# Radiation from Particles Accelerated in Relativistic Jet Shocks and Shear-
flows

K.-I. Nishikawa  Department of Physics, University of Alabama in Huntsville,
ZP12, Huntsville, AL 35899, USA  P. Hardee  Department of Physics and
Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA  I. Dutan
Institute of Space Science, Atomistilor 409, Bucharest-Magurele RO-077125,
Romania  B. Zhang  Department of Physics, University of Nevada, Las Vegas, NV
89154, USA  A. Meli  Department of Physics and Astronomy, University of Gent,
Proeftuinstraat 86 B-9000, Gent, Belgium  E. J. Choi, and K. Min  Korea
Advanced Institute of Science and Technology, Daejeon 305-701, South Korea  J.
Niemiec  Institute of Nuclear Physics PAN, ul. Radzikowskiego 152, 31-342
Kraków, Poland  Y. Mizuno  Institute for Theoretical Physics, University of
Frankfurt am Main, 60438, Germany  M. Medvedev  Department of Physics and
Astronomy, University of Kansas, KS 66045, USA  Å Nordlund, and J. T.
Frederiksen  Niels Bohr Institute, University of Copenhagen, Juliane Maries
Vej 30, 2100 Copenhagen , Denmark  H. Sol  LUTH, Observatore de Paris-Meudon,
5 place Jules Jansen, F-92195 Meudon Cedex, France  M. Pohl  Institut fur
Physik und Astronomie, Universitäat Potsdam, D-14476 Potsdam-Golm, Germany;
DESY, Platanenallee 6, D-15738 Zeuthen, Germany  D. Hartmann  Department of
Physics and Astronomy, Clemson University, Clemson, SC 29634, USA

###### Abstract

We have investigated particle acceleration and emission from shocks and shear
flows associated with an unmagnetized relativistic jet plasma propagating into
an unmagnetized ambient plasma. Strong electro-magnetic fields are generated
in the jet shock via the filamentation (Weibel) instability. Shock field
strength and structure depend on plasma composition (($e^{\pm}$ or $e^{-}$\-
$p^{+}$ plasmas) and Lorentz factor. In the velocity shear between jet and
ambient plasmas, strong AC ($e^{\pm}$ plasmas) or DC ($e^{-}$\- $p^{+}$
plasmas) magnetic fields are generated via the kinetic Kelvin-Helmholtz
instability (kKHI), and the magnetic field structure also depends on the jet
Lorentz factor. We have calculated, self-consistently, the radiation from
electrons accelerated in shock generated magnetic fields. The spectra depend
on the jet’s initial Lorentz factor and temperature via the resulting particle
acceleration and magnetic field generation. Our ongoing “Global” jet
simulations containing shocks and velocity shears will provide us with the
ability to calculate and model the complex time evolution and/or spectral
structure observed from gamma-ray bursts, AGN jets, and supernova remnants.

##  I Introduction

Blazars and Gamma-Ray Bursts (GRBs) are the most luminous phenomena in the
universe. Despite extensive observational and theoretical programs, our
understanding of the physics remains quite limited. There is broad consensus
that both are powered by relativistic jets, which are directly imaged by
interferometry in the case of blazars, and that the jets are launched and
collimated mainly by magnetic forces e.g., Vlah03 ; mck06 ; komisetal07 ;
tchmn08 . However, there is uncertainty regarding details such as (1) magnetic
versus kinetic domination, (2) rapid acceleration of particles to GeV and TeV
energies, (3) location of highly variable gamma-ray emission and (4) source of
seed photons if inverse Compton, (5) the scale of magnetic field turbulence in
the radiation zone(s), and (6) the role of large and small scale instabilities
in jet structure, dynamics, magnetization, particle energization and
radiation.

For blazars, the observational data are now quite rich, with dense time
sampling of flux at many frequencies from radio to GeV and TeV $\gamma$-ray,
linear polarization at radio to optical, and images with sub-parsec linear
resolution at mm wavelengths e.g., Krawcznski (2007); Narayan, & Piran (2012).
For GRBs, the basic measurements of $\gamma$-ray and X-ray flux vs. time
during the burst have been supplemented by observations of the afterglow at
soft X-ray, optical, IR, and radio frequencies. What is missing is a
comprehensive theoretical framework for interpreting this wealth of
observational data. There are numerous studies that consider only radiative
processes, instabilities, or particle acceleration (see Hardee (2010)), and a
smaller number that pair the first with one of the latter two e.g., Dexter,
McKinney, &Agol (2005a). While separating analyses into soluble parts is a
valuable technique, in GRB and AGN jets the dynamics, instabilities, and
energy gains and losses are coupled processes. Here we present recent progress
in shock and velocity shear simulations using a relativistic particle-in-cell
(RPIC) code.

