly reduce the image production computational time, with a speedup factor of up to approximately 1200. The significant speedup facilitates, e.g., dynamic model fitting to the EHT data, including polarimetric data. The method extension may enable studies of emission from plasma with nonthermal particle distribution functions for which accurate approximate synchrotron emissivities are not available. The significant speedup reduces the carbon footprint of the generation of the EHT image libraries by at least an order of magnitude.

Non-thermal particle acceleration from maximum entropy in collisionless plasmas

Dissipative processes cause collisionless plasmas in many systems to develop non-thermal particle distributions with broad power-law tails. The prevalence of power-law energy distributions in space/astrophysical observations and kinetic simulations of systems with a variety of acceleration and trapping (or escape) mechanisms poses a deep mystery. We consider the possibility that such distributions can be modelled from maximum-entropy principles, when accounting for generalizations beyond the Boltzmann–Gibbs entropy. Using a dimensional representation of entropy (related to the Renyi and Tsallis entropies), we derive generalized maximum-entropy distributions with a power-law tail determined by the characteristic energy scale at which irreversible dissipation occurs. By assuming that particles are typically energized by an amount comparable to the free energy (per particle) before equilibrating, we derive a formula for the power-law index as a function of plasma parameters for magnetic dissipation in systems with sufficiently complex topologies. The model reproduces several results from kinetic simulations of relativistic turbulence and magnetic reconnection.

Flare echoes from relaxation shocks in perturbed relativistic jets

Context. One of the main scenarios to account for the multiwavelength flux variability observed in relativistic jets of active galactic nuclei (AGNs) is based on the diffusive shock acceleration of a population of relativistic electrons on internal shocks of various origins. Any complete AGN emission scenario has to be able to explain the wide range of observed variability timescales, which change over several orders of magnitude between the radio and gamma-ray band. In addition to observations of flux variability, constraints are also provided by very-long-baseline interferometry (VLBI), which shows a large variety of moving and standing emission zones with distinct behaviors. Aims. Combining dynamic hydrodynamic jet simulations with radiative transfer, we aim to characterize the evolution of stationary and moving emission zones in the jet and study their multiwavelength signatures through emission maps and light curves. We focus our study on flare events that occur during strong interactions between moving ejecta and stationary recollimation shocks. Such events are shown to lead to a significant perturbation of the stationary jet structure. Methods. We simulate relativistic jets with the magneto-hydrodynamic code MPI-AMRVAC and inject nonthermal particle distributions of electrons into shock regions. We follow the propagation of a moving shock and its interactions with a structure of standing recollimation shocks in the jet. Synchrotron emission and radiative transfer are calculated in the post-processing code RIPTIDE for given observation angles and frequencies, assuming a turbulent magnetic field and taking the light crossing effect into account. Results. In the case of strong shock–shock interactions, we demonstrate the appearance of trailing components behind the leading moving shock. The latter destabilizes the jet, causing the emergence of oscillating standing shocks and relaxation shocks. Emissions from these regions can dominate the overall flux or lead to “flare echoes” in the light curve. Another observational marker for the presence of relaxation shocks appears in time-distance plots of bright VLBI components of the jet. Our scenario provides a plausible explanation for radio VLBI observations of the radio galaxy 3C 111, where trailing components were observed during a radio outburst event in 1997, and may be applicable to other sources with similar features.

Impact of non-thermal particles on the spectral and structural properties of M87

Context. The recent 230 GHz observations of the Event Horizon Telescope are able to image the innermost structure of M 87 and show a ring-like structure that agrees with thermal synchrotron emission generated in a torus surrounding a supermassive black hole. However, at lower frequencies, M 87 is characterised by a large-scale and edge-brightened jet with clear signatures of non-thermal emission. In order to bridge the gap between these scales and to provide a theoretical interpretation of these observations, we perform general relativistic magnetohydrodynamic simulations of accretion onto black holes and jet launching. Aims. M 87 has been the target for multiple observations across the entire electromagnetic spectrum. Among these, very large baseline interferometry (VLBI) observations provide unique details of the collimation profile of the jet down to several gravitational radii. We aim to model the observed broad-band spectrum of M 87 from the radio to the near-IR regime and at the same time, fit the jet structure as observed with global millimeter-VLBI at 86 GHz. Methods. We used general relativistic magnetohydrodynamics and simulated the accretion of the magnetised plasma onto Kerr black holes in 3D. The radiative signatures of these simulations were computed taking different electron distribution functions into account, and a detailed parameter survey was performed in order to match the observations. Results. The results of our simulations show that magnetically arrested disks around fast-spinning black holes (a⋆ ≥ 0.5) together with a mixture of thermal and non-thermal particle distributions are able to simultaneously model the broad-band spectrum and the innermost jet structure of M 87.

