# Inverse Compton Emission and Cooling of Relativistic Particles Accelerated
at Shear Boundary Layers in Relativistic Jets

Tej Chand Centre for Space Research
North-West University
Potchefstroom, 2520, South Africa  Markus Böttcher Centre for Space Research
North-West University
Potchefstroom, 2520, South Africa

###### Abstract

Both observational evidence and theoretical considerations from MHD
simulations of jets suggest that the relativistic jets of active galactic
nuclei (AGN) are radially stratified, with a fast inner spine surrounded by a
slower-moving outer sheath. The resulting relativistic shear layers are a
prime candidate for the site of relativistic particle acceleration in the jets
of AGN and gamma-ray bursts (GRBs). In this article, we present outcomes of
particle-in-cell simulations of magnetic-field generation and particle
acceleration in the relativistic shear boundary layers (SBLs) of jets in AGN
and GRBs. We investigate the effects of inverse Compton cooling on
relativistic particles that are accelerated in the SBLs of relativistic jets
including the self-consistent calculation of the radiation spectrum produced
by inverse Compton scattering of relativistic electrons in an isotropic
external soft photon field. We find that the Compton cooling can be
substantial, depending on the characteristic energy (blackbody temperature and
energy density) of the external radiation field. The produced Compton emission
is highly anisotropic and more strongly beamed along the jet direction than
the characteristic $1/\Gamma$ pattern expected from intrinsically isotropic
emission in the comoving frame of an emission region moving along the jet with
bulk Lorentz factor $\Gamma$. We suggest that this may resolve the long-
standing problem of the Doppler Factor Crisis.

relativistic jets: shear boundary layers, shear acceleration, radiation drag,
radiation mechanism

##  1 Introduction

The existence of jets associated with accretion onto compact objects is
ubiquitous in astrophysical systems. Those collimated relativistic outflows
emanating from a variety of astrophysical sources like a stellar mass compact
object, such as a white dwarf (WD), a neutron star (NS), or a black hole (BH),
in close binaries, as well as jets from gamma-ray bursts (GRBs) can travel
over pc scales whereas those from some active galactic nuclei (AGNs) extend
over kpc scales without showing significant loss of momentum and kinetic
energy. Many processes in and properties of relativistic jets, like jet
composition, collimation, particle acceleration, stabilization, mass-loading
mechanism, and radiative mechanisms, are still poorly understood. A recent
review on relativistic jets can be found in, e.g., Romero et al. (2017).

Both observational evidence and theoretical considerations from Magneto-
Hydrodynamic (MHD) simulations indicate that astrophysical jets have radial or
transverse structure (e.g., Hardee, 2007; McKinney, 2006; Sol et al., 1989;
Giroletti et al., 2004). When the powerful jet from the central engine travels
through the interstellar plasma, sharp boundary layers can be formed due to
the velocity difference between the jet and the surrounding plasma. Also, the
jet may accelerate to different intrinsic bulk speeds at different radial
distances from the jet axis. The ensuing shear boundary layers (SBLs) are
promising sites for the particle acceleration (e.g., Stawarz & Ostrowski,
2002; Sahayanathan, 2009; Liang et al., 2013a; Chand et al., 2019). The
formation of SBLs is expected in a variety of astrophysical environments like
AGN jets (e.g., Alves et al., 2012; Liang et al., 2013a; Rieger, 2019) and
also in the jets involving ultra-relativistic outflows of GRBs (e.g., Piran,
2000). The majority of such astrophysical systems are collisionless, wherein
the electron mean free path is much larger than the system size, requiring
Particle-in-Cell (PiC) methods to study the kinetic effects. The angular
variations in the jet Lorentz factor indicated from the recent observations of
the afterglow of the neutron star-neutron star merger GW170817 (Abbott et al.,
2017) also support the picture of a structured jet in which the Lorentz factor
of the jet and its energetics vary with the angle from the central axis (e.g.,
Lamb & Kobayashi, 2017; Coughlin & Begelman, 2020). Ghisellini et al. (2005)
proposed a spine-layer structure of blazar and radio galaxy jets in order to
interpret the longstanding issue of the Bulk Lorentz Factor Crisis (BLFC) in
some blazar jets (e.g., Henri & Sauge, 2006; Lyutikov & Lister, 2010a). Recent
studies of SBLs using Particle-in-Cell (Birdsall, 1991) simulations of
relativistic jets have demonstrated that from initially unmagnetized plasma
electric and magnetic fields can be efficiently generated leading to particle
acceleration (e.g., Alves et al., 2012; Liang et al., 2013b; Rieger, 2019).
Recent studies (e.g., Grismayer et al., 2013a, b; Alves et al., 2014) have
provided theoretical predictions and a physical picture of plasma
instabilities like Kelvin-Helmholtz instability, KHI (Chandrasekhar, 1961), or
the Weibel instability (Weibel, 1959), that can develop within shear layers.
These studies suggest that the development of magnetic fields in SBLs is due
to those plasma instabilities. Although the shear interface, in the
hydrodynamic limit, is unstable in the case of the classical KHI, the
instability is suppressed by the longitudinal B-field in the presence of
ambient magnetic fields (Chandrasekhar, 1961). Thus, the shear interface in
the kpc jets of AGN is stable against the KHI. The impact of the transverse
dynamics of the KHI in unmagnetized shearing flow, which is not accounted for
in 2D simulation, has recently been explored through the use of 3D PiC
simulations (e.g., Alves et al., 2012). While the longitudinal dynamics of the
KHI hold significance in both subrelativistic and relativistic regimes, the
transverse dynamics become particularly notable in the relativistic regime, in
contrast to the subrelativistic regime where longitudinal dynamics dominate.
In relativistic regimes, the complete 3D evolution of the KHI is governed by
transverse dynamics, characterized by the aggregation of a Weibel-like
electron bunching process. This results in the creation of electron current
filaments, ultimately driving the acceleration of electrons across the shear
interface.

The most widely accepted acceleration mechanism in the scenario of
relativistic jets is the first-order Fermi acceleration mechanism.
Considerable attention is given to shock acceleration as a particle
acceleration mechanism in relativistic jets. However, the Fermi process is
also possible without a shock where scattering centres flow at different
speeds despite the fact that the flows are parallel. The particles attain
different velocities between the fast-moving spine and the slow-moving layer.
This mechanism, called shear acceleration in shear flows, occurs as a result
of energetic particles encountering different local velocities in the
collisionless background flow (e.g., Rieger & Duffy, 2004). Recent high-
resolution studies of extragalactic jets indicate that the first-order Fermi
mechanism alone cannot fully reckon the detection of extended high energy
emission (e.g., Rieger & Duffy, 2006). The strong synchrotron cooling process
which is expected in the shock acceleration scenarios is not observed, e.g.,
in the case of the quasar 3C 273 (e.g., Jester et al., 2001), suggesting that
there is the need for a continuous re-acceleration mechanism in the
relativistic jets to resolve these issues. Thus, shear layer acceleration is
likely to be significant in high-energy astrophysical phenomena. The energetic
particles may sample the velocity difference while moving across the shear
flow. Magnetic field inhomogeneities embedded in different layers of the shear
flow scatter energetic particles, leading to energy gain due to bulk velocity
differences across the shear flow (e.g., Liu et al., 2017). This eventually
converts the bulk motion kinetic energy of the background flows to nonthermal
particle energies. Previous studies have indicated that the efficiency of the
shear acceleration depends on the strength of the velocity shear and hence the
shear acceleration is employed more efficiently in relativistic jets than in
non-relativistic ones (e.g., Webb, 1989; Ostrowski, 1990).

The spectral energy distributions (SEDs) of blazars consist typically of two
broad, non-thermal components. The lower energy bump, attributed to the
synchrotron emission from relativistic leptons, is observed in the radio
through optical/UV bands and in some cases up to X-rays. In leptonic models,
the higher energy bump is attributed to the Inverse Compton (IC) scattering by
the same leptons of the synchrotron photons (SSC) or External photons (EC).
High energy particles energized at SBLs across magnetic field lines have an
anisotropic momentum distribution with efficient emission of Synchrotron
radiation (e.g., Liang et al., 2013a). Several investigations employing PiC
simulations have examined the influence of synchrotron cooling (e.g., Hakobyan
et al., 2019) and IC cooling (e.g., Werner et al., 2018) on relativistic jet
particles within the framework of magnetic reconnection, including the
resulting radiation spectra. However, despite the potential for radiative
cooling to significantly influence particle dynamics, the effect of radiation
drag on radiating particles and the resulting radiation spectra largely remain
unexplored in studies of particle acceleration at SBLs in jets.

In this study, we investigate the acceleration mechanism and radiation
properties of relativistic jets’ SBLs. We demonstrate that SBL turbulence
generated by self-generated electromagnetic fields produces high-energy
particles. Furthermore, we calculate the radiation spectra by considering the
IC scattering of relativistic electrons in an angle-averaged and angle-
dependent soft photon field, including the impact of IC cooling on particle
dynamics and energetics. In section 2, the physical and simulation setup are
discussed with the initial simulation parameters. In section 3, we discuss the
theoretical background of an IC scattering of an external blackbody photon
field and how we evaluate the radiation spectra. In section 4, we present our
PiC simulation results of self-generated electric and magnetic fields,
particle spectra, and radiative output. In section 5, we summarize the main
results.

##  2 Model Setup

In this section, we present the physical and simulation setups used for our
study of particle acceleration and inverse Compton emission in spine-sheath
plasma jet structures of relativistic jets.

###  2.1 Physical Model

In our simulations, we examine the mechanisms behind particle acceleration and
radiation in SBLs of relativistic jets using a spine-sheath plasma jet
structure. The jet consists of two parts moving in opposite directions along
the $\rm x$-axis with the same velocity and Lorentz factor of $\Gamma=15$. The
spine moves in the positive $\rm x$-direction while the sheath moves in the
negative $\rm x$-direction. The plasma in this simulation is initially
unmagnetized ($\rm B=0$) and has a temperature of $\mathrm{2.5~{}keV}$. Table
1 contains a list of the physical parameters that we use in our simulations.

Table 1: Table of physical parameters Physical parameters | Values
---|---
Initial magnetic field (B) | 0
Ion temperature ($\mathrm{K_{B}T_{i}}$) | 2.5 keV
Electron temperature ($\mathrm{K_{B}T_{e}}$) | 2.5 keV
Bulk Lorentz factor of the spine ($\Gamma_{\rm sp}$) | 15
Bulk Lorentz factor of the sheath ($\Gamma_{\rm sh}$) | 15

###  2.2 Simulation Setup

To study the particle acceleration and radiation mechanism in SBLs of
relativistic jets, we perform fully kinetic 2.5D electromagnetic Particle-in-
Cell (PiC) simulations using the TRISTAN-MP code (Spitkovsky, 2005). The
rectangular computational domain we consider in the $\rm XY$ plane has
periodic boundaries with a total of $\mathrm{L_{x}=1024}$ grid points in the
$\rm x$-direction and $\mathrm{L_{y}=2L_{x}}$ in the $\rm y$-direction. All
distances are measured in units of electron skin depth,
$\mathrm{d_{e}=c/\omega_{p,e}}$, where $\rm c$ is the speed of light and
$\mathrm{\omega_{p,e}=\sqrt{4\pi ne^{2}/m_{e}}}$ is the electron plasma
frequency ($\mathrm{n,e}$, and $\mathrm{m_{e}}$ represent electron number
density, elementary charge, and electron mass, respectively). The simulation
involves 20 particles per cell (PPC) per species with a reduced proton-to-
electron mass ratio of $\mathrm{m_{p}/m_{e}=16}$, and the simulation time is
measured in terms of $\mathrm{1/\omega_{p,e}}$. In PiC simulations of
relativistic plasma, the use of a reduced $\mathrm{m_{p}/m_{e}}$ is a common
practice to enhance computational efficiency and numerical stability. The
large mass disparity between electrons and protons can lead to significant
timestep discrepancies, where electrons require much smaller time steps than
protons to accurately capture their dynamics. If we need to take the ion
contribution into account, we should adjust the plasma frequency as
$\mathrm{\omega_{p,e}^{2}\rightarrow\omega_{p,e}^{2}(1+m_{e}/m_{p})}$.
However, this modification is practically insignificant when dealing with the
real $\mathrm{m_{p}/m_{e}}$ (e.g., Grismayer et al., 2013b). As we aim to
study the underlying physics at electron scales and conduct analyses of
simulation results covering time scales relevant for protons, utilizing the
actual mass ratio is beyond our computational capabilities. Consequently, we
opt for a reduced proton-to-electron mass ratio. PiC simulations have been
employed to study magnetic field generation and particle acceleration within
electron-proton plasmas in relativistic shear flows, considering both actual
(e.g., Alves et al., 2014) and reduced (e.g., Grismayer et al., 2013a) mass
ratios. We set the speed of light as $\mathrm{c=0.45\Delta x/\Delta t}$, where
$\mathrm{\Delta x}$ and $\mathrm{\Delta t}$ are the spatial and temporal
resolutions, respectively. A correction factor for the speed of light is
introduced, which causes electromagnetic waves to travel a little bit faster
than the maximum speed of the particles. This helps ultra-relativistic flows
to avoid the numerical Cerenkov instability, enabling stable and accurate
simulations (e.g., Spitkovsky, 2007; Sironi & Spitkovsky, 2009).