##  II Shock Simulations

.pdf

###  II.1 Particle Acceleration and Magnetic Field Generation

RPIC simulations have been used to study particle acceleration, magnetic field
generation, and emission from collisionless shocks. Simulations reveal that
the filamentation (Weibel) instability, which generates magnetically wrapped
current filaments, dominates relativistic shock processes Dieckmann et al.
(2006). There are significant differences between electron-positron pair and
electron-ion shocks as the ion filamentation instability enhances shock
magnetic field generation and thermal energy density relative to pair plasmas
e.g., Nishikawa et al. (2003, 2005, 2006, 2007, 2009a); Frederiksen et al.
(2003, 2004); Hededal & Nishikawa (2005); Ramirezet et al. (2007); Spitkovsky
(2005, 2008).

Our 3-D MPI parallelized RPIC code has been used to simulate relativistic
electron-ion jet propagation into an unmagnetized ambient electron-ion plasma
($m_{\rm i}/m_{\rm e}=16$) with equal jet and ambient electron number density,
and jet thermal velocity $v^{\rm e}_{\rm j,th}=0.2c$ where $c$ is the speed of
light, and the jet Lorentz factor is $\gamma=15$. The simulation used a system
with dimension ($L_{\rm x},L_{\rm y},L_{\rm z})=(8192\Delta,$
$64\Delta,64\Delta)$, where $\Delta$ is the cell size, and a total of $\sim 1$
billion particles (16 particles$/$cell$/$species for the ambient plasma) Choi,
Min, & Nishkawa (2014). This computational domain is twelve times longer than
in our previous simulations (Nishikawa et al. (2006); Ramirezet et al.
(2007)).

Figure 1 shows the averaged (in the $y-z$ plane) ion density and
electromagnetic field energy along the electron-ion jet at simulation time
$t=7372\omega_{\rm pe}^{-1}$.

Figure 1: Panel (a) shows the ion density (black: total, red: ambient, and
blue: jet), and panel (b) shows the magnetic field components at the end of
the simulation ($T=7372/\omega_{\rm pe}$). Electromagnetic field energy
densities are normalized by the jet kinetic energy density, and quantities are
averaged over the $yz-$plane.

The resulting profiles of jet (red), ambient (blue), and total (black) ion
density are shown in Figure 1a. Warm jet ions are thermalized and ambient
electrons are accelerated in the resulting leading (bow) and trailing (jet)
shock system. Ambient ions are accelerated and pile up towards the jet front.
The ambient plasma density increases behind the jet front, with additional
increase to a higher plateau farther behind the jet front indicating the
leading shock. The jet ion density remains approximately constant. The
strongest electromagnetic fields are located at $x/\Delta=4,500$ as shown in
Figure 1b and are associated with the trailing shock. These strong fields may
lead to the observed time dependent GRB afterglow emission. The longer
simulation system has allowed significant non-linear shock Weibel instability
and associated particle acceleration development.

###  II.2 Self-Consistent Synthetic Spectra

We have calculated the radiation spectra directly from our simulations by
integrating the expression for the retarded power, derived from Liénard-
Wiechert potentials, for a large number of representative particles in the PIC
representation of the plasma Jackson (1999); Hededal (2005); Nishikawa et al.
(2009b, 2010a, 2010b, 2010c, 2011); Martins et al. (2009); Sironi & Spitkovsky
(2009); Frederiksen et al. (2010); Medvedev et al. (2011).

The synthetic spectra shown in Figure 2a are obtained for emission from
electrons in jets with Lorentz factors of $\gamma=$ 10, 20, 50, 100, and 300.
Spectra are obtained from an ensemble of electrons selected from the region
where the Weibel instability, particle acceleration, and magnetic field
generation are strongest.

Figure 2: Panel (a) shows synthetic spectra for jets with Lorentz factors of
$\gamma=$ 10, 20, 50, 100, and 300 and cold (thin lines) or warm (thick lines)
jet electrons. Panel (b) shows modeled Fermi spectra in $\nu F_{\nu}$ units at
early (a) to late (e) times Abdo et al. (2009). Straight red lines indicate a
slope of $\nu F_{\nu}=1$.

For each jet Lorentz factor we computed spectra for cold, $v_{\rm
jet,th}=0.01c$, (thin lower lines) and warm, $v_{\rm jet,th}=0.1c$, (thick
upper lines) jet electrons Nishikawa et al. (2011b, 2012a, 2012b, 2012c). Here
the spectra are calculated for radiation beamed along the jet axis. We note
that radiation losses are not included Jaroschek & Hoshino (2009); Medvedev &
Spitkovsky (2009).

Synthetic spectra are Bremsstrahlung-like at low frequencies (Hededal (2005))
because the magnetic fields generated by the Weibel instability are weak and
electron acceleration is modest. Synthetic spectra low frequency slopes are
very similar to those of the spectra shown in Fig. 2b from Abdo et al. (2009).
Comparison between our synthetic spectra and the spectra from Abdo et al.Abdo
et al. (2009) suggest that the spectral evolution observed from early to late
times is mimicked by our synthetic spectra evolution from higher to lower jet
Lorentz factor. However, further investigation is necessary and this is one of
our future research efforts.