A Shock-in-Jet Synchrotron Mirror Model for Blazars

Reinhard Schlickeiser has made groundbreaking contributions to various aspects of blazar physics, including diffusive shock acceleration, the theory of synchrotron radiation, the production of gamma-rays through Compton scattering in various astrophysical sources, etc. This paper, describing the development of a self-consistent shock-in-jet model for blazars with a synchrotron mirror feature, is therefore an appropriate contribution to a Special Issue in honor of Reinhard Schlickeiser’s 70th birthday. The model is based on our previous development of a self-consistent shock-in-jet model with relativistic thermal and non-thermal particle distributions evaluated via Monte-Carlo simulations of diffusive shock acceleration, and time-dependent radiative transport. This model has been very successful in modeling spectral variability patterns of several blazars, but has difficulties describing orphan flares, i.e., high-energy flares without a significant counterpart in the low-frequency (synchrotron) radiation component. As a solution, this paper investigates the possibility of a synchrotron mirror component within the shock-in-jet model. It is demonstrated that orphan flares result naturally in this scenario. The model’s applicability to a recently observed orphan gamma-ray flare in the blazar 3C279 is discussed and it is found that only orphan flares with mild (≲ a factor of 2–3) enhancements of the Compton dominance can be reproduced in a synchrotron-mirror scenario, if no additional parameter changes are invoked.

Nonlinear particle trapping by coherent waves in thermal and nonthermal plasmas

We perform a test particle simulation to study the resonant interaction of electrons with a stationary electrostatic wave. In the simulation, we use a new method to generate nonthermal particle distributions efficiently. This method utilizes inverse cumulative distribution function, and compared with the existing methods in the literature, found to have a negligible deviation from the analytical form of the distribution, particularly in the tail side, where nonthermal particles are situated. The simulation revealed that the trapped particle density varies highly with the width (≈10%–60%) and amplitude (≈13%–19%) of the wave potential. The particle trapping is found to be more efficient in nonthermal plasmas than thermal plasma. It implies that particle transport through waves could be more effective in nonthermal plasmas than thermal plasmas. In addition, the simulation sheds light on the particle trapping time and its relation to the characteristics of the wave potential. Overall, the present study will be helpful to understand the particle trapping by the coherent waves, which are frequently seen in space and astrophysical plasmas.

Particle acceleration and non-thermal emission in colliding-wind binary systems

ABSTRACT We present a model for the creation of non-thermal particles via diffusive shock acceleration in a colliding-wind binary. Our model accounts for the oblique nature of the global shocks bounding the wind–wind collision region and the finite velocity of the scattering centres to the gas. It also includes magnetic field amplification by the cosmic ray induced streaming instability and the dynamical back reaction of the amplified field. We assume that the injection of the ions and electrons is independent of the shock obliquity and that the scattering centres move relative to the fluid at the Alfvén velocity (resulting in steeper non-thermal particle distributions). We find that the Mach number, Alfvénic Mach number, and transverse field strength vary strongly along and between the shocks, resulting in significant and non-linear variations in the particle acceleration efficiency and shock nature (turbulent versus non-turbulent). We find much reduced compression ratios at the oblique shocks in most of our models compared to our earlier work, though total gas compression ratios that exceed 20 can still be obtained in certain situations. We also investigate the dependence of the non-thermal emission on the stellar separation and determine when emission from secondary electrons becomes important. We finish by applying our model to WR 146, one of the brightest colliding wind binaries in the radio band. We are able to match the observed radio emission and find that roughly 30 per cent of the wind power at the shocks is channelled into non-thermal particles.