The simulations are carried out in the equal Lorentz factor frame of reference
(ELF) with an initial bulk Lorentz factor of $\Gamma_{\rm sp}=\Gamma_{\rm
sh}=15$. The right-moving spine plasma occupies the central 50% of the
numerical Y-grid, while the left-moving sheath plasma occupies the lower and
upper quarters of the Y-grid. The shear interfaces are located at
$\mathrm{Y=512}$ and $\mathrm{Y=1536}$ of the simulation box. Table 2
summarizes the simulation parameters and their respective values used in this
study.

Figure 1: 2D simulation setup involving an electron-ion plasma with an initially unmagnetized shear flow: The plasma is composed of a central region in the Y-grid where right-moving plasma (spine) is located, where the top 25% and bottom 25% of the Y-grid are occupied by left-moving plasmas (sheath). The spine and sheath move in opposite directions with equal and opposite x-momenta $\mathrm{p_{x}/mc=\pm 15}$ in the ELF. Table 2: Table of simulation parameters Simulation parameters | Values
---|---
PPC | 20
Plasma skin depth ($\mathrm{d_{e}}$) | 10 $\mathrm{\Delta x}$
$\mathrm{m_{i}/m_{e}}$ | 16
speed of light (c) | 0.45 $\mathrm{\Delta x/\Delta t}$
Correction factor for c | 1.025
$\mathrm{L_{x}}$ | 1024 $\mathrm{\Delta x}$
$\mathrm{L_{y}}$ | 2048 $\mathrm{\Delta x}$
Plasma density ($\mathrm{n_{0}}$) | PPC $\mathrm{\Delta x^{-3}}$
Plasma frequency ($\omega_{p,e}$) | 0.045$\mathrm{\Delta t^{-1}}$
Spatial resolution ($\mathrm{\Delta x}$) | 0.1 $\mathrm{d_{e}}$
Temporal resolution ($\mathrm{\Delta t}$) | $0.045\mathrm{\omega_{p,e}^{-1}}$

##  3 Inverse Compton scattering of External Blackbody Photons: Theory

In this section, we give a theoretical outline of our paper in which we derive
the equations describing the inverse-Compton scattering of external photons.

###  3.1 Angle-averaged Inverse-Compton Emission in the Thomson Regime

We consider inverse Compton cooling of accelerated electrons at SBLs in a
thermal blackbody external photon field of varying temperatures with a
characteristic frequency of $\mathrm{h\nu=2.7K_{B}T}$, where $\rm K_{B}$ and T
are Boltzmann constant and radiation temperature, respectively. We first
consider inverse-Compton cooling in the Thomson regime in which the scattering
cross section is independent of the seed photon energy and the scattered
photons are beamed along the direction of motion of the electrons. However, we
use the full Klein-Nishina cross-section later while evaluating the angle-
dependent radiation spectra (see section 3.2). The differential Compton cross-
section for a relativistic electron (e.g., Dermer & Menon, 2009; Boettcher et
al., 2012) is approximated using a hard cut-off at the transition to the
Klein-Nishina regime:

$\displaystyle\begin{split}\frac{d\sigma}{d\Omega
d\epsilon_{s}}&=\sigma_{T}\delta(\epsilon_{s}-\gamma^{2}\epsilon_{0})\delta(\Omega_{s}-\Omega_{e})~{}H(1-\epsilon_{0}\gamma)\end{split}$
(1)

The inverse-Compton emissivity of a single electron in the Thomson limit can
be expressed as

$\displaystyle\begin{split}\operatorname{j_{\epsilon_{0}}^{head-
on}(\gamma,\theta,\epsilon_{0})}&=\sigma_{T}m_{e}c^{3}\epsilon_{s}\int_{0}^{\infty}n_{ph}(\epsilon_{0},\theta)\\\
&\hskip
10.00002pt\delta(\epsilon_{s}-\epsilon_{0}\gamma^{2})\delta(\Omega_{s}-\Omega_{e})~{}d\epsilon_{0},\end{split}$
(2)

where $\mathrm{\sigma_{T}}$ is Thomson cross section,
$\mathrm{\epsilon_{0}=h\nu/m_{e}c^{2}}$ and
${\rm\epsilon_{s}=h\rm\nu_{s}/\rm{m_{e}c^{2}}}$ refer to the initial and
scattered photon energies, $\mathrm{\nu}$ and $\mathrm{\nu_{s}}$ being photon
frequencies before and after the scattering. $\mathrm{\gamma}$ is the Lorentz
factor of the relativistic electron scattering the soft target photons. The
term $\mathrm{\delta(\Omega_{s}-\Omega_{e})}$ specifies that the scattered
photon travels in the direction of the incoming electron. The Heaviside
function $\rm{H}$ cuts the cross-section off at the transitions to the Klein-
Nishina regime. $\mathrm{\theta=K_{B}T/m_{e}c^{2}}$ is the temperature of soft
photon field normalized by electron rest mass energy, where c and
$\mathrm{m_{e}}$ refer to the speed of light and electronic mass respectively.
The density of the soft photon field $\mathrm{n_{ph}}$ is given by

$n_{ph}(\epsilon_{0})=K\frac{\epsilon_{0}^{2}}{\exp(\frac{\epsilon_{0}}{\theta})-1}$
(3)

where, $\mathrm{K=8\pi/\lambda_{C}^{3}}$ and $\mathrm{\lambda_{C}}$ is Compton
wavelength. Equation 2, when using a Dirac-delta approximation for the
scattered photon energy, results in

$\displaystyle\operatorname{j_{\epsilon_{s}}^{head-on}(\gamma,\theta)}$
$\displaystyle=\sigma_{T}m_{e}c^{3}K\delta(\Omega_{s}-\Omega_{e})\frac{1}{\gamma^{6}}\frac{\epsilon_{s}^{3}}{\exp(\frac{\epsilon_{s}}{\gamma^{2}\theta})-1}.$
(4)

The next step involves calculating the rate at which energy is lost by the
electron as a result of inverse Compton cooling. To calculate this, we
integrate Equation 4 over the range of scattered photon energy
($\epsilon_{s}$). This integration yields the standard expression for the
inverse Compton cooling rate of high-energy electrons in the Thomson regime
within the sheath rest frame, where the radiation field is isotropic.

$\displaystyle\frac{d\gamma}{dt}$
$\displaystyle=\frac{\pi^{4}}{15}\sigma_{T}cK\gamma^{2}\theta^{4}$ (5)

The radiation cooling term for inverse Compton scattering of relativistic
electrons in the angle-integrated blackbody photon field in the Thomson regime
in the spine frame is

$\displaystyle\frac{d\gamma}{dt}$
$\displaystyle=\Gamma_{rel}^{2}\frac{\pi^{4}}{15}\sigma_{T}cK\gamma^{2}\theta^{4},$
(6)

where $\mathrm{\Gamma_{rel}=2\Gamma^{2}+1}$ is the relative bulk Lorentz
factor between the spine and the sheath.

###  3.2 Angle-dependent Radiation Spectra and Electron Cooling Rates

In the evaluation of angle-dependent radiation spectra, we adopt a head-on
approximation for the Compton cross section (e.g., Dermer & Menon, 2009;
Boettcher et al., 2012), where all photons travel in the direction of the
incoming electron. This allows us to simplify the Compton cross-section as
follows:

$\frac{d\sigma}{d\Omega_{s}d\epsilon_{s}}=\delta(\Omega_{s}-\Omega_{e})\frac{d\sigma}{d\epsilon_{s}}$
(7)

The differential Compton cross section can be integrated over $\Omega_{s}$ to
get

$\displaystyle\frac{d\sigma_{C}}{d\epsilon_{s}}$ $\displaystyle=\frac{\pi
r_{e}^{2}}{\gamma\epsilon^{{}^{\prime}}}~{}\bigg{(}y+\frac{1}{y}-\frac{2\epsilon_{s}}{\gamma\epsilon^{{}^{\prime}}y}+\big{(}\frac{\epsilon_{s}}{\gamma\epsilon^{{}^{\prime}}y}\big{)}^{2}\bigg{)}$
(8) $\displaystyle\quad
H\big{(}\epsilon_{s}~{};~{}\frac{\epsilon^{{}^{\prime}}}{2\gamma}~{},~{}\frac{2\gamma\epsilon^{{}^{\prime}}}{1+2\epsilon^{{}^{\prime}}}\big{)},$

where $\epsilon^{{}^{\prime}}=\gamma\epsilon_{0}(1-\beta\cos\psi)$ is the
photon energy in electron rest frame, and $y=1-(\epsilon_{s}/\gamma)$. The
emissivity of the electron is obtained by using the Compton cross section in
equation (8) as:

$\displaystyle\operatorname{j_{C}^{head-
on}(\epsilon_{s},\epsilon^{{}^{\prime}},\gamma)}$
$\displaystyle=\frac{3m_{e}c^{3}\sigma_{T}\epsilon_{s}(1-\beta\mu)}{8\gamma}\int_{0}^{\infty}\frac{1}{\epsilon^{{}^{\prime}}}\bigg{\\{}y+\frac{1}{y}$
(9)
$\displaystyle\quad-\frac{2\epsilon_{s}}{\gamma\epsilon^{{}^{\prime}}y}+\bigg{(}\frac{\epsilon_{s}}{\gamma\epsilon^{{}^{\prime}}y}\bigg{)}^{2}\bigg{\\}}~{}n_{ph}(\epsilon_{0},\theta)$
$\displaystyle\quad
H\bigg{(}\epsilon_{s}~{};~{}\frac{\epsilon^{{}^{\prime}}}{2\gamma}~{},~{}\frac{2\gamma\epsilon^{{}^{\prime}}}{1+2\epsilon^{{}^{\prime}}}\bigg{)}$

Figure 2: Various angles in spherical geometry depicting Compton upscattering
of photons with energy $\epsilon_{0}$ to $\epsilon_{s}$ off electrons with
energies $\gamma$.

The blackbody radiation spectrum is strongly peaked at a mean photon energy of
$\mathrm{<\epsilon_{0}(\theta)>=2.7\theta}$. Hence, a monochromatic
$\delta$-function approximation can further simplify the spectral calculation
to sufficient accuracy for a first exploration of the spectral and beaming
patterns of such Compton emission (e.g., Dermer & Menon, 2009). Thus, the
blackbody photon density can be approximated as,
$\mathrm{n_{ph}(\epsilon_{0},\theta)=n_{ph}(\theta)\delta(\epsilon_{0}-2.7\theta)=2.4K\theta^{3}}\delta(\epsilon_{0}-2.7\theta)$.
With this simplification, the Compton emission coefficient in Equation (9) can
be expressed as:

$\displaystyle\operatorname{j(\epsilon_{s},\theta,\gamma)}$
$\displaystyle=\frac{0.3m_{e}c^{3}\sigma_{T}K\epsilon_{s}\theta^{2}}{\gamma^{2}}\bigg{[}y+\frac{1}{y}-\frac{2\epsilon_{s}}{2.7y\gamma^{2}\theta(1-\beta\mu)}$
(10)
$\displaystyle\quad+\frac{\epsilon_{s}^{2}}{(2.7)^{2}\gamma^{4}\theta^{2}(1-\beta\mu)^{2}y^{2}}\bigg{]}$
$\displaystyle\quad
H\bigg{(}\epsilon_{s};\frac{2.7\theta(1-\beta\mu)}{2},\frac{5.4\gamma^{2}(1-\beta\mu)\theta}{1+5.4\gamma\theta(1-\beta\mu)}\bigg{)}$

Equation (10) needs to be integrated over scattered photon energies,
$\epsilon_{s}$ within the limits constrained by the Heaviside function to get
the Compton cooling term. Equation (10) is the general expression for Compton
emissivity that is applicable in both the Thomson and

Figure 3: Comparison between equations (10) and (11), for
$\mathrm{\gamma=10^{3}}$ and $\mathrm{\cos\psi=-1}$: The top panel shows the
Compton emissivity due to a single electron vs. scattered photon energy for
$\mathrm{\theta=4.58\times 10^{-10}}$ (left) and $\mathrm{\theta=10^{-8}}$
(right), respectively whereas the lower panel shows the same for
$\mathrm{\theta=10^{-6}}$ (left) and $\mathrm{\theta=10^{-5}}$ (right),
respectively.