##  III Velocity Shear Simulations

###  III.1 Slab Jet Velocity Shear

In this simulation study we used a core-sheath plasma jet structure instead of
the counter-streaming plasma setups used in previous simulations by Alves
(2010); Alves et al. (2012, 2014); Grismayer et al. (2013a, b); Liang et al.
(2013a, b). The basic setup and illustrative results are shown in Figure 3. In
our setup, a jet core with velocity $v_{\rm core}$ in the positive $x$
direction resides in the middle of the computational box (see Figure 3a). The
upper and lower quarters of the box contain a sheath plasma that can be
stationary or moving with velocity $v_{\rm sheath}$ Nishikawa et al. (2013a,
b, 2014a, 2014b). This setup is similar to that in our RMHD simulations
(Mizuno et al. (2007)) that used a cylindrical jet core. Overall, this
structure is similar in spirit, although not in scale to that proposed for AGN
relativistic jet cores surrounded by a slower moving sheath, and is also
relevant to GRB jets. However, here we represent the jet core and sheath as
plasma slabs. Initially, the system is charge and current neutral.

The simulations were performed using a numerical grid with dimension $(L_{\rm
x},L_{\rm y},L_{\rm z})=(1005\Delta,205\Delta,205\Delta)$, where $\Delta$ is
the cell size, and periodic boundary conditions in all directions. The jet and
sheath (electron) plasma number density measured in the simulation frame is
$n_{\rm jt}=n_{\rm am}=8$. The electron skin depth, $\lambda_{\rm
s}=c/\omega_{\rm pe}=12.2\Delta$, where $\omega_{\rm pe}=(e^{2}n_{\rm
am}/\epsilon_{0}m_{\rm e})^{1/2}$ is the electron plasma frequency and the
electron Debye length for the ambient electrons $\lambda_{\rm D}$ is
$1.2\Delta$. The jet-electron thermal velocity is $v_{\rm jt,th,e}=0.014c$ in
the jet reference frame, where $c$ is the speed of light. The electron thermal
velocity in the ambient plasma is $v_{\rm am,th,e}=0.03c$, and ion thermal
velocities are smaller by $(m_{\rm i}/m_{\rm e})^{1/2}$. Simulations were
performed using an electron-positron ($e^{\pm}$) plasma or an electron-proton
($e^{-}$\- $p^{+}$ with $m_{\rm p}/m_{\rm e}=1836$) plasma for jet Lorentz
factors of 1.5, 5.0, and 15.0 with the sheath plasma at rest ($v_{\rm
sheath}=0$) Nishikawa et al. (2014a).

Figure 3: Panel (a) shows our three-dimensional simulation setup. Panels (b) &
(c) show the magnetic field component $B_{\rm y}>0$ (red) and $B_{\rm y}<0$
(blue) plotted in the $x-z$ plane (jet flow indicated by large arrows) at the
center of the simulation box, $y=100\Delta$ at $t=300\,\omega_{\rm pe}^{-1}$,
(b) for the $e^{-}$\- $p^{+}$ case and (c) for the $e^{\pm}$ case, both with
$\gamma_{\rm jt}=15$. The smaller arrows indicate the magnetic field direction
in the plane. Panels (b) & (c) cover one fifth of the simulation system length
in the $x$ direction.

The development of the velocity shear surfaces is shown in Figure 3b for
$e^{-}$\- $p^{+}$ and Figure 3c for $e^{\pm}$ plasmas with $v_{\rm
core}=0.9978c~{}(\gamma_{\rm jt}=15)$. For the $e^{-}$\- $p^{+}$ case, a
nearly DC magnetic field is generated at the shear-surfaces with negative
(blue) $B_{\rm y}$ at $z=150\Delta$ and positive (red) $B_{\rm y}$ at
$z=50\Delta$. Additionally, a $B_{\rm z}$ (and $B_{\rm x}$) magnetic field
component, shown by the small arrows in Figure 3b, is generated at the shear
surfaces by current filaments. On the other hand, for the $e^{\pm}$ case a
relatively long wavelength ($\sim 100\Delta$) AC magnetic field is generated
at the shear surfaces. Note the alternating $B_{\rm y}>0$ (red) and $B_{\rm
y}<0$ (blue) along the flow direction. While our results are similar to those
found by Liang et al. (2013a, b), there are significant important structural
differences because their simulations were two-dimensional and used a counter-
streaming setup.

###  III.2 Cylindrical Jet Velocity Shear

Since relativistic jets and internal filamentary structures are more suitably
modeled as intrinsically cylindrical, we have investigated velocity shear in
cylindrical geometry for a pair ($e^{\pm}$) and an electron-proton ($e^{-}$\-
$p^{+}$) jet. Figure 4 shows isocontour images of the $x$ component of the
current along with magnetic field lines generated by the kKHI for $e^{\pm}$
and $e^{-}$\- $p^{+}$ jets.