Colliding-wind binary systems: diffusive shock acceleration and non-thermal emission

ABSTRACT We present a model for the non-thermal emission from a colliding-wind binary. Relativistic protons and electrons are assumed to be accelerated through diffusive shock acceleration (DSA) at the global shocks bounding the wind–wind collision region. The non-linear effects of the backreaction due to the cosmic ray pressure on the particle acceleration process and the cooling of the non-thermal particles as they flow downstream from the shocks are included. We explore how the non-thermal particle distribution and the keV−GeV emission changes with the stellar separation and the viewing angle of the system, and with the momentum ratio of the winds. We confirm earlier findings that DSA is very efficient when magnetic field amplification is not included, leading to significantly modified shocks. We also find that the non-thermal flux scales with the binary separation in a complicated way and that the anisotropic inverse Compton emission shows only a moderate variation with viewing angle due to the spatial extent of the wind–wind collision.

Detection of ultra-high-energy gamma rays from the Crab Nebula: physical implications

ABSTRACT The Crab Nebula is an extreme particle accelerator that boosts the energy of electrons up to a few PeV ($10^{15} \ \rm eV$), close to the maximum energy allowed theoretically. The physical conditions in the acceleration site and the nature of the acceleration process itself remain highly uncertain. The key information about the highest-energy accelerated particles is contained in the synchrotron and inverse Compton (IC) channels of radiation at energies above 1 MeV and 100 TeV, respectively. A recent report of the detection of an ultra-high-energy gamma-ray signal from the Crab Nebula up to 300 TeV allows us to determine the energy distribution of the highest-energy electrons and to derive the magnetic field strength in the acceleration region, $B\le 120\rm \, \mu G$, in a parameter-free way. This estimate brings new constraints on the properties of non-thermal particle distributions and places important constraints on the magnetohydrodynamic models for the Crab Nebula, in particular on the feasible magnetization and anisotropy of the pulsar wind. The calculations of synchrotron and IC emission show that future observations with instruments that allow detection of the Crab Nebula above 300 TeV and above 1 MeV will clarify the conditions that allow acceleration of electrons beyond PeV energies in the Crab Nebula. In particular, we will be able to verify the hypothetical multicomponent composition of the electron energy distribution, and we will determine the magnetic field strength in the regions responsible for the acceleration of PeV electrons.

System-size Convergence of Nonthermal Particle Acceleration in Relativistic Plasma Turbulence

Abstract We apply collisionless particle-in-cell simulations of relativistic pair plasmas to explore whether driven turbulence is a viable high-energy astrophysical particle accelerator. We characterize nonthermal particle distributions for varying system sizes up to L/2πρ e0 = 163, where L/2π is the driving scale and ρ e0 is the initial characteristic Larmor radius. We show that turbulent particle acceleration produces power-law energy distributions that, when compared at a fixed number of large-scale dynamical times, slowly steepen with increasing system size. We demonstrate, however, that convergence is obtained by comparing the distributions at different times that increase with system size (approximately logarithmically). We suggest that the system-size dependence arises from the time required for particles to reach the highest accessible energies via Fermi acceleration. The converged power-law index of the energy distribution, α ≈ 3.0 for magnetization σ = 3/8, makes turbulence a possible explanation for nonthermal spectra observed in systems such as the Crab Nebula.

High-energy Emission from Nonrelativistic Radiative Shocks: Application to Gamma-Ray Novae