Klein-Nishina regimes. The term $\mathrm{y+1/y}$ can be rewritten as
$\mathrm{2+\frac{\epsilon_{s}^{2}}{\gamma(\gamma-\epsilon_{s})}}$. In the
Thomson regime, when $\epsilon^{{}^{\prime}}<<1$, $\gamma>>\epsilon_{s}$ and
$\gamma-\epsilon_{s}\cong\gamma$. In this limit, equation (10) can be
simplified by binomially expanding the terms containing
($\gamma-\epsilon_{s}$) as

$\displaystyle\operatorname{j(\epsilon_{s},\theta,\gamma)}$
$\displaystyle=\frac{0.3c\sigma_{T}K\theta^{2}}{\gamma^{2}}\Bigg{[}2\epsilon_{s}+\epsilon_{s}\bigg{\\{}\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}^{2}+\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}^{3}$
(11)
$\displaystyle\quad+....\bigg{\\}}-\frac{2}{2.7\theta(1-\beta\mu)}\bigg{\\{}\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}^{2}+\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}^{3}$
$\displaystyle\quad+....\bigg{\\}}+\frac{1}{2.7^{2}\theta^{2}(1-\beta\mu)^{2}}\frac{1}{\gamma}\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}^{3}\bigg{\\{}1+$
$\displaystyle\quad
2\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}+3\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}^{2}+4\bigg{(}\frac{\epsilon_{s}}{\gamma}\bigg{)}^{3}+....\bigg{\\}}\Bigg{]}$
$\displaystyle\quad
H\bigg{(}\epsilon_{s}~{};~{}\epsilon_{\mathrm{s_{min}}},\epsilon_{\mathrm{s_{max}}}\bigg{)},$

where
$\mathrm{\epsilon_{s_{max}}=5.4\gamma^{2}(1-\beta\mu)\theta/(1+5.4\gamma\theta(1-\beta\mu))}$
and $\mathrm{\epsilon_{s_{min}}=2.7\theta(1-\beta\mu)/2}$ are the maximum and
minimum energies of the upscattered photons. Equations (10) and (11) were
compared to assess the impact of the binomial approximation. Figure 3
indicates that the approximation has no significant effect on the result for
electrons with Lorentz factors of $\gamma=10^{3}$, even when scattering UV
target photons with $\mathrm{\mu=-1}$. However, when dealing with higher
values of $\gamma$ and $\epsilon_{0}$, it is essential to consider the higher
order terms of the binomial expansion that contain
($\mathrm{\gamma-\epsilon_{s}}$) to avoid significant deviations from the full
results. Due to the presence of the Heaviside function in the emissivity
equations, (10) and (11), a sudden drop in emissivity occurs beyond the
scattered photon energy, $\mathrm{\epsilon_{s}=\epsilon_{s_{max}}}$. By
employing the aforementioned approximations, the integration of equation (11)
over $\mathrm{\epsilon_{s}}$ becomes feasible, considering the limits imposed
by the Heaviside function.

Figure 4: Variation of cooling rate with the cosine of collision angle
($\mathrm{\cos\psi}$) and observer’s angle
($\mathrm{\cos(\theta_{\text{obs}}}$)) in the ELF at a Lorentz factor
($\mathrm{\gamma}$) of $\mathrm{10^{3}}$ and radiation temperature
($\mathrm{\theta}$) of $\mathrm{10^{-5}}$: the plot on the right depicts the
inverse cooling term, obtained by integrating equation (12) over the photon
distribution angle ($\mathrm{\mu_{\text{ph}}}$). The figure illustrates that
the maximum cooling occurs during head-on collisions between electrons and
photons, or when electrons, after scattering, travel parallel to the jet axis.
Conversely, the minimum cooling is observed for tail-on collisions or when
electrons travel perpendicular to the jet axis.

The cooling term associated with inverse Compton scattering in the sheath
frame, taking into account an angle-dependent photon field, can be expressed
as

$\displaystyle\frac{d\gamma}{dt}(\mu,\gamma,\theta)$
$\displaystyle=A\Bigg{[}\big{(}\epsilon_{s_{\mathrm{max}}}^{2}-\epsilon_{s_{\mathrm{min}}}^{2}\big{)}+\frac{\epsilon_{s_{\mathrm{max}}}^{4}-\epsilon_{s_{\mathrm{min}}}^{4}}{4\gamma^{2}}$
(12)
$\displaystyle\quad+\frac{\epsilon_{s_{\mathrm{max}}}^{5}-\epsilon_{s_{\mathrm{min}}}^{5}}{5\gamma^{3}}-\frac{2}{B}\bigg{\\{}\frac{(\epsilon_{s_{\mathrm{max}}}^{3}-\epsilon_{s_{\mathrm{min}}}^{3})}{3\gamma^{2}}$
$\displaystyle\quad+\frac{\epsilon_{s_{\mathrm{max}}}^{4}-\epsilon_{s_{\mathrm{min}}}^{4}}{4\gamma^{3}}\bigg{\\}}+\frac{1}{\gamma^{4}B^{2}}\bigg{\\{}\frac{\epsilon_{s_{\mathrm{max}}}^{4}-\epsilon_{s_{\mathrm{min}}}^{4}}{4}$
$\displaystyle\quad+\frac{2\big{(}\epsilon_{s_{\mathrm{max}}}^{5}-\epsilon_{s_{\mathrm{min}}}^{5}\big{)}}{5\gamma}+\frac{\epsilon_{s_{\mathrm{max}}}^{6}-\epsilon_{s_{\mathrm{min}}}^{6}}{2\gamma^{2}}$
$\displaystyle\quad+\frac{4\big{(}\epsilon_{s_{\mathrm{max}}}^{7}-\epsilon_{s_{\mathrm{min}}}^{7}\big{)}}{7\gamma^{3}}\bigg{\\}}\Bigg{]},$

where $\mathrm{A\equiv 0.3~{}c~{}\sigma_{T}K\theta^{2}/\gamma^{2}}$ and
$\mathrm{B\equiv 2.7\theta(1-\beta\mu)}$. To calculate the cooling rate in the
spine frame, a factor $\mathrm{\Gamma_{rel}^{2}}$ needs to be added to the
r.h.s. The angular distribution of inverse Compton scattering is described by
the expression
$\mu=\mu_{obs}\mu_{ph}+\sqrt{1-\mu_{obs}^{2}}\sqrt{1-\mu_{ph}^{2}}\cos(\phi_{obs}-\phi_{ph})$.
Here, $\mathrm{\mu_{obs}}$ and $\mathrm{\mu_{ph}}$ correspond to the cosines
of the angles $\mathrm{\theta_{obs}}$ and $\mathrm{\theta_{ph}}$,
respectively. These angles indicate the directions of the electron and photon
relative to the jet axis. By assuming the azimuthal symmetry of the system and
performing the numerical integration of equation (12) over
$\mathrm{\mu_{ph}}$, the electron cooling rate can be obtained as a function
of $\mathrm{\mu_{obs}}$. Figure 4 illustrates the dependency of the cooling
rate on the cosine of $\mathrm{\psi}$ and $\mathrm{\theta_{obs}}$,
highlighting the variation of the electron cooling rate with respect to these
angular parameters. The plots are generated in the ELF at electron Lorentz
factor of $\mathrm{\gamma=10^{3}}$ and the radiation temperature of
$\mathrm{\theta=10^{-5}}$.

##  4 Results

This section presents the outcomes of our study concerning self-generated
electric and magnetic fields in SBLs of relativistic jets, along with the
effects of particle anisotropy, radiative output, and IC cooling on particle
dynamics obtained through PiC simulations.

###  4.1 Self-generated magnetic and electric fields

Plasma instabilities like the Weibel instability self-generate a magnetic
field along the z-direction (perpendicular to the jet axis). The self-
generated electromagnetic fields in SBLs create turbulence in the spine-sheath
interface which eventually leads to particle acceleration up to TeV energy in
relativistic jets. Figures 5(a), 5(b), and 5(c) show the dominant components
of electric and magnetic fields and the current density produced from the PiC
simulation.

(a) |  (b)
---|---
(c) |  (d)
(e) |  (f)
Figure 5: First row: Figure (a) illustrates the xy-cut of the self-generated
electric field and Figure (b) presents the magnetic field in SBLs, with the
red and magenta colors representing opposite polarities. Second row: Figure
(c) depicts the current distribution ($\mathrm{J_{x}}$). The ions dominate the
stronger outer current sheet, while the weaker inner current sheet is
dominated by electrons. Figure (d) shows the total charge density
($\mathrm{\rho}$). Third row: Figures (e) and (f) respectively show the
density distribution of electrons and ions. Notably, a discernible depletion
in particle density is observed in the vicinity of the boundary layers. All
figures correspond to $\mathrm{t=3000\omega_{p,e}^{-1}}$. The units displayed
are arbitrary. Figure 6: The electron number densities as a function of y at
$t=3000\omega_{p,e}^{-1}$: while the ions are fully expelled from the shear
interface, the electrons create a layer near the interface, resulting in the
formation of a triple layer due to charge separation.

Figure 5(d) illustrates the net charge distribution in the SBLs, showing a
notable concentration of negatively charged particles (electrons) within these
regions. Positively charged particles (ions), on the other hand, tend to have
a slightly more dispersed distribution and are found slightly further from the
SBLs, thereby forming a triple-layer structure. Figures 5(e) and 5(f) present
the spatial distribution of electron and ion number densities, respectively.
These figures highlight higher electron density within the SBLs compared to
the ion density. This disparity in densities is further evident in the density
profiles of both species (Figure 6). The density profiles of ions, electrons,
and net charge as a function of the y-coordinate are demonstrated in Figure 6.
Due to the different velocities and densities of ions and electrons, the
current density along the x-direction ($\mathrm{J_{x}}$) (as depicted in
Figure 5(c)) produces the uniform magnetic field ($\mathrm{B_{z}}$) along the
shear interface within the shear flows. Because the electron gyroradius is
smaller than that of ions, ions are more displaced than electrons, leading to
their expulsion from the shear interface due to the extra magnetic pressure,
resulting in the formation of a density trough. This process induces charge
separation and the formation of a triple layer, as illustrated in Figure 6.
Ion-dominated current sheets are formed farther from the shear interface,
while electron-dominated current sheets are formed near it, with an electric
field ($\mathrm{E_{y}}$) pointing from the ion-dominated current sheet to the
electron-dominated current sheet. Ultimately, the electrons undergo efficient
acceleration driven by the cross fields ($\mathrm{E\times B}$) in the
x-direction, which aligns with the jet-propagation direction. As the ions are
more massive than electrons, ions undergo free streaming while electrons
behave like fluid (e.g., Gruzinov, 2008; Liang et al., 2017). The kinetic KHI
plays a vital role in initially cold electrons. However, the electrons attain
a finite transverse momentum either by plasma instabilities or due to the
finite temperature achieved by electrons. Eventually, electrons with finite
transverse momentum cross over to the sheath moving in the opposite direction,
but heavy ions do not. This mechanism, referred to as the electron counter-
current instability (ECCI) (e.g., Liang et al., 2017), dominates, leading to
the formation of opposite current layers on the two sides of the shear
interface, creating a monopolar slab of magnetic field.

###  4.2 Particle Anisotropy in SBLs

Figure 7: Spine electrons’ $\mathrm{p_{y}}$ versus $\mathrm{p_{xLab}}$ at time
$\mathrm{t=3000\omega_{p,e}^{-1}}$: $\mathrm{p_{x}}$ are Lorentz boosted to
the Laboratory frame by $\mathrm{\Gamma=15}$. The figure illustrates that some
of the spine electrons diffuse into the sheath region, leading to the
deceleration of those spine electrons, represented by the low-energy arc-
shaped electron population. The high-energy electron population towards the
right-hand side of the figure corresponds to electrons that remain in the
spine region without crossing over into the sheath region.