Figure 4: Isocontour plots of the $J_{\rm x}$ magnitude with magnetic filed
lines (one fifth of the jet size) for (a) an $e^{-}$\- $p^{+}$ and (b) an
$e^{\pm}$ jet at simulation time $t=300\Omega_{\rm pe}^{-1}$. The 3D displays
are clipped along the jet and perpendicular to the jet in order to view the
interior.

The isocontour images show that in the $e^{-}$\- $p^{+}$ jet case currents are
generated in sheet like layers and the magnetic fields are wrapped around the
jet. On the other hand, in the $e^{\pm}$ jet case many distinct current
filaments are generated near the velocity shear and the individual current
filaments are wrapped by the magnetic field. The clear difference in the
magnetic field structure between these two cases may make it possible to
distinguish different jet compositions via differences in circular and linear
polarization.

##  IV A Combined Shock and Velocity Shear Simulation

We have begun “global” simulations involving injection of a cylindrical jet
into an ambient plasma in order to investigate shock (Weibel instability) and
velocity shear (kKHI) simultaneously. Previously these two processes have been
investigated separately. In reality a jet or internal filament is injected
into an ambient plasma resulting in velocity shear and shocks in a potentially
complicated shock/shear system.

In order to begin investigation of the combined processes we have performed a
simulation where a relativistic cylincrical jet is injected into an ambient
plasma. We used a small system size of $(L_{\rm x},L_{\rm y},L_{\rm
z})=(1005\Delta,131\Delta,131\Delta$) with jet radius $r_{\rm j}=20\Delta$ and
Lorentz factor $\gamma_{\rm jt}=5$ to examine the fundamental differences
between $e^{\pm}$ and $e^{-}$\- $p^{+}$ jet cases and to test synthetic
spectra computations. Previous synthetic spectra computations can be found in
Nishikawa et al. (2011, 2012a, 2012b, 2012c, 2013a, 2013b).

###  IV.1 2D Density, Current and Magnetic Field Structure

Figure 5 shows 2D mid-plane slices of the electron density and the transverse
magnetic field. Current filaments at the jet front are excited by fast
current-driven instability in the shock precursor Lemoine et al. (2014a, b).

Figure 5: Mid-plane slices of the electron density for jet Lorentz factor
$\gamma_{\rm jt}=5$ at simulation time $t=500\,\omega_{\rm pe}^{-1}$. The jet
is injected at $x/\Delta=100$, propagates to the right, and the jet front is
located at $x/\Delta=600$. The upper panel (a) shows the electron density
structure for the mass ratio $m_{\rm i}/m_{\rm e}=1836$, and the lower panel
(b) for the electron-positron case. Associated current structures are shown in
Figure 6.

Figure 6: Slices at $x/\Delta=480$ (left column) and at $x/\Delta=580$ (right
column) at simulation time $t=500\,\omega_{\rm pe}^{-1}$ showing curents and
magnetic fields for ((a) and (b)) mass ratio $m_{\rm i}/m_{\rm e}=1836$ and
for ((c) and (d)) the electron-positron case. Jets come out of the page at
plane center.

Comparing the electron-proton and electron-positron cases reveals significant
electron density structure differences. Jet electrons and protons remain
within the jet in the electron-proton case, but are found outside the jet in
the electron-positron case due to mixing with ambient electrons and positrons
as the positrons have more mobility than heavy protons.

Our previous simulations of the Weibel instability (e.g., Nishikawa et al.
(2009a)), showed current filaments associated with the growing instability and
Figure 6 shows the structure of current filaments in cross sections of the jet
at two locations. For the electron-proton case the current filaments lie
within the jet and a negative current is dominant outside the jet (as in a
previous simulation shown in Fig. 7a and 7b in Nishikawa et al.Nishikawa et
al. (2014a). However, for the electron-positron case, current filaments are
found both inside and outside the jet. In particular, large current filaments
can be seen at $x/\Delta=480$ outside the jet, similar to what was observed in
the slab model (Nishikawa et al.Nishikawa et al. (2013a, b, 2014a, 2014b)).

###  IV.2 3D Current and Magnetic Field Structure

Figure 7 shows 3D current filament isosurfaces along with magnetic field lines
that are generated by the Weibel instability and by the kKHI. Only the front
part of the jet is displayed ($120<x/\Delta<520$) and the jet is propagating
from back left to forward right. The cube is

Figure 7: Global jet simulations for $e^{-}$\- $p^{+}$ and $e^{\pm}$ jets at
time $t=500\Omega_{\rm pe}^{-1}$ . Panel (a) shows the $e^{-}$\- $p^{+}$ jet
where long and continuous current filaments are confined within the jet. Panel
(b) shows the $e^{\pm}$ jet where short current filaments are found within and
outside the jet.

clipped at $x/\Delta=320$ and $y/\Delta=66$ in order to view cross sections
parallel and perpendicular to the jet axis. For both cases, compact current
filaments are confined mostly within the jet at the jet front (see also Figure
6b and 6d). The magnetic fields wrap around the current filaments. Since the
simulation system is small, current structures are at an early stage of
development, and a fully developed shock system is not yet formed at the jet
front. In the $e^{-}$\- $p^{+}$ case long and continuous current filaments
remain confined within the jet along the velocity shear surface behind the jet
front(Fig. 7a). For the $e^{\pm}$ case short current filaments are found
within and outside the jet along the velocity shear surface behind the jet
front (Fig. 7b). This presence of current filaments inside and outside the jet
in the $e^{\pm}$ case is also observed in slab geometry (see Fig. 3a and
Nishikawa et al.Nishikawa et al. (2014a)), where current filaments were found
farther outside the jet at smaller jet Lorentz factors.