Abstract The observation of GeV gamma-rays from novae by Fermi/LAT demonstrates that the nonrelativistic radiative shocks in these systems can accelerate particles to energies of at least ∼10 GeV. The low-energy extension of the same nonthermal particle distribution inevitably gives rise to emission in the hard X-ray band. Above , this radiation can escape the system without significant absorption/attenuation, and can potentially be detected by NuSTAR. We present theoretical models for hard X-ray and gamma-ray emission from radiative shocks in both leptonic and hadronic scenarios, accounting for the rapid evolution of the downstream properties due to the fast cooling of thermal plasma. We find that due to strong Coulomb losses, only a fraction of of the gamma-ray luminosity is radiated in the NuSTAR band; nevertheless, this emission could be detectable simultaneously with the LAT emission in bright gamma-ray novae with a ∼50 ks exposure. The spectral slope in hard X-rays is for typical nova parameters, thus serving as a testable prediction of the model. Our work demonstrates how combined hard X-ray and gamma-ray observations can be used to constrain properties of the nova outflow (velocity, density, and mass outflow rate) and particle acceleration at the shock. A very low X-ray to gamma-ray luminosity ratio ( ) would disfavor leptonic models for the gamma-ray emission. Our model can also be applied to other astrophysical environments with radiative shocks, including SNe IIn and colliding winds in massive star binaries.

Erratum: Reconnection and particle acceleration in interacting flux ropes – II. 3D effects on test particles in magnetically dominated plasmas

We analyze particle acceleration in explosive reconnection events in magnetically dominated proton-electron plasmas. Reconnection is driven by large-scale magnetic stresses in interacting current-carrying flux tubes. Our model relies on development of current-driven instabilities on macroscopic scales. These tilt-kink instabilities develop in an initially force-free equilibrium of repelling current channels. Using MHD methods we study a 3D model of repelling and interacting flux tubes in which we simultaneously evolve test particles, guided by electromagnetic fields obtained from MHD. We identify two stages of particle acceleration; Initially particles accelerate in the current channels, after which the flux ropes start tilting and kinking and particles accelerate due to reconnection processes in the plasma. The explosive stage of reconnection produces non-thermal energy distributions with slopes that depend on plasma resistivity and the initial particle velocity. We also discuss the influence of the length of the flux ropes on particle acceleration and energy distributions. This study extends previous 2.5D results to 3D setups, providing all ingredients needed to model realistic scenarios like solar flares, black hole flares and particle acceleration in pulsar wind nebulae: formation of strong resistive electric fields, explosive reconnection and non-thermal particle distributions. By assuming initial energy equipartition between electrons and protons, applying low resistivity in accordance with solar corona conditions and limiting the flux rope length to a fraction of a solar radius we obtain realistic energy distributions for solar flares with non-thermal power law tails and maximum electron energies up to 11 MeV and maximum proton energies up to 1 GeV.

Reconnection and particle acceleration in interacting flux ropes – II. 3D effects on test particles in magnetically dominated plasmas

We analyze particle acceleration in explosive reconnection events in magnetically dominated proton-electron plasmas. Reconnection is driven by large-scale magnetic stresses in interacting current-carrying flux tubes. Our model relies on development of current-driven instabilities on macroscopic scales. These tilt-kink instabilities develop in an initially force-free equilibrium of repelling current channels. Using MHD methods we study a 3D model of repelling and interacting flux tubes in which we simultaneously evolve test particles, guided by electromagnetic fields obtained from MHD. We identify two stages of particle acceleration; Initially particles accelerate in the current channels, after which the flux ropes start tilting and kinking and particles accelerate due to reconnection processes in the plasma. The explosive stage of reconnection produces non-thermal energy distributions with slopes that depend on plasma resistivity and the initial particle velocity. We also discuss the influence of the length of the flux ropes on particle acceleration and energy distributions. This study extends previous 2.5D results to 3D setups, providing all ingredients needed to model realistic scenarios like solar flares, black hole flares and particle acceleration in pulsar wind nebulae: formation of strong resistive electric fields, explosive reconnection and non-thermal particle distributions. By assuming initial energy equipartition between electrons and protons, applying low resistivity in accordance with solar corona conditions and limiting the flux rope length to a fraction of a solar radius we obtain realistic energy distributions for solar flares with non-thermal power law tails and maximum electron energies up to 11 MeV and maximum proton energies up to 1 GeV.