In order to illustrate the particle acceleration and radiation properties in
the stationary (sheath) frame of the source, we Lorentz-transform the particle
momenta and Compton emission into the sheath frame, utilizing a bulk Lorentz
factor of $\mathrm{\Gamma=15}$. The results demonstrate that spine electrons
undergo significant acceleration,

Figure 8: $\mathrm{y}$ ($\mathrm{c/\omega_{p,e}}$) versus $\mathrm{p_{xLab}}$
of spine electrons at $\mathrm{t=3000\omega_{p,e}^{-1}}$: $\mathrm{p_{x}}$ are
Lorentz boosted to the laboratory frame. As depicted in the figure, some of
the spine electrons cross over through SBLs and enter the sheath region, where
they experience deceleration. The spine electrons that remain strictly within
the spine region undergo acceleration. Figure 9: The distribution of the
tangent of the beam angle of spine electrons versus electron Lorentz factor in
the laboratory frame at $\mathrm{t=3000\omega_{p,e}^{-1}}$: all high-energy
spine electrons, which did not cross over to the sheath region, possess beam
angles significantly smaller than 1/$\mathrm{\Gamma}$, as indicated by the red
line. The figure further illustrates an anticorrelation between beam angle and
electron energy.

attaining high energies and exhibiting pronounced momentum anisotropy
($\mathrm{p_{xlab}>>p_{y}}$) in the laboratory frame (Figure 7). A portion of
the spine electrons diffuses into the sheath region, experiencing deceleration
and giving rise to a low-energy population, as depicted at the left in Figures
7 and 8. Moreover, Figure 8 illustrates the large particle momenta along the
jet axis in the spine. Furthermore, we observe an anti-correlation between the
beam angle $\mathrm{|p_{y}/p_{\text{xLab}}|}$ and electron energies, with the
beam angle decreasing as electron energies increase (Figure 9).

Notably, the beam angles of all high-energy spine electrons, which remained in
the spine region and did not transition to the sheath region, are
significantly smaller than those resulting from Doppler boosting an isotropic
particle distribution from the spine rest frame to the laboratory frame
($\mathrm{1/\Gamma}$), as depicted by the red dashed line in Figure 9.

###  4.3 Numerical Cerenkov Instability

The Numerical Cerenkov Instability (NCI) is a significant concern in PiC
simulations that involve relativistic drifts. NCI is a computational artifact
that arises in PiC simulations due to the discretization scheme employed. It
is not a physical instability but a numerical phenomenon. In PiC simulations,
electromagnetic fields and particle motions are discretized on a computational
grid. When the discrete grid points interact with the chosen spatial and
temporal resolutions, NCI occurs generating spurious wave modes. These modes
can distort simulation results and potentially impact the accuracy of the
physical processes under the study. The momentum plot of the magnetic field
provides insight into the presence and effects of NCI.

To mitigate the NCI, the TRISTAN-MP code incorporates a range of specialized
numerical schemes that help suppress the instability. It employs both time-
centered and space-centered finite difference schemes to ensure second-order
accuracy in both space and time. Time-centered schemes calculate values at
time step midpoints for enhanced time integration precision, while space-
centered schemes compute spatial derivatives at midpoint grid locations for
improved spatial accuracy. Magnetic and Electric fields are stored within a 3D
Yee mesh (e.g., Yee, 1966), effectively capturing their spatial distribution.
A tri-linear interpolation function relates these fields to particle
positions, ensuring accurate values for simulated particle positions. A three-
point digital binomial filter (e.g., Birdsall, 1991; Hockney & Eastwood, 2021)
is applied along each spatial dimension to source terms, using weights of
0.25, 0.5, and 0.25 to mitigate NCI and other non-physical high-frequency
field modes arising from finite difference calculations. The code

(a)
---
(b)
Figure 10: 2D color contour plots of the Fourier amplitude of $\mathrm{B_{z}}$
at two different sample times, (a) $\mathrm{t=200\omega_{p,e}^{-1}}$ and (b)
$\mathrm{t=3000\omega_{p,e}^{-1}}$: It can be observed that the effects of NCI
are minimal in the simulation, and they tend to gradually decrease as the
simulation progresses.

offers a choice between a second-order method and a fourth-order stencil
approach (e.g., Greenwood et al., 2004). We opt for the latter approach as it
is aimed at minimizing NCI effects, particularly in relativistic scenarios.
The code adheres to the Courant-Friedrichs-Lewy (CFL) condition, ensuring that
the product of the timestep and the speed of light is smaller than the minimum
cell size. In this context, the speed of light is set as $\rm{c=0.45\Delta
x/\Delta t}$. This also serves to mitigate the effects of the NCI. As a
result, the current simulations are able to contain the NCI within acceptable
levels, leading to minimizing adverse effects on particle heating. The
$\mathrm{kx-ky}$ plot in Figure (10) represents the distribution of the
$\mathrm{z}$-component of the magnetic field in momentum space, with
$\mathrm{kx}$ and $\mathrm{ky}$ representing the wave numbers in the x and y
directions, respectively. The smooth and well-behaved distribution of the
magnetic field in the $\mathrm{kx-ky}$ plot suggests a minimal effect of NCI
indicating that the chosen parameters in the simulations adequately mitigate
numerical artifacts.

###  4.4 Electron Spectra and Effect of IC Cooling

(a)
---
(b)
Figure 11: Comparison of electron Spectra with and without inverse Compton
cooling by blackbody photons: Figure (a) displays the impact of IC cooling due
to angle-averaged UV photons, while Figure (b) exhibits the effect of IC
cooling induced by angle-dependent UV and optical photons at various values of
$\mathrm{\cos(\theta_{obs}}$).

This section focuses on electron acceleration with PiC simulations
demonstrating the impact of IC cooling on particle spectra. Initially, the
energy distribution of electrons follows a Maxwellian distribution.
Acceleration occurs due to the electromagnetic field generated by the relative
displacement of ions and electrons. The thermal electron bulk Lorentz factor
reaches its peak when energy equipartition between the ions and electrons is
attained (e.g., Lyubarsky, 2006; Iwamoto et al., 2019). Thus, the maximum
electron Lorentz factor can be expressed as:

$\displaystyle\gamma_{max}\simeq\frac{1}{2}\frac{\gamma_{i}m_{i}}{m_{e}}$

The spectra illustrated in Figure 11 are measured in the simulation frame,
where a pure $e^{-}$-ion plasma shows no evidence of a power law. Initially,
the electron spectra exhibit a Maxwellian distribution that evolves into a
non-Maxwellian distribution, with a remnant of the original distribution.
Ultimately, the system reaches a steady state with a peak around
$\mathrm{\gamma\cong\Gamma m_{i}/m_{e}}$.

To investigate the impact of radiation cooling on the electron spectra, the
cooling terms presented in equations 6 and 12 can be incorporated into the
electron subroutine of the PiC code. Figure 11 illustrates the impact of
Inverse Compton (IC) cooling on electron spectra induced by angle-averaged and
angle-dependent photons. The strength of IC cooling depends on the radiation
energy density and the Lorentz factor of the electrons. At lower electron
energies, the cooling effect is negligible, while at higher energies, it
significantly affects electron acceleration and the energy spectrum. Due to
radiation drag, the high-energy end of the spectrum shifts towards lower
energies (as seen in Fig. 11). Compared to IC cooling induced by UV photons,
the effect of Compton drag for photons with temperatures $\theta\leq 10^{-6}$
is subdominant, and thus does not appreciably affect particle dynamics and the
electron energy spectrum.

###  4.5 Self-consistent Radiation Spectra

We now discuss the radiation spectra resulting from Compton upscattering of
different radiation backgrounds. Specifically, we consider the cosmic
microwave background, infrared, optical, and ultraviolet photons with
corresponding values of $\theta$ equal to $\mathrm{4.58\times 10^{-10}}$,
$\mathrm{10^{-8}}$, $\mathrm{10^{-6}}$, and $\mathrm{10^{-5}}$, respectively.

We observe sharp spectral peaks in the emissivity of Compton-scattered
blackbody photons by mono-energetic relativistic electrons as evident in
Figure 3. Leveraging the pronounced spectral peaks, we apply a monochromatic
approximation to convert the particle energy loss rate to a monochromatic
radiation spectrum. This simplification significantly streamlines our
calculations and reduces computational time, enabling more efficient analysis
of the radiation spectrum. The obtained results, encompassing both angle-
averaged and angle-dependent photon distributions, are presented in

(a)
---
(b)
(c)
Figure 12: Time-integrated angle-averaged Compton spectra in the sheath frame
obtained from simulations conducted for different radiation temperatures of
angle-averaged blackbody photon fields: the spectra correspond to three
distinct phases: (a) early phase of the simulations at
$\mathrm{t=1500~{}\omega_{p,e}^{-1}}$, (b) intermediate phase at
$\mathrm{t=2500~{}\omega_{p,e}^{-1}}$, and (c) later steady state at
$\mathrm{t=3000~{}\omega_{p,e}^{-1}}$.

Figures 12 and 13.

(a)
---
(b)
(c)
Figure 13: Time-integrated angle-dependent Compton spectra in the sheath
frame, resulting from simulations conducted at radiation temperature
$\mathrm{\theta=10^{-5}}$ and different viewing angles: spectra are obtained
at simulation time of (a) $\mathrm{t=2000~{}\omega_{p,e}^{-1}}$, (b)
$\mathrm{t=4000~{}\omega_{p,e}^{-1}}$, and (c)
$\mathrm{t=5000~{}\omega_{p,e}^{-1}}$.

Throughout the various stages of the simulations, we observe distinct
evolutions in the Compton spectra of relativistic electrons interacting with
photon fields. In the early phases of the simulations, both angle-integrated
and angle-dependent cases exhibit a quasi-thermal inverse Compton spectrum
with a single dominant component. We observe a quasi-thermal inverse Compton
spectrum with a single component around simulation time of
$\mathrm{t=1500/\omega_{p,e}}$ for an angle-integrated photon field and
$\mathrm{t=2000/\omega_{p,e}}$ for an angle-dependent photon distribution, as
depicted in Figures 12(a) and 13(a). As the simulation progresses, the Compton
spectra undergo noteworthy changes. The single-component spectrum develops
into a double-component spectrum, around $\mathrm{t=2500/\omega_{p,e}}$ and
$\mathrm{t=4000/\omega_{p,e}}$ for the angle-integrated and angle-dependent
photon distributions, respectively (see Figures 12(b) and 13(b)). In the later
steady state of the simulations, observed beyond
$\mathrm{t=3000/\omega_{p,e}}$ and $\mathrm{t=5000/\omega_{p,e}}$,
respectively, a distinct spectral pattern emerges. The Compton spectrum
exhibits a quasi-thermal low-frequency spectrum with a cut-off power-law tail
(see Figures 12(c) and 13(c)).

The steady state is achieved later in the case of the inverse Compton spectra
from scattering an angle-dependent photon field compared to the angle-averaged
photon field. The angle-dependent Compton spectra exhibit a flatter peak in
comparison to the angle-averaged Compton spectra. The observed peak energy in
the angle-dependent radiation spectra is approximately
$\mathrm{2.7\theta\gamma^{2}}$ for $\mathrm{\cos(\theta_{obs})=1}$. As the
value of $\mathrm{\cos(\theta_{obs}})$ decreases, the peak energy of the
radiation spectra progressively shifts towards lower energies. This observed
correlation between the peak energy and $\mathrm{\cos(\theta_{obs})}$
highlights the pronounced angular dependence intrinsic to the radiative output
in relativistic jets. The radiation Spectra obtained in the case of an angle-
dependent photon distribution exhibit similarities to those obtained for
angle-averaged photons.

In our simulations, we observe the angle-averaged radiation spectra attaining
a steady state beyond $\rm{t=3000/\omega_{p,e}}$, while angle-dependent
radiation spectra beyond $\rm{t=5000/\omega_{p,e}}$. Beyond this stage, there
are no significant fluctuations in their shapes, reaffirming the reliability
and accuracy of our results and reflecting the dynamic equilibrium achieved in
our simulated system. The time delay in reaching a steady state for the angle-
dependent photon field’s inverse Compton spectra, in comparison to the angle-
averaged photon field, can be attributed to the increased complexity and
larger parameter space resulting from the inclusion of angle-dependent
photons. This introduces a higher computational demand for thorough sampling
and evaluation of radiation outputs, leading to extended times needed to
establish a stable equilibrium and achieve consistent radiation outputs. The
broader and flatter spectral peak observed in the angle-dependent Compton
spectra

(a)
---
(b)
Figure 14: Radiation intensity resulting from Compton scattering of an angle-
averaged photon field with varying temperatures as a function of the viewing
angle of the jet at a simulation time of $\mathrm{t=3000\omega_{p,e}^{-1}}$:
panel (a) displays the global radiative energy distribution per unit solid
angle in the ELF, while panel (b) shows the same distribution in the
laboratory frame (sheath). The dashed curve represents the $\delta^{3}$
boosting pattern, characteristic of co-moving isotropic photon emission. The
violet dashed line represents $\mathrm{1/\Gamma}$. (a)
---
(b)
Figure 15: Radiation intensity of an angle-dependent photon field, subject to
Compton upscattering by relativistic electrons as a function of the viewing
angle of the jet at a simulation time of $\mathrm{t=5000\omega_{p,e}^{-1}}$
for varying radiation temperatures: panel (a) exhibits the global radiative
energy distribution per unit solid angle in the ELF. Panel (b) shows the
corresponding distribution in the laboratory frame. The dashed curve
represents the $\mathrm{\delta^{3}}$ boosting pattern.

signifies a greater number of scattering events, indicating a more intricate
interplay between photons and particles in this scenario. This difference in
spectral characteristics further contributes to the increased time required to
attain a steady state in the angle-dependent case.