Small system simulations cannot fully distinguish between current filaments
generated by the Weibel instability and those generated by the kKHI, and
velocity shear effects are insufficiently resolved. Larger simulations need to
be performed to clearly reveal the differences between the two cases, e.g.,
Nishikawa et al.Nishikawa et al. (2009a) and Choi et al.Choi, Min, & Nishkawa
(2014), and allow proper evaluation of emission from the jet boundary.

##  V Future Work: Helical Magnetic fields and Reconnection

Our preliminary 3D global structure simulations indicate the importance of
using global simulations to investigate the combined evolution of
collisionless shocks and velocity shears. In order to resolve collisionless
shock and velocity shear structures we will perfume global jet simulations
using much larger systems with $(L_{\rm x},L_{\rm y},L_{\rm
z})=(8005\Delta,405\Delta,405\Delta$). Additionally, a very large system is
needed in order to scale kinetic processes to larger scale jet structures. A
larger system will allow us to obtain a much more reasonable picture of the
evolution of the magnetic field and subsequently the dependence of spectra and
polarization on jet composition through synthetic emission computations.

Recently, Markidis et al.Markidis et al. (2014) performed three-dimensional
PIC simulations of a flux rope instability using a single flux rope with a
simple screwpinch configuration Freidberg (2007), i.e., the helical magnetic
field has a constant pitch (e.g., Mizuno et al. (2012)). An artificial ion to
electron mass ratio equal to 25 was chosen to reduce the simulation execution
time, and the initial current was carried by the electrons (ions were
initially stationary). The simulation revealed magnetic reconnection during
the kink instability of the flux rope. Secondary signatures of magnetic
reconnection included a quadrupolar structure in the density, a bipolar
structure in the Hall field, and a reconnection associated electric field in
proximity to the reconnection region.

In our future work we will inject jets like those shown in Figure 7 but
containing a helical magnetic field like that implemented in Markidis et al.
Markidis et al. (2014) and using a setup like that used in Wieland et al.
Wieland et al. (2014). This setup avoids transient phenomena due to an
infinitely sharp contact discontinuity at the colliding front and avoids an
artificial magnetic field pileup. In our setup we will generate a helical
magnetic field via faster jet ions (protons or positrons) instead of
electrons. This configuration will allow investigation of (1) the effect of
helical magnetic field on growth of the Weibel instability and the kKHI, (2)
the possible development of MHD-like kink and/or global KHI, and (3) the
development of magnetic reconnection.

##  VI Conclusions

We have presented 3D PIC simulations of collisionless shock and velocity shear
development mediated by the Weibel instability and the kKHI for both electron-
positron and electron-ion plasmas. The processes studied here are important in
AGN and GRB jets that are expected to have shocks and velocity shears between
faster and slower moving plasmas both within the jet and at the jet external
medium interface.

We have shown via shock simulations that shock structure depends on the
composition of the plasma, e.g., Choi et al.Choi, Min, & Nishkawa (2014) and
Nishikawa et al.Nishikawa et al. (2009a). The collisionless electron-ion
($m_{\rm i}/m_{\rm e}=16$) shock shows a sharper rise in the electron density
at the forward shock than the electron-positron case (see Fig.1a and compare
to Fig. 1a in Nishikawa et al.Nishikawa et al. (2009a)). This sharper rise
occurs because in the electron-positron case jet electrons propagate through
the forward shock to the jet front but in the electron-ion case the jet
electron density declines in front of the forward shock. This decline in the
electron-ion case is due to the ambipolar electric fields created at the jet
front by the heavier ions.

We have shown via velocity shear simulations that velocity shear structure
depends on the composition of the plasma and the jet Lorentz factor, e.g.,
Nishikawa et al.Nishikawa et al. (2013b, 2014a, 2014b). The growth rate for
the kKHI for the mildly relativistic jet case ($\gamma_{\rm j}=1.5$) is larger
than the relativistic jet case ($\gamma_{\rm j}=15$). In particular, the
different magnetic field velocity shear structure associated with electron-
positron composition versus electron-proton composition should have
consequences for the polarization of jets in very high-resolution radio
imaging. For a simple cylindrical geometry velocity shear case an electron-
proton jet primarily builds magnetic field in the toroidal direction at the
velocity shear surface. In contrast, a pair-plasma jet generates sizable
radial field components that are only about a factor of two weaker than the
toroidal field. In either case, strong electric and magnetic fields in the
velocity shear zone will also be conducive to particle acceleration.