Reconnection and particle acceleration in interacting flux ropes I. Magnetohydrodynamics and test particles in 2.5D

Magnetic reconnection and non-thermal particle distributions associated with current-driven instabilities are investigated by means of resistive magnetohydrodynamics (MHD) simulations combined with relativistic test particle methods. We propose a system with two parallel, repelling current channels in an initially force-free equilibrium, as a simplified representation of flux ropes in a stellar magnetosphere. The current channels undergo a rotation and separation on Alfv\'enic timescales, forming secondary islands and (up to tearing unstable) current sheets in which non-thermal energy distributions are expected to develop. Using the recently developed particle module of our open-source grid-adaptive MPI-AMRVAC software, we simulate MHD evolution combined with test particle treatments in MHD snapshots. We explore under which plasma-$\beta$ conditions the fastest reconnection occurs in two-and-a-half dimensional (2.5D) scenarios and in these settings test particles are evolved. We quantify energy distributions, acceleration mechanisms, relativistic corrections to the particle equations of motion and effects of resistivity in magnetically dominated proton-electron plasmas. Due to large resistive electric fields and indefinite acceleration of particles in the infinitely long current channels, hard energy spectra are found in 2.5D configurations. Solutions to these numerical artifacts are proposed for both 2.5D setups and future 3D work. We discuss the magnetohydrodynamics of an additional kink instability in 3D setups and the expected effects on energy distributions. The obtained results hold as a proof-of-principle for test particle approaches in MHD simulations, relevant to explore less idealised scenarios like solar flares and more exotic astrophysical phenomena, like black hole flares, magnetar magnetospheres and pulsar wind nebulae.

A comparative study of the filamentation and two-stream instabilities in weakly relativistic counter-streaming plasmas

Counter-streaming plasma structures are ubiquitous in astrophysical sources of non-thermal radiations. We discuss the dispersion properties and the stability of this non-thermal particle distribution, which is modeled on the basis of the relativistic Jüttner-Maxwell distribution function in the correct laboratory frame of reference. In this work, we aim to construct analytical solutions of the dispersion relations and investigate the properties of the growth rate of the filamentation and two-stream instabilities in an unmagnetized and homogeneous counter-propagating plasma. The Maxwell and the relativistic Vlasov equations are used to derive the covariant dispersion relations that are valid in any (conveniently chosen) reference frame. Aperiodic solutions (ℜ(ω)≃0) to the covariant dispersion relations of the growing modes (ℑ(ω)>0) are demonstrated with the aid of analytical calculations. The dependence of the growth rate on the normalized bulk velocity β0=V0/c and thermal parameter μ=mc2/KBT is shown in graphic illustrations. We found that for both kinds of instabilities, growth rates are decreased by increasing the temperature and decreasing the bulk velocity. Therefore, the electrons at sufficiently low temperatures and with relativistic streams are capable of increasing the range of unstable wave numbers and consequently prevent the instability to cease at small wave numbers. The results indicate that under the same condition and in contrast to the non-relativistic regime, the filamentation instability has the largest growth rate and the electrostatic two-stream instability is in the next place.

A time dependent approach to model X-ray and γ–ray light curves of Mrk 421 observed during the flare in February 2010

Strong X-ray and γ–ray flares have been detected in February 2010 from the high synchrotron peaked blazar Mrk 421 (z = 0.031). With the motivation of understanding the physics involved in this flaring activity, we study the variability of the source in X-ray and γ–ray energy bands during the period February 10–23, 2010 (MJD 55237–55250). We use near simultaneous X-ray data collected by MAXI, Swift-XRT and γ–ray data collected by Fermi-LAT and TACTIC along with the optical V-band observations by SPOLat Steward Observatory. We observe that the variation in the one day averaged flux from the source during the flare is characterized by fast rise and slow decay. Besides, the TeV γ–ray flux shows a strong correlation with the X-ray flux, suggesting the former to be an outcome of synchrotron self Compton emission process. To model the observed X-ray and γ–ray light curves, we numerically solve the kinetic equation describing the evolution of particle distribution in the emission region. The injection of particle distribution into the emission region, from the putative acceleration region, is assumed to be a time dependent power law. The synchrotron and synchrotron self Compton emission from the evolving particle distribution in the emission region are used to reproduce the X-ray and γ–ray flares successfully. Our study suggests that the flaring activity of Mrk 421 can be an outcome of an efficient acceleration process associated with the increase in underlying non-thermal particle distribution.