###  4.6 Observable Features of Radiation Spectra

To establish a connection between the radiation output obtained from the
simulations and the observational data, we can plot the radiation intensity
($\mathrm{dE/d\Omega}$) as a function of the viewing angle. Here,
$\mathrm{dE}$ represents the radiation energy and $\mathrm{d\Omega}$ denotes
the differential solid angle, defined as
$\mathrm{d\Omega=\sin\theta_{ph}d\theta_{ph}d\phi_{ph}}$. The angle
$\mathrm{\theta_{ph}}$ represents the angle between the final direction of
radiation and the axis of the jet (i.e., the viewing angle), while
$\mathrm{\phi_{ph}}$ represents the azimuthal angle. The system is azimuthally
symmetric in $\mathrm{\phi}$, so we may set $\mathrm{d\phi_{ph}=2\pi}$. In the
ELF, as depicted in Figures 14(a) and 15(a), the spine electrons emit the
highest radiation intensities along the jet axis (i.e. at
$\mathrm{\theta_{ph}=\theta_{elf}=0^{\circ}}$).

In order to determine the radiation intensity as observed by an observer, we
apply a Lorentz transformation into the sheath frame. Figures 14 and 15 show
plots of the radiation intensity versus viewing angle and demonstrate that the
inverse Compton radiation emitted from the jet’s SBL experiences a strong
boost in the forward direction, with a characteristic angle much smaller than
$\mathrm{1/\Gamma}$. This is indicated by the violet dashed lines in Figures
14(b) and 15(b).

In the case of isotropic photon emission in the co-moving frame of an emission
region moving with Lorentz factor $\mathrm{\Gamma}$ along the jet viewed at an
angle $\mathrm{\theta_{lab}}$, the radiative energy flux $\mathrm{dF/dE}$ is
boosted by a factor $\mathrm{\delta^{3}}$, with
$\mathrm{\delta=1/\Gamma(1-\beta_{\Gamma}\cos\theta_{lab})}$ being the Doppler
factor. The dashed curve depicted in Figures 14(b) and 15(b) illustrate the
comparison of the simulated beaming characteristic to this $\delta^{3}$
pattern.

##  5 Conclusion

In this article, we have presented the results of our study on the self-
generated electric and magnetic fields and particle acceleration occurring at
relativistic SBLs in relativistic jets associated with AGN and GRBs, and the
subsequent inverse-Compton radiation output. Through the utilization of fully
kinetic PiC simulations incorporating electron-ion plasma, we have shown that
the electric and magnetic fields are self-generated and play a vital role in
particle acceleration in SBLs. High energy particles are accelerated along the
magnetic field lines, resulting in an anisotropic momentum distribution. This
anisotropic particle distribution contributes effectively to the radiative
output, along with the exchange of particles between the spine and sheath
regions.

We employed the monochromatic approximation to compute the radiation spectra
motivated by the sharp spectral peaks in the emissivity resulting from the
Compton scattering of blackbody photons by monoenergetic relativistic
electrons, as illustrated in Figure 3. The difference between the
monochromatic approximation and a full integration over the emission profiles
is expected to be very small, as the spectral shape is dominated by the shape
of the broad, thermal+non-thermal electron spectra and not the emissivity
profiles of mono-energetic electrons. The recent work by (Del Gaudio et al.,
2020) has introduced a promising numerical prospect to investigate radiative
processes like inverse Compton emission. Their benchmarked algorithm
seamlessly integrates with the standard PiC loop, leading to improved
computational efficiency. We plan to experiment with this scheme in future
work. The radiative output in our study has been directly obtained from the
PiC simulation, from which we have calculated the radiation intensity to study
the angle-dependent inverse-Compton spectra, self-consistently taking into
account inverse-Compton cooling of relativistic electrons. To our knowledge,
investigations into the angular dependence of radiative emissions from shear
boundary layers (SBLs) within relativistic jets have not been conducted
previously. Our research marks the pioneering effort to explore particle
acceleration, the ensuing inverse Compton spectra, and the impact of inverse
Compton cooling on the dynamics of relativistic electrons, while considering
the impact of angular variations within these shear layers. We looked at both
angle-averaged and angle-dependent distributions of target photons, using a
$\mathrm{\delta}$-function approximation for the target photon field. Our
findings showed that IC cooling can have a non-negligible impact on the
acceleration of relativistic particles at SBLs, causing high-energy electrons
to cool down to slightly lower energies. As expected, the inverse-Compton
cooling effect becomes more pronounced for higher target-photon blackbody
temperatures. In the early stages of simulations, both angle-integrated and
angle-dependent inverse Compton radiation spectra exhibit a single component,
quasi-thermal radiation spectrum. As the simulations progress, the spectrum
evolves into a two-component spectrum, which eventually becomes a quasi-
thermal low-frequency spectrum with a cut-off power-law tail.

The radiation emitted is strongly boosted along the jet axis, with a
characteristic opening angle much less than 1/$\mathrm{\Gamma}$. This boosting
is more powerful than what would be expected from Doppler boosting of an
isotropic radiation field in the co-moving frame of the spine. The Doppler
factor estimates for TeV blazars using one-zone models are inconsistent with
observations made with VLBI. Our findings suggest that the extreme beaming
patterns of particles accelerated at SBLs could resolve the long-standing
problem of the Doppler factor crisis (e.g., Lyutikov & Lister, 2010b). In
future work, we will explore the effects of a non-zero initial magnetic field
on the plasma in relativistic jets, which could involve investigating a
relativistic magnetically dominated electron-positron jet interacting with a
weakly magnetized electron-ion ambient plasma. Recent work by Sironi et al.
(2021) has shown that such systems can develop Kelvin-Helmholtz instabilities
and kinetic-scale reconnection layers, which provide a first-principles
mechanism for particle injection into shear-driven acceleration. This could
offer valuable insights into the complex dynamics of these shear-structured
jets.

We would like to thank Anatoly Spitkovsky for initial discussions on the
implementation of shear-driven particle acceleration in PiC codes. This work
was supported through the South African Research Chair Initiative (SARChI) of
the Department of Science and Technology and the National Research Foundation
(NRF) of South Africa under SARChI chair grant no. 64789.

## References

  * Abbott et al. (2017) Abbott, B., Abbott, R., Abbott, T., et al. 2017, Physical Review Letters, 119, doi: 10.1103/physrevlett.119.161101
  * Alves et al. (2014) Alves, E. P., Grismayer, T., Fonseca, R. A., & Silva, L. O. 2014, New Journal of Physics, 16, 035007, doi: 10.1088/1367-2630/16/3/035007
  * Alves et al. (2012) Alves, E. P., Grismayer, T., Martins, S. F., et al. 2012, 746, L14, doi: 10.1088/2041-8205/746/2/l14
  * Birdsall (1991) Birdsall, C. 1991, IEEE Transactions on Plasma Science, 19, 65, doi: 10.1109/27.106800
  * Boettcher et al. (2012) Boettcher, M., Harris, D. E., & Krawczynski, H. 2012, Relativistic Jets from Active Galactic Nuclei
  * Chand et al. (2019) Chand, T. B., Böttcher, M., & Kilian, P. 2019, PoS, HEASA2018, 046
  * Chandrasekhar (1961) Chandrasekhar, S. 1961, Hydrodynamic and hydromagnetic stability
  * Coughlin & Begelman (2020) Coughlin, E. R., & Begelman, M. C. 2020, Monthly Notices of the Royal Astronomical Society, 499, 3158, doi: 10.1093/mnras/staa3026
  * Del Gaudio et al. (2020) Del Gaudio, F., Grismayer, T., Fonseca, R. A., & Silva, L. O. 2020, Journal of Plasma Physics, 86, 905860516, doi: 10.1017/S002237782000118X
  * Dermer & Menon (2009) Dermer, C. D., & Menon, G. 2009, High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos
  * Ghisellini et al. (2005) Ghisellini, G., Tavecchio, F., & Chiaberge, M. 2005, A&A, 432, 401, doi: 10.1051/0004-6361:20041404
  * Giroletti et al. (2004) Giroletti, M., Giovannini, G., Feretti, L., et al. 2004, ApJ, 600, 127, doi: 10.1086/379663
  * Greenwood et al. (2004) Greenwood, A. D., Cartwright, K. L., Luginsland, J. W., & Baca, E. A. 2004, Journal of Computational Physics, 201, 665, doi: 10.1016/j.jcp.2004.06.021
  * Grismayer et al. (2013a) Grismayer, T., Alves, E. P., Fonseca, R. A., & Silva, L. O. 2013a, Physical Review Letters, 111, 015005, doi: 10.1103/PhysRevLett.111.015005
  * Grismayer et al. (2013b) —. 2013b, Plasma Physics and Controlled Fusion, 55, 124031, doi: 10.1088/0741-3335/55/12/124031
  * Gruzinov (2008) Gruzinov, A. 2008, GRB: magnetic fields, cosmic rays, and emission from first principles?  https://arxiv.org/abs/0803.1182
  * Hakobyan et al. (2019) Hakobyan, H., Philippov, A., & Spitkovsky, A. 2019, The Astrophysical Journal, 877, 53, doi: 10.3847/1538-4357/ab191b
  * Hardee (2007) Hardee, P. E. 2007, The Astrophysical Journal, 664, 26, doi: 10.1086/518409
  * Henri & Sauge (2006) Henri, G., & Sauge, L. 2006, The Astrophysical Journal, 640, 185, doi: 10.1086/500039
  * Hockney & Eastwood (2021) Hockney, R. W., & Eastwood, J. W. 2021, Computer simulation using particles (crc Press)
  * Iwamoto et al. (2019) Iwamoto, M., Amano, T., Hoshino, M., et al. 2019, The Astrophysical Journal, 883, L35, doi: 10.3847/2041-8213/ab4265
  * Jester et al. (2001) Jester, S., Röser, H.-J., Meisenheimer, K., Perley, R., & Conway, R. 2001, Astronomy & Astrophysics, 373, 447, doi: 10.1051/0004-6361:20010593
  * Lamb & Kobayashi (2017) Lamb, G. P., & Kobayashi, S. 2017, Monthly Notices of the Royal Astronomical Society, 472, 4953, doi: 10.1093/mnras/stx2345
  * Liang et al. (2013a) Liang, E., Boettcher, M., & Smith, I. 2013a, The Astrophysical Journal, 766, L19, doi: 10.1088/2041-8205/766/2/l19
  * Liang et al. (2013b) Liang, E., Fu, W., Boettcher, M., Smith, I., & Roustazadeh, P. 2013b, The Astrophysical Journal, 779, L27, doi: 10.1088/2041-8205/779/2/l27
  * Liang et al. (2017) Liang, E., Fu, W., & Böttcher, M. 2017, 847, 90, doi: 10.3847/1538-4357/aa8772
  * Liu et al. (2017) Liu, R.-Y., Rieger, F. M., & Aharonian, F. A. 2017, 842, 39, doi: 10.3847/1538-4357/aa7410
  * Lyubarsky (2006) Lyubarsky, Y. 2006, Astrophys. J., 652, 1297, doi: 10.1086/508606
  * Lyutikov & Lister (2010a) Lyutikov, M., & Lister, M. 2010a, 722, 197, doi: 10.1088/0004-637x/722/1/197
  * Lyutikov & Lister (2010b) —. 2010b, The Astrophysical Journal, 722, 197, doi: 10.1088/0004-637x/722/1/197
  * McKinney (2006) McKinney, J. C. 2006, Monthly Notices of the Royal Astronomical Society, 368, 1561, doi: 10.1111/j.1365-2966.2006.10256.x
  * Ostrowski (1990) Ostrowski, M. 1990, A&A, 238, 435
  * Piran (2000) Piran, T. 2000, Physics Reports, 333-334, 529–553, doi: 10.1016/s0370-1573(00)00036-3
  * Rieger (2019) Rieger, F. M. 2019, Galaxies, 7, doi: 10.3390/galaxies7030078
  * Rieger & Duffy (2004) Rieger, F. M., & Duffy, P. 2004, The Astrophysical Journal, 617, 155, doi: 10.1086/425167
  * Rieger & Duffy (2006) —. 2006, Proceedings of the International Astronomical Union, 2, 103, doi: 10.1017/S1743921307010034
  * Romero et al. (2017) Romero, G. E., Boettcher, M., Markoff, S., & Tavecchio, F. 2017, Space Science Reviews, 207, 5, doi: 10.1007/s11214-016-0328-2
  * Sahayanathan (2009) Sahayanathan, S. 2009, Monthly Notices of the Royal Astronomical Society: Letters, 398, L49, doi: 10.1111/j.1745-3933.2009.00707.x
  * Sironi et al. (2021) Sironi, L., Rowan, M. E., & Narayan, R. 2021, The Astrophysical Journal Letters, 907, L44, doi: 10.3847/2041-8213/abd9bc
  * Sironi & Spitkovsky (2009) Sironi, L., & Spitkovsky, A. 2009, The Astrophysical Journal, 698, 1523, doi: 10.1088/0004-637x/698/2/1523
  * Sol et al. (1989) Sol, H., Pelletier, G., & Asséo, E. 1989, Monthly Notices of the Royal Astronomical Society, 237, 411, doi: 10.1093/mnras/237.2.411
  * Spitkovsky (2005) Spitkovsky, A. 2005, AIP Conference Proceedings, doi: 10.1063/1.2141897
  * Spitkovsky (2007) —. 2007, The Astrophysical Journal, 673, L39–L42, doi: 10.1086/527374
  * Stawarz & Ostrowski (2002) Stawarz, ., & Ostrowski, M. 2002, Publications of the Astronomical Society of Australia, 19, 22, doi: 10.1071/AS01092
  * Webb (1989) Webb, G. M. 1989, ApJ, 340, 1112, doi: 10.1086/167462
  * Weibel (1959) Weibel, E. S. 1959, Phys. Rev. Lett., 2, 83, doi: 10.1103/PhysRevLett.2.83
  * Werner et al. (2018) Werner, G. R., Philippov, A. A., & Uzdensky, D. A. 2018, Monthly Notices of the Royal Astronomical Society: Letters, 482, L60, doi: 10.1093/mnrasl/sly157
  * Yee (1966) Yee, K. 1966, IEEE Transactions on Antennas and Propagation, 14, 302, doi: 10.1109/TAP.1966.1138693