When global jet injection simulations are performed the combination of shock
and velocity shear structures makes the situation more complicated but our
preliminary simulations show clear differences between electron-positron and
electron-proton plasmas. Our preliminary simulations are too short for
definitive statements on the efficacy of the process and the resulting
spectra. However, it is clear that the magnetic field structure along with
particle acceleration and transport in compact regions will be necessary for a
realistic assessment and interpretation of observed emission spectra and
polarization.

###### Acknowledgements.

K.-I. Nishikawa and/or P. Hardee have been supported by NSF awards AST-0908010
and AST-0908040, and by NASA awards NNG05GK73G, NNX07AJ88G, NNX08AG83G,
NNX08AL39G, NNX09AD16G, NNX12AH06G, NNX13A P-21G, and NNX13AP14G. J. Niemiec
has been supported by the Polish National Science Centre via awards
DEC-2011/01/B/ST9/03183 and DEC-2012/04/A/ST9/00083. Y. Mizuno has been
supported by the Ministry of Science and Technology of Taiwan via awards NSC
100-2112-M-007-022-MY3 and MOST 103-2112-M-007-023-MY3. M. Pohl is supported
via award PO 1508/1-2 from the Deutsche Forschungsgemeinschaft. Simulations
were performed on Columbia and Pleiades at the NASA Advanced Supercomputing
(NAS), Kraken and Nautilus at The National Institute for Computational
Sciences (NICS), and Stampede at The Texas Advanced Computing Center. This
research was begun during the program “Chirps, Mergers and Explosions: The
Final Moments of Coalescing Compact Binaries” at the Kavli Institute for
Theoretical Physics, which is supported by the NSF via grant PHY05-51164.
Initial kKHI work began at the Aspen Center for Physics workshop
“Astrophysical Mechanisms of Particle Acceleration and Escape from the
Accelerators” (September 1-15, 2013).

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11institutetext: ABBYY, Ottawa, Canada
22institutetext: Chair of Process and Data Science, RWTH Aachen University,
Aachen, Germany
33institutetext: XPAN Law Partners, Boston, USA

# Trustworthy Artificial Intelligence and Process Mining: Challenges and
Opportunities

Andrew Pery  11 Majid Rafiei  22 Michael Simon  33 Wil M.P. van der Aalst  22

###### Abstract

The premise of this paper is that compliance with Trustworthy AI governance
best practices and regulatory frameworks is an inherently fragmented process
spanning across diverse organizational units, external stakeholders, and
systems of record, resulting in process uncertainties and in compliance gaps
that may expose organizations to reputational and regulatory risks. Moreover,
there are complexities associated with meeting the specific dimensions of
Trustworthy AI best practices such as data governance, conformance testing,
quality assurance of AI model behaviors, transparency, accountability, and
confidentiality requirements. These processes involve multiple steps, hand-
offs, re-works, and human-in-the-loop oversight. In this paper, we demonstrate
that process mining can provide a useful framework for gaining fact-based
visibility to AI compliance process execution, surfacing compliance
bottlenecks, and providing for an automated approach to analyze, remediate and
monitor uncertainty in AI regulatory compliance processes.

###### Keywords:

AI ethics Fairness Artificial intelligence Trust mining Process mining

##  1 Introduction

AI-based technologies are becoming pervasive, impacting virtually every facet
of our lives. While AI has a lot of promise, not all of its impacts are good.
There is growing evidence that AI models can embed human and societal biases
and deploy them at scale. As such, the ever-increasing growth of AI highlights
the vital importance of balancing AI utility with the fairness of outcomes,
thereby engender a culture of trustworthy AI. Fairness is the foundation for
Trustworthy AI. Intuitively, fairness seems like a simple concept. However, it
embodies consideration of a number of dimensions, such as trade-offs between
algorithmic accuracy versus human values, demographic parity versus policy
outcomes and power-focused questions such as who gets to decide what is fair.

These are vexing challenges for AI developers, policy-makers and consumers
alike. For AI developers, clarity of what constitutes AI fairness is a key
consideration given the juxtaposition of ethical, legal, and reputational
issues. For policy-makers and regulators, the challenge is how to promote
innovation while protecting consumers from the harmful impacts of AI. For
consumers of AI, its about trustworthiness, whether they can rely upon AI
outputs to be accurate and transparent, with safeguards in place to protect
them from adverse outcomes.

This paper explores the challenges and opportunities associated with fostering
a culture of Trustworthy AI, with particular focus on: (1) The current state
of Trustworthy AI, including a survey of key industry and standards
organization initiatives with emphasis on the proposed EU Artificial
Intelligence Act, (2) The relationship between Trustworthy AI and Responsible
Data Science (RDS), and (3) Contribution of trust aware process mining to
facilitate a data-driven analytical framework to surface uncertainties,
variabilities, and vulnerabilities in Trustworthy AI compliance processes.