# MetaMax: Improved Open-Set Deep Neural Networks via Weibull Calibration

Zongyao Lyu, Nolan B. Gutierrez, and William J. Beksi
The University of Texas at Arlington
Arlington, TX, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Open-set recognition refers to the problem in which classes that were not seen
during training appear at inference time. This requires the ability to
identify instances of novel classes while maintaining discriminative
capability for closed-set classification. OpenMax was the first deep neural
network-based approach to address open-set recognition by calibrating the
predictive scores of a standard closed-set classification network. In this
paper we present MetaMax, a more effective post-processing technique that
improves upon contemporary methods by directly modeling class activation
vectors. MetaMax removes the need for computing class mean activation vectors
(MAVs) and distances between a query image and a class MAV as required in
OpenMax. Experimental results show that MetaMax outperforms OpenMax and is
comparable in performance to other state-of-the-art approaches.

##  1 Introduction

Image classification with deep neural networks has made significant progress
[14, 26, 8, 9]. However, the majority of the work is based on a closed-set
assumption where training datasets are expected to include all the classes
that may be encountered in the environments in which the vision system will be
deployed. Yet, this assumption cannot be guaranteed in real-world environments
where samples from unknown classes not seen during training may appear during
testing and cause system failure [21, 28, 31].

Figure 1: MetaMax utilizes extreme value theory-based meta-recognition for
post-recognition analysis to discover the presence of unknown classes while
preserving closed-set classification accuracy.

To address the limitation of closed-set classification, open-set recognition
(OSR) has been introduced [23]. OSR describes the scenario where incomplete
knowledge of the world is present during training, and new classes can appear
during testing. Not only does this require the model to maintain the
capability of accurately classifying known classes, but it must also be able
to effectively identify unknown classes. OSR has recently gained significant
attention in the research community [7, 10, 24, 2, 3, 5, 1].

With the introduction of the formal definition of OSR, Scheirer et al. [23]
presented a “1-vs-Set Machine” which extends the 1-class and binary SVM in a
way that better supports OSR. Bendale and Boult [3] proposed the first deep
learning-based solution for OSR by introducing OpenMax as an alternative to
SoftMax. OpenMax can be applied directly on a standard closed-set
classification network to calibrate its output score vectors, thus enabling a
closed-set classifier to perform OSR. It does not require accessing additional
training data, updating the training procedure, or modification of the network
architecture.

Inspired by OpenMax, we introduce MetaMax, a method that utilizes extreme
value theory-based meta-recognition [25, 22] as a post-recognition analysis
technique to assist the identification of unknown classes, Fig. 1. More
specifically, we directly model non-match scores from the network outputs as
opposed to OpenMax’s method of explicitly calculating and storing mean
activation vectors for each class. By modeling the non-match distributions, we
can assemble additional data points for building a Weibull model and therefore
achieve more accurate parameters for each constructed model.

To the best of our knowledge, MetaMax is the only alternative to OpenMax.
Different from most work in this area that either modifies the network
architecture or requires auxiliary training data that can be hard to acquire
in practice, we present a calibration tool that can be readily plugged into
various classification networks thus enabling them to perform OSR without
additional overhead.

##  2 Preliminaries

###  2.1 OpenMax

OpenMax introduced the concept of activation vectors, which are the output
values from the penultimate layer of a neural network. Concretely, let
$\bm{v}(x)=v_{1}(x),\ldots,v_{K}(x)$ be the level of activation for each of
the $K$ classes seen during training and let the unknown classes begin at
index 0. OpenMax obtains the activation vectors by saving the output of
$C_{\bm{\theta}}$’s layer immediately before the SoftMax layer. It then
separates the activation vectors into $K$ clusters representing $K$ known
classes. For each of the clusters, the distance between every activation
vector and the cluster mean (i.e., class mean activation vector) is
calculated.

OpenMax proceeds to compute $K$ Weibull models by fitting each model to the
largest distances of the correctly classified samples. During inference, the
$K$ Weibull models are used to revise the top few highest activations to
obtain a new vector $\hat{\bm{v}}(x)$. OpenMax then defines an additional
activation for class $K+1$ thereby producing a score vector for all $K+1$
classes. The class corresponding to the highest probability is the prediction
of the OpenMax algorithm. If the predicted class label for an input is indexed
at 0, then it is considered to be an unknown class.

##  3 MetaMax

MetaMax is derived from contributions in class descriptors and the modeling of
non-match distributions for the purpose of calibrating OSR.

###  3.1 Class Representations

Object recognition can be mapped to the problem of determining _match_ scores
between an input image and a class descriptor such as the class mean
activation vector (MAV) used in OpenMax. Given an activation vector for a
correctly classified image by a closed-set classifier, we consider the largest
score as the match score and the rest of the scores in the vector as the non-
match scores. The match and non-match scores from all the training data
constitute the match distribution and non-match distribution, respectively.

###  3.2 Modeling of Rare Events

Given a particular input, OpenMax finds the distance between each sample and
its respective MAV. The farthest, i.e., the rarest, distances are used to
build per class Weibull models. These models are then utilized to compute
revised activation vectors that OpenMax uses to provide probability estimates
for the unknown rejection. Yet, if data is limited these largest distances may
misrepresent the data manifold, produce incorrect activation vectors, and thus
hinder the Weibull model’s accuracy.

When building the Weibull model for each class, OpenMax finds a single data
point by modeling the largest distance. We eliminate this limitation by
modeling the non-match distribution to provide multiple data points per input.
Concretely, we model the non-match distribution via non-match scores to obtain
$K-1$ data points per vector.

The first step of MetaMax is to obtain a Weibull model fitted for each class
as shown in Algorithm 1. To do this, we first train a regular classification
network. Next, we collect all the activation vectors and separate them into
$K$ sets, one for each known class. Each of these sets have shape $N_{j}$ by
$K$, where $N_{j}$ is the number of samples belonging to class $j$. We proceed
to remove column $j$ for each set, where $j$ indicates the index of the match
score (line 1 of Algorithm 1).

The next step is to concatenate all of the non-match scores to obtain $\bm{W}$
resulting in $N\cdot(K-1)$ data points (line 2). We employ the FitHigh
function of LibMR [4] to find the parameters of the Weibull distribution using
only the $q$ highest activations among the non-match class activations. LibMR
is an open-scource library for Meta-Recognition, and FitHigh is the function
in LibMR for building Weibull models. Lastly, we pass $\bm{W}$ to FitHigh to
obtain the model $p_{j}$ for class $j$ (line 3).

1:LibMR FitHigh function

2:Activation set $\bm{A}\in\mathcal{R}^{N\times K}$ for inputs
$\bm{X}\in\mathcal{R}^{H\times W\times c}$ belonging to class $j$

3:$q$, the number of highest activations to model

4:Weibull model for class $j$

5:Obtain only non-match scores by removing column $j$ from activation set
$\bm{A}$ to acquire new activation vectors
$\tilde{\bm{A}}\in\mathcal{R}^{N\times(K-1)}$

6:Concatenate all non-match scores to obtain $N\cdot(K-1)$ data points
resulting in $\bm{W}$

7:Weibull Fit $p_{j}=(\rho_{j},\kappa_{j},\lambda_{j})$ = FitHigh$(\bm{W},q)$

Algorithm 1 Weibull calibration for class activations.

We derived a working algorithm by analyzing how non-match activations affect
calibration in MetaMax’s inference step. MetaMax’s inference step (Algorithm
2) uses the per-class Weibull models to make a decision on whether to accept
the network’s SoftMax scores or to reject the input. In lines 3 through 6, we
modify the modulation vector $\bm{m}$, which has been initialized to all ones.
We obtain the key second step of MetaMax by extending OpenMax to allow for the
modeling of non-match activations. The main difference concerns the modulation
vector $\bm{m}$’s construction, where the Weibull cumulative distribution
function is applied to modify the activation vector. Since we do not model the
highest class activation, we effectively remove the influence of the highest
activation on the unknown activation $a_{0}$ by skipping the index associated
with the highest class activation. The consequence of this modification is
that the unknown class activation (defined in line 8) simply adds zero when
index $i$ is equal to the index of the highest class activation. The new
activations are then passed to the standard SoftMax function to produce the
class probabilities (line 9). Finally, a prediction is found and a decision to
accept or reject is made (line 11).

1:$\beta$, the number of activation scores to revise

2:LibMR models $p_{j}$

3:Activation vector $\bm{a}\in\mathcal{R}^{K\times 1}$

4:Obtain sorted indices $\bm{b}=\textrm{argsort}(\bm{a})$

5:for $i=0,\dots,K-1$ do $m_{i}=1$

6:for $i=1,\ldots,\beta$ do

7: if $a_{b_{i}}>a_{j}\forall j\neq i$ then continue

8: else

9:
$m_{\bm{b}_{i}}=1-\frac{\beta-i}{\beta}e^{-(\frac{\bm{a}_{\bm{b}_{i}}}{\lambda_{\bm{b}_{i}}})^{\kappa_{\bm{b}_{i}}}}$

10:Revise activation vector $\hat{\bm{a}}=\bm{a}\cdot\bm{m}$

11:Define $a_{K}=\sum_{i}(a_{i}-a_{i}\cdot m_{i})$

12:$\hat{P}(y=j\,|\,\bm{x})=\frac{e^{a_{j}}}{\sum\limits_{i=0}^{K}e^{a_{i}}}$

13:Let $\hat{y}=\mathop{\mathrm{argmax}}\limits_{j}\hat{P}(y=j\,|\,\bm{x})$

14:if $\hat{y}$ is $K$ then reject input

Algorithm 2 MetaMax probability estimation using non-match activations.

##  4 Experiments

In this section, we present the experimental details for testing MetaMax,
compare it against OpenMax and other recent work, and report the measured
performance on several public datasets. Additionally, to demonstrate the
applicability of MetaMax to enable closed-set classifiers to perform OSR, we
apply and report the performance of MetaMax to the following popular
classification networks: DenseNet [9], ResNet [8], and VGGNet [26].

###  4.1 Experimental Setup

Following experimental setup in [29], we use DenseNet [9] as the backbone
network for classification. We evaluate our method on the following common
benchmark datasets: MNIST [16], SVHN [18], CIFAR10 [13], and TinyImageNet
(TIN) [15]. To test under open-set conditions, we follow the most common data
partition protocol [17]. Specifically, for MNIST, SVHN, and CIFAR10, we split
each dataset at random such that 6 classes are chosen to be known and the
remaining 4 classes to be unknown. For the TinyImageNet dataset, we train on
$K=20$ known classes and test on the full 200-class set. We repeat the
experiment over 5 runs and report the average score. For the evaluation
metrics, we use the area under receiver operating characteristic (AUROC) curve
and the F1-scores.

###  4.2 Results

In the first set of experiments, we demonstrate the effectiveness of MetaMax.
We compare the AUROC score of MetaMax against recent methods in this area. We
also show the multiclass ROC curves of our method on each dataset in the
appendix. The ROC curves demonstrate the magnitude of difficulty with each
dataset.

As shown in Table 1, MetaMax outperforms most recent methods and is on par
with the state of the art. Note that most of these OSR methods either need to
modify the network architecture or require auxiliary training data, which can
be nontrivial to obtain in practice. On the contrary, MetaMax is much simpler
and it can be directly applied post-hoc to a classifier with little-to-no
extra overhead.

In the second set of experiments, to verify the wide applicability of allowing
various classification networks to perform OSR using MetaMax, we apply and
report the F1-scores for two other networks: VGGNet [26] and ResNet [8]. The
results can be found in the appendix, which shows that MetaMax outperforms
OpenMax and the baseline network consistently. This demonstrates the
significance of our work in that MetaMax can potentially be applied to any
classification network and enable it to operate under open-set conditions.

Method | MNIST | SVHN | CIFAR10 | TIN
---|---|---|---|---
G-OpenMax [6] | 0.984 | 0.896 | 0.675 | 0.580
OSRCI [17] | 0.988 | 0.910 | 0.699 | 0.586
CROSR [29] | 0.991 | 0.899 | 0.883 | 0.589
C2AE [19] | 0.989 | 0.922 | 0.895 | 0.748
GDFR [20] | - | 0.935 | 0.807 | 0.608
CGDL [27] | 0.994 | 0.935 | 0.903 | 0.762
OpenHybrid [30] | 0.995 | 0.947 | 0.950 | 0.793
PROSR [32] | - | 0.943 | 0.891 | 0.693
OpenGAN [12] | 0.999 | 0.988 | 0.973 | 0.907
MetaMax (Ours) | 0.997 | 0.977 | 0.938 | 0.846
Table 1: AUROC scores of MetaMax and the compared methods using DenseNet. TIN
stands for TinyImageNet.

##  5 Conclusion

In this work we introduced MetaMax, an approach to calibrate deep neural
network-based classifiers by modeling non-match class activations for OSR.
Experiments on four standard image datasets demonstrate the effectiveness of
the proposed method. As a simpler and more effective alternative to OpenMax,
the modularization and general applicability of our method can have a wide
impact in the community and benefit future research by applying MetaMax to
calibrate standard closed-set classification networks for open-set conditions.

## References

  * [1] Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané.  Concrete problems in ai safety.  arXiv preprint arXiv:1606.06565, 2016.
  * [2] Abhijit Bendale and Terrance Boult.  Towards open world recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1893–1902, 2015.
  * [3] Abhijit Bendale and Terrance E Boult.  Towards open set deep networks.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1563–1572, 2016.
  * [4] Terrance E Boult, Walter J Scheirer, Vijay Iyer, Ethan Rudd, and Manuel Gunther.  LibMR, 2018.  https://pypi.org/project/libmr.
  * [5] Thomas G Dietterich.  Steps toward robust artificial intelligence.  AI Magazine, 38(3):3–24, 2017.
  * [6] Zongyuan Ge, Sergey Demyanov, Zetao Chen, and Rahil Garnavi.  Generative openmax for multi-class open set classification.  In Proceedings of the British Machine Vision Conference. British Machine Vision Association and Society for Pattern Recognition, 2017.
  * [7] Chuanxing Geng, Sheng-jun Huang, and Songcan Chen.  Recent advances in open set recognition: A survey.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020\.
  * [8] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 770–778, 2016.
  * [9] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger.  Densely connected convolutional networks.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4700–4708, 2017.
  * [10] Lalit P Jain, Walter J Scheirer, and Terrance E Boult.  Multi-class open set recognition using probability of inclusion.  In Proceedings of the European Conference on Computer Vision, pages 393–409. Springer, 2014.
  * [11] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  arXiv preprint arXiv:1412.6980, 2014.
  * [12] Shu Kong and Deva Ramanan.  Opengan: Open-set recognition via open data generation.  In Proceedings of the IEEE International Conference on Computer Vision, pages 813–822, 2021.
  * [13] Alex Krizhevsky and Geoffrey Hinton.  Learning multiple layers of features from tiny images.  Technical report, University of Toronto, Toronto, 2009.
  * [14] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Proceedings of the Advances in Neural Information Processing Systems, volume 25, pages 1097–1105, 2012.
  * [15] Ya Le and Xuan Yang.  Tiny imagenet visual recognition challenge.  CS 231N, 7(7):3, 2015.
  * [16] Yann LeCun, Corinna Cortes, and Chris Burges.  Mnist handwritten digit database, 2010.
  * [17] Lawrence Neal, Matthew Olson, Xiaoli Fern, Weng-Keen Wong, and Fuxin Li.  Open set learning with counterfactual images.  In Proceedings of the European Conference on Computer Vision, pages 613–628, 2018.
  * [18] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng.  Reading digits in natural images with unsupervised feature learning.  In Proceedings of the Conference on Neural Information Processing Systems Workshops, 2011.
  * [19] Poojan Oza and Vishal M Patel.  C2ae: Class conditioned auto-encoder for open-set recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2307–2316, 2019.
  * [20] Pramuditha Perera, Vlad I Morariu, Rajiv Jain, Varun Manjunatha, Curtis Wigington, Vicente Ordonez, and Vishal M Patel.  Generative-discriminative feature representations for open-set recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11814–11823, 2020.
  * [21] Nicolas Pinto, David D Cox, and James J DiCarlo.  Why is real-world visual object recognition hard?  PLOS Computational Biology, 4(1):e27, 2008.
  * [22] Walter J Scheirer, Anderson de Rezende Rocha, Jonathan Parris, and Terrance E Boult.  Learning for meta-recognition.  IEEE Transactions on Information Forensics and Security, 7(4):1214–1224, 2012.
  * [23] Walter J Scheirer, Anderson de Rezende Rocha, Archana Sapkota, and Terrance E Boult.  Toward open set recognition.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(7):1757–1772, 2012.
  * [24] Walter J Scheirer, Lalit P Jain, and Terrance E Boult.  Probability models for open set recognition.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(11):2317–2324, 2014.
  * [25] Walter J Scheirer, Anderson Rocha, Ross J Micheals, and Terrance E Boult.  Meta-recognition: The theory and practice of recognition score analysis.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(8):1689–1695, 2011.
  * [26] Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition.  arXiv preprint arXiv:1409.1556, 2014.
  * [27] Xin Sun, Zhenning Yang, Chi Zhang, Keck-Voon Ling, and Guohao Peng.  Conditional gaussian distribution learning for open set recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 13480–13489, 2020.
  * [28] A Torralba and AA Efros.  Unbiased look at dataset bias.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1521–1528, 2011.
  * [29] Ryota Yoshihashi, Wen Shao, Rei Kawakami, Shaodi You, Makoto Iida, and Takeshi Naemura.  Classification-reconstruction learning for open-set recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4016–4025, 2019.
  * [30] Hongjie Zhang, Ang Li, Jie Guo, and Yanwen Guo.  Hybrid models for open set recognition.  In Proceedings of the European Conference on Computer Vision, pages 102–117. Springer, 2020.
  * [31] Peng Zhang, Jiuling Wang, Ali Farhadi, Martial Hebert, and Devi Parikh.  Predicting failures of vision systems.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3566–3573, 2014.
  * [32] Da-Wei Zhou, Han-Jia Ye, and De-Chuan Zhan.  Learning placeholders for open-set recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4401–4410, 2021.

## Appendix

In this appendix, we provide additional experimental details on MetaMax.
First, we describe the training procedure and show results that highlight the
sensitivity of MetaMax to different hyperparameters. Next, we illustrate the
performance differences among SoftMax, OpenMax, and MetaMax by analyzing their
respective ROC curves. We also use scatter plots to illustrate the correlation
between class activations and mean distances. Finally, we demonstrate the
applicability of MetaMax by providing comparisons against SoftMax and OpenMax
on DenseNet [9], ResNet [8], and VGGNet [26].

##  Appendix A Training Parameters

In the first set of experiments where we showed the superiority of MetaMax
over OpenMax, we trained DenseNet121 for 200 epochs with a batch size of 64
for all three datasets. DenseNet121 has a total of 6,960,006 parameters in its
121 layers. Training was performed using the Adam [11] optimization method.
The learning rate decayed by a multiplicative factor $\gamma$ at the end of
each epoch. In the second set of experiments where we demonstrated the
applicability of MetaMax on different classification networks, we conducted
experiments using ResNet and VGGNet following the same experimental protocol.

##  Appendix B Sensitivity to Hyperparameters

The hyperparameter $q$ in Algorithm 1 is passed to the FitHigh function along
with the obtained non-match scores. $q$ controls the number of top non-match
activations we choose to fit the per-class Weibull model during the inference
step. We performed an experiment to show the sensitivity of MetaMax to this
hyperparameter. To do this, we varied the range of $q$ from 2 to 30. As shown
in Table 2, the results are very robust to these changes.

$q$ | F1-Score $\uparrow$ | AUROC $\uparrow$
---|---|---
2 | 0.71107668 | 0.93882572
5 | 0.71114378 | 0.93882158
10 | 0.71109837 | 0.93880752
20 | 0.71141039 | 0.93878481
30 | 0.71089159 | 0.93872475
Table 2: The F1 and AUROC scores of MetaMax using DenseNet on CIFAR-10 with
different values of the hyperparameter $q$.

##  Appendix C ROC Curves

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 2: The multiclass ROC curves for the (a-c) MNIST, (d-f) SVHN, (g-i)
CIFAR10, and (j-l) TinyImageNet datasets. The left, middle, and right columns
illustrate the ROC curves for SoftMax, OpenMax, and MetaMax, respectively.

ROC curves demonstrating the magnitude of difficulty SoftMax, OpenMax, and
MetaMax have with each dataset are shown in Fig. 2. In the first column of
Fig. 2, the SoftMax activation function has steeper ROC curves for all classes
except the unknown class. However, SoftMax’s unknown ROC curve lies directly
on top of the non-discrimination line, which dramatically hurts performance.
OpenMax and MetaMax both have steep ROC curves for the unknown class.
Nevertheless, MetaMax has a steeper curve when compared against OpenMax. In
comparison to SoftMax and OpenMax, this aligns with the higher AUROC scores
for MetaMax as shown in the main paper. The area under each class’s ROC curve
can be interpreted as being inversely related to the difficulty of picking a
sample from that class against all other classes.

##  Appendix D Scatter Plots

(a)

(b)

Figure 3: An illustration of the correlation between class activations and
mean distances for (a) non-match and (b) match activation vectors. Both plots
correspond to the activation vectors associated with class 4 in the SVHN [18]
dataset. Figures (a) and (b) show plots of the activations of class 3 and 4,
respectively, on the x-axis.

Intuitively, class activations offer a measure of similarity between a sample
and its implicitly-stored mean activation vectors in the latent space. We
present empirical evidence of this in Fig. 3 using the SVHN dataset. Fig. 3a
shows a scatter plot for the activation vectors of class 4. The distance of
each activation vector to the mean activation for class 4 is plotted on the
y-axis. On the x-axis are the class activations for class 3. We can see that
as the non-match activation increases, the distance from this sample to the
class 4 descriptor also increases. We believe this correlation justifies the
use of implicit over explicit class descriptors. Furthermore, it suggests that
activations are highly correlated with the similarity between a sample and a
class descriptor. In Fig. 3b we show a plot of class-4 class activations
versus mean distances to the class-4 descriptor.

##  Appendix E ResNet and VGGNet Results

To verify the wide applicability of various classification networks in
performing open-set recognition using MetaMax, we utilize and report the
F1-scores on DenseNet (Table 3), ResNet (Table 4), and VGGNet (Table 5). We
can see from these results that MetaMax consistently outperforms both OpenMax
and the baseline network. This demonstrates the significance of our work in
that MetaMax can be applied to any classification network, hence allowing it
to operate under open-set conditions. Moreover, these results provide evidence
that other open-set recognition methods can achieve performance gains using
MetaMax alongside any backbone network.

Method | MNIST | SVHN | CIFAR10 | TinyImageNet
---|---|---|---|---
SoftMax | 0.644 | 0.682 | 0.547 | 0.512
OpenMax | 0.758 | 0.815 | 0.669 | 0.627
MetaMax (Ours) | 0.813 | 0.846 | 0.711 | 0.683
Table 3: The F1-scores of MetaMax against other methods using DenseNet. Method | MNIST | SVHN | CIFAR10 | TinyImageNet
---|---|---|---|---
SoftMax | 0.637 | 0.684 | 0.542 | 0.510
OpenMax | 0.735 | 0.828 | 0.654 | 0.618
MetaMax (Ours) | 0.801 | 0.854 | 0.696 | 0.669
Table 4: The F1-scores of MetaMax against other methods using ResNet. Method | MNIST | SVHN | CIFAR10 | TinyImageNet
---|---|---|---|---
SoftMax | 0.623 | 0.671 | 0.528 | 0.496
OpenMax | 0.695 | 0.793 | 0.607 | 0.583
MetaMax (Ours) | 0.752 | 0.817 | 0.654 | 0.618
Table 5: The F1-scores of MetaMax against other methods using VGGNet.