The remainder of the paper is organized as follows. In Section 2, we define
the contours of Trustworthy AI principles. In Section 3, we explore the
proposed EU Artificial Intelligence Act (AIA) that intends to operationalize
and implement rigorous risk-based prescriptive processes for ensuring a
culture of Trustworthy AI. In Section 4, we map the relationship between RDS
and Trustworthy AI, including a discussion of challenges associated with
contextualizing AI fairness as a foundation for Trustworthy AI. In Section 5,
we discuss the applications and benefits of process mining as an important
tool to enable organizations to make data-driven decisions relating to the
obligations and conformance requirements inherent in the proposed EU AI
regulation.

##  2 Trustworthy AI

Surveys reveal an undercurrent of pervasive distrust of AI systems. Cathy
O’Neil, a leading advocate for AI algorithmic fairness, highlighted three main
reasons behind consumer distrust of AI: opacity, scale, and damage [12].
Fairness is the foundation for trustworthy AI. It is the connective tissue
that binds together the principles of ethical use, interpretability,
transparency, accountability, and confidentiality that engenders trust and
promotes the use of AI for social good. Trustworthy AI is a governance
framework designed to mitigate potential adverse impacts on consumers as AI is
poised to profoundly and indelibly change our lives. As mentioned in [17],
Trustworthy AI is changing the dynamic between user and system into a
relationship.

###  2.1 Achieving Trust in AI

Trustworthy AI starts with human agency and autonomy. Trust in AI systems is
enhanced when there is a human-in-the-loop who monitors the overall
performance of AI systems and when circumstances dictate, remediates potential
adverse outcomes. Trust in AI is strengthened by giving users the ability to
make informed decisions about the impact of AI on their personal and economic
well-being.

AI is perceived by consumers to be a black box. Data inputs to the AI systems,
their learning models, and how they arrive at decisions are neither visible,
nor understood by consumers. Furthermore, many AI developers defensively
protect their algorithms as proprietary and a competitive differentiator.
Interpretability and explainability of AI are two important elements that
strengthen trust in AI. Interpretability of AI provides insight into the cause
and effect between inputs and outputs of an AI system and how AI predicts
outcomes. Explainability of AI goes one step further by providing users with
not only insight into how AI models work but also traceability of AI decisions
and documentation relating to the process of data gathering, labeling, and
methods used for training AI algorithms.

Consumers have limited recourse to hold AI developers accountable for the
adverse impacts of AI systems. While there is sectoral legislation, e.g.,
Section 5 of the FTC (Federal Trade Commission)
Act111https://www.federalreserve.gov/boarddocs/supmanual/cch/ftca.pdf,
available for consumers to remedy disparate treatment attributable to AI
systems it is an onerous process to prevail. Moreover, for the disparate
impact, the burden of proof requires statistical analysis that a protected
class is treated differently from others, which is hardly something that would
be accessible to average consumers. For these reasons, accountability,
including redress mechanisms in the event of demonstrated harmful impact need
to be addressed to achieve trust in AI.

###  2.2 The Emergence of Trustworthy AI Principles

We can see efforts being made, to varying degrees, that recognize and deal
with issues relating to trust in AI by the data sciences community (see
Section 4), standards organizations, e.g., IEEE [16], NIST (National Institute
of Standards and Technology) [13], and by public sector organizations.

In 2019, OECD member countries adopted OECD Council Recommendation on
Artificial
Intelligence222https://legalinstruments.oecd.org/en/instruments/OECD-
LEGAL-0449 consisting of five principles of human centered values of fairness
of AI, inclusive investments in AI, transparency, accountability, and
robustness of AI systems. The OECD recommendations were subsequently endorsed
by the G20 with particular reference to the view that the “digital society
must be built on trust among all stakeholders including governments, civil
society, international organizations, academics, and businesses through
sharing common values and principles including equality, justice,
transparency, and accountability taking into account the global economy and
interoperability”.

While Trustworthy AI principles serve as a helpful framework, they are just
that. Adherence to Trustworthy AI is fragmented at best and they lack
effective enforcement mechanisms to safeguard against potentially harmful
impacts. For this reason, the momentum has shifted towards the regulation of
AI: “The calls for modest regulation that lets industry take the lead are part
of a failed regulatory philosophy, one that saw its natural experiment over
the past several decades come up lacking. AI is too important and too
promising to be governed in a hands-off fashion, waiting for problems to
develop and then trying to fix them after the
fact”.333https://www.brookings.edu/research/ai-needs-more-regulation-not-less/

##  3 The Proposed EU Regulation of AI

On April 20, 2021 the European Commission released the proposal for the
regulation of artificial
intelligence444https://ec.europa.eu/commission/presscorner/detail/en/ip_21_1682,
the ambition of which is to balance the socio-economic benefits of AI and new
risks or negative consequences for individuals or society. The proposed
Artificial Intelligence Act (AIA) takes a risk-based approach to regulate AI
by fostering an “ecosystem of trust that should give citizens the confidence
to take up AI applications and give companies and public organisations the
legal certainty to innovate using AI”. In the following, we demonstrate five
governing principles for trustworthy AI proposed by AIA.

###  3.1 Scope of the Proposed Regulation

The proposed AIA applies to all providers, i.e., natural or legal persons,
public authorities, agencies, or any other body that develops an AI system,
that places or makes available on the market or puts into service AI systems
or services in the EU (cf. Article 3). The AIA also assigns responsibility to
users, importers, distributors, and operators who make use of or make
substantial modifications to the functionality and performance of AI systems
(cf. Article 26-29). The geographic scope for the AIA will operate
irrespective of whether such providers are established in the EU or a third
country, and so will cover where the system users are in the EU or the output
of the systems is used in the EU (cf. Article 2). AI systems under the
regulation encompass a wide range of methods and algorithms including
supervised, unsupervised, and reinforcement machine learning for a given set
of human-defined objectives that generate outputs such as content,
predictions, recommendations, or decisions influencing the environments they
interact with (cf. Article 3).

###  3.2 Risk-Based Approach

The foundation of the AIA is a risk-based approach that classifies AI systems
into three categories based on a combination of factors that include the
intended purpose, the number of impacted persons, and the potential risk of
harms (cf. Article 5-7):

  * •

Prohibited AI: Systems that use subliminal techniques that cause physiological
or psychological harm, exploit vulnerable groups, effectuate social scoring by
public authorities that may result in discrimination or unfavorable treatment,
and remote biometric systems used by law enforcement in public spaces (subject
to well-defined exceptions) (cf. Article 5).

  * •

High Risk: Annex III provides a list of systems that are used in critical
infrastructures, educational or vocational training, human resources,
essential private and public services, law enforcement, migration, asylum and
border control management, and administration of justice and democratic
processes (cf. Article 7).

  * •

Low Risk: While not explicitly named (we use the term low risk of our own
choosing), by default, all systems not categorized as prohibited or high-risk.
Providers of such systems are encouraged to institute responsible use of AI
best practices on a voluntary basis (cf. Article 69).

###  3.3 Promote Fair and Trustworthy AI Best Practices

The AIA sets forth a comprehensive legislative mandate to ensure fairness in
the application of AI systems that safeguards fundamental human values and
promotes socio-economic rights. Some of these mandates are as follows:
obligation on providers to implement appropriate risk management measures
throughout the entire lifecycle of AI systems (cf. Article 9), rigorous data
governance processes (cf. Article 10), technical documentation, and record-
keeping processes to enable monitoring of compliance (cf. Article 11-12),
transparency that enables full interpretation of outputs (cf. Article 13), and
Human-in-the-loop oversight (cf. Article 14).

###  3.4 Transparency and Accountability

According to the AIA, providers of AI systems will be required to implement a
range of processes to ensure full transparency into and accountability for AI
systems (cf. Article 19-23) such as (1) conformity assessment and
certification processes, (2) auditability, including accessible event logs,
and (3) Explainability, potentially to coordinate with the human-in-the-loop
for adjudication and remediation.

###  3.5 Enforcement

The AIA incorporates an onerous enforcement mechanism that even surpasses the
fines under the GDPR (cf. Article 71). Some examples are as follows: up to
€10m or 2% of the total worldwide annual turnover for the supply of incorrect,
incomplete or misleading information to the authorities, up to €20m or 4% of
the total worldwide annual turnover for non-compliance with any other AIA
requirement or obligation, and up to €30m or 6% of the total worldwide annual
turnover for violations of prohibited practices.

While the proposed AIA is far from ratification and still subject to vigorous
debate within the EU Parliament and Council, the momentum towards its adoption
is inevitable. Like the GDPR, the AIA will serve as a model for other
jurisdictions that will seek to finally exert control over what has been the
unregulated, hyperbolic growth of AI across the globe.

##  4 Responsible Data Science and Trustworthy AI

Responsible Data Science (RDS) is a discipline that is influential in shaping
Trustworthy AI best practices. RDS refers to the collection of techniques and
approaches trying to reap the benefits of data science and big data while
ensuring fairness, accuracy, confidentiality and transparency [2]. To minimize
adverse AI outcomes of AI the role of RDS is to: (1) Avoid unfair conclusions
even if they are true, i.e., the fairness principle, (2) Answer questions with
a guaranteed level of accuracy, i.e., the accuracy principle, (3) Answer
questions without revealing secrets, i.e., the confidentiality principle, and
(4) Clarify Answers such that they become indisputable, i.e., the transparency
principle.

RDS applies a methodology throughout the entire life cycle of information to
support trustworthy AI best practices by applying these four principles of
fairness, accuracy, confidentiality, and transparency to the data science
pipeline resulting in rigorous data governance as illustrated in Figure 1.

Figure 1: The data science pipeline facing the four FACT challenges [2].

RDS delivers a robust framework for the ethical design of AI systems that
addresses the following key areas: (1) Unbiased outcomes through the
application of appropriate fairness constraints to the training data, (2)
Algorithmic outcomes interpreted