* [22] Z. Li, P. Liang, P. Avgeriou, Architectural technical debt identification based on architecture decisions and change scenarios, in: Proeedings of the 12th Working IEEE/IFIP Conference on Software Architecture (WICSA), IEEE, Montreal, QC, Canada, 2015, pp. 65–74.
  * [23] D. Vietz, J. Barzen, F. Leymann, K. Wild, On decision support for quantum application developers: categorization, comparison, and analysis of existing technologies, in: Proceedings of the 21st International Conference on Computational Science (ICCS), Springer, Krakow, Poland, 2021, pp. 127–141.
  * [24] A. Ahmad, A. A. Khan, M. Waseem, M. Fahmideh, T. Mikkonen, Towards process centered architecting for quantum software systems, in: Proceedings of the 1st IEEE International Conference on Quantum Software (QSW), IEEE, Barcelona, Spain, 2022, pp. 26–31.
  * [25] B. Sodhi, R. Kapur, Quantum computing platforms: Assessing the impact on quality attributes and sdlc activities, in: Proceedings of 18th IEEE International Conference on Software Architecture (ICSA), IEEE, Stuttgart, Germany, 2021, pp. 80–91.
  * [26] J. Barzen, F. Leymann, S. Feld, M. Wimmer, 2nd workshop on quantum software architecture (QSA), in: Proceedings of the 19th IEEE International Conference on Software Architecture Companion (ICSA-C), IEEE, Los Alamitos, CA, USA, 2022, pp. 128–128.
  * [27] T. Yue, W. Mauerer, S. Ali, D. Taibi, Challenges and opportunities in quantum software architecture, in: Software Architecture: Recent Trends in Software Architecture, Springer, 2023, pp. 45–52.
  * [28] A. Di Sorbo, S. Panichella, C. A. Visaggio, M. Di Penta, G. Canfora, H. C. Gall, Development emails content analyzer: Intention mining in developer discussions, in: Proceedings of the 30th IEEE/ACM International Conference on Automated Software Engineering (ASE), IEEE, Lincoln, NE, USA, 2015, pp. 12–23.
  * [29] R. W. Grove, An analysis of the constant comparative method, Internation Journal of Qualitative Studies in Education 1 (3) (1988) 273–279.
  * [30] M. S. Aktar, P. Liang, M. Waseem, A. Tahir, A. Ahmad, B. Zhang, Dataset for the paper: Architecture decisions in quantum software systems: An empirical study on stack exchange and github, https://doi.org/10.5281/zenodo.10050560, 2023\.
  * [31] R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics 21 (6/7).
  * [32] D. A. Sofge, A survey of quantum programming languages: History, methods, and tools, in: Proceedings of the 2nd International Conference on Quantum, Nano and Micro Technologies (ICQNM), IEEE, Sainte Luce, Martinique, France, 2008, pp. 66–71.
  * [33] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, Y. Su, Toward the first quantum simulation with quantum speedup, Proceedings of the National Academy of Sciences 115 (38) (2018) 9456–9461.
  * [34] B. Coecke, G. de Felice, K. Meichanetzidis, A. Toumi, Foundations for near-term quantum natural language processing (2020), arXiv preprint arXiv:2012.03755.
  * [35] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, S. Lloyd, Quantum machine learning, Nature 549 (7671) (2017) 195–202.
  * [36] D. Tosh, O. Galindo, V. Kreinovich, O. Kosheleva, Towards security of cyber-physical systems using quantum computing algorithms, in: Proceedings of the 15th International Conference of System of Systems Engineering (SoSE), IEEE, Budapest, Hungary, 2020, pp. 313–320.
  * [37] N. Quetschlich, L. Burgholzer, R. Wille, Mqt bench: Benchmarking software and design automation tools for quantum computing, Quantum 7 (2023) 1062.
  * [38] F. T. Chong, D. Franklin, M. Martonosi, Programming languages and compiler design for realistic quantum hardware, Nature 549 (7671) (2017) 180–187.
  * [39] A. A. Khan, M. A. Akbar, A. Ahmad, M. Fahmideh, M. Shameem, V. Lahtinen, M. Waseem, T. Mikkonen, Agile practices for quantum software development: Practitioners’ perspectives, in: Proceedings of the 2nd IEEE International Conference on Quantum Software (QSW), IEEE, Chicago, IL, USA, 2023, pp. 9–20.
  * [40] M. A. Akbar, A. A. Khan, S. Mahmood, S. Rafi, Quantum software engineering: A new genre of computing (2022), arXiv preprint arXiv:2211.13990.
  * [41] S. Ali, T. Yue, R. Abreu, When software engineering meets quantum computing, Communications of the ACM 65 (4) (2022) 84–88.
  * [42] M. A. Akbar, A. A. Khan, S. Rafi, A systematic decision-making framework for tackling quantum software engineering challenges, Automated Software Engineering 30 (2) (2023) 22.
  * [43] S. Ducasse, D. Pollet, Software architecture reconstruction: A process-oriented taxonomy, IEEE Transactions on Software Engineering 35 (4) (2009) 573–591.
  * [44] A. Alreshidi, A. Ahmad, Architecting software for the internet of thing based systems, Future Internet 11 (7) (2019) 153.
  * [45] H. Muccini, K. Vaidhyanathan, Software architecture for ml-based systems: What exists and what lies ahead, in: Proceedings of 1st IEEE/ACM Workshop on AI Engineering - Software Engineering for AI (WAIN), IEEE, Madrid, Spain, 2021, pp. 121–128.
  * [46] M. Waseem, P. Liang, A. Ahmad, M. Shahin, A. A. Khan, G. Márquez, Decision models for selecting patterns and strategies in microservices systems and their evaluation by practitioners, in: Proceedings of the 44th IEEE/ACM International Conference on Software Engineering: Software Engineering in Practice (ICSE-SEIP), Pittsburgh, PA, USA, 2022, pp. 135–144.
  * [47] M. Shahin, P. Liang, M. R. Khayyambashi, Architectural design decision: Existing models and tools, in: Proceedings of the Joint 8th Working IEEE/IFIP Conference on Software Architecture and 3rd European Conference on Software Architecture (WICSA/ECSA), IEEE, Pittsburgh, PA, USA, 2009, pp. 293–296.
  * [48] A. A. Khan, M. A. Akbar, A. Ahmad, M. Fahmideh, M. Shameem, V. Lahtinen, M. Waseem, T. Mikkonen, Agile practices for quantum software development: Practitioners perspectives (2022), arXiv preprint arXiv:2210.09825.
  * [49] V. R. B. G. Caldiera, H. D. Rombach, The goal question metric approach, Encyclopedia of Software Engineering (1994) 528–532.
  * [50] S. Easterbrook, J. Singer, M.-A. Storey, D. Damian, Selecting empirical methods for software engineering research, in: Guide to Advanced Empirical Software Engineering, Springer, 2008, pp. 285–311.
  * [51] V. Cosentino, J. L. C. Izquierdo, J. Cabot, A systematic mapping study of software development with github, IEEE Access 5 (2017) 7173–7192.
  * [52] A. Barua, S. W. Thomas, A. E. Hassan, What are developers talking about? an analysis of topics and trends in stack overflow, Empirical Software Engineering 19 (2014) 619–654.
  * [53] H. Li, F. Khomh, M. Openja, et al., Understanding quantum software engineering challenges an empirical study on stack exchange forums and github issues, in: Proceedings of the 37th IEEE International Conference on Software Maintenance and Evolution (ICSME), IEEE, Luxembourg, 2021, pp. 343–354.
  * [54] M. Waseem, P. Liang, M. Shahin, A. Ahmad, A. R. Nassab, On the nature of issues in five open source microservices systems: An empirical study, in: Proceedings of 25th International Conference on Evaluation and Assessment in Software Engineering (EASE), ACM, Trondheim, Norway, 2021, pp. 201–210.
  * [55] J. Di Rocco, D. Di Ruscio, C. Di Sipio, P. Nguyen, R. Rubei, Topfilter: an approach to recommend relevant github topics, in: Proceedings of the 14th ACM/IEEE International Symposium on Empirical Software Engineering and Measurement (ESEM), ACM, Bari, Italy, 2020, pp. 1–11.
  * [56] J. Cohen, A coefficient of agreement for nominal scales, Educational and Psychological Measurement 20 (1) (1960) 37–46.
  * [57] A. R. da Silva, D. Savić, Linguistic patterns and linguistic styles for requirements specification: Focus on data entities, Applied Sciences 11 (9) (2021) 4119.
  * [58] L. R. Hallberg, The “core category” of grounded theory: Making constant comparisons, International Journal of Qualitative Studies on Health and Well-being 1 (3) (2006) 141–148.
  * [59] J. L. Campbell, C. Quincy, J. Osserman, O. K. Pedersen, Coding in-depth semistructured interviews: Problems of unitization and intercoder reliability and agreement, Sociological Methods & Research 42 (3) (2013) 294–320.
  * [60] A. Koziolek, Automated improvement of software architecture models for performance and other quality attributes, Vol. 7, KIT Scientific Publishing, 2014\.
  * [61] P. Ralph, E. Tempero, Construct validity in software engineering research and software metrics, in: Proceedings of the 22nd International Conference on Evaluation and Assessment in Software Engineering (EASE), ACM, Christchurch, New Zealand, 2018, pp. 13–23.
  * [62] A. R. Nasab, M. Shahin, S. A. H. Raviz, P. Liang, A. Mashmool, V. Lenarduzzi, An empirical study of security practices for microservices systems, Journal of Systems and Software 198 (2023) 111563.
  * [63] F. Tian, P. Liang, M. A. Babar, How developers discuss architecture smells? an exploratory study on stack overflow, in: Proceedings of the 16th IEEE International Conference on Software Architecture (ICSA), IEEE, Hamburg, Germany, 2019, pp. 91–100.
  * [64] F. Truger, J. Barzen, M. Bechtold, M. Beisel, F. Leymann, A. Mandl, V. Yussupov, Warm-starting and quantum computing: A systematic mapping study (2023), arXiv preprint arXiv:2303.06133.
  * [65] M. Haghparast, T. Mikkonen, J. K. Nurminen, V. Stirbu, Quantum software engineering challenges from developers’ perspective: Mapping research challenges to the proposed workflow model (2023), arXiv preprint arXiv:2308.01141.
  * [66] M. Felderer, D. Taibi, F. Palomba, M. Epping, M. Lochau, B. Weder, Software engineering challenges for quantum computing: Report from the first working seminar on quantum software engineering (WSQSE 22), ACM SIGSOFT Software Engineering Notes 48 (2) (2023) 29–32.
  * [67] P. Zhao, Z. Miao, S. Lan, J. Zhao, Bugs4q: A benchmark of existing bugs to enable controlled testing and debugging studies for quantum programs, Journal of Systems and Software 205 (2023) 111805.
  * [68] X. Li, P. Liang, T. Liu, Decisions and their making in oss development: An exploratory study using the hibernate developer mailing list, in: Proceedings of the 26th Asia-Pacific Software Engineering Conference (APSEC), IEEE, Putrajaya, Malaysia, 2019, pp. 323–330.
  * [69] T. Bi, P. Liang, A. Tang, Architecture patterns, quality attributes, and design contexts: How developers design with them, in: Proceedings of the 25th Asia-Pacific Software Engineering Conference (APSEC), IEEE, Nara, Japan, 2018, pp. 49–58.
  * [70] B. Zhang, T. Liu, P. Liang, C. Wang, M. Shahin, J. Yu, Architecture decisions in ai-based systems development: An empirical study, in: Proceedings of the 30th IEEE International Conference on Software Analysis, Evolution and Reengineering (SANER), IEEE, Taipa, Macao, 2023, pp. 616–626.
  * [71] M. Shahin, P. Liang, M. R. Khayyambashi, Improving understandability of architecture design through visualization of architectural design decision, in: Proceedings of the 5th Workshop on Sharing and Reusing Architectural Knowledge (SHARK), ACM, Cape Town, South Africa, 2010, pp. 88–95.
  * [72] T. Liu, P. Liang, C. Yang, Z. Xiong, C. Wang, R. Li, Understanding the decision-making of students in requirements engineering course projects, in: Proceedings of the 1st Software Engineering Education Workshop (SEED) co-located with the 26th Asia-Pacific Software Engineering Conference (APSEC), CEUR-WS.org, Putrajaya, Malaysia, 2019, pp. 1–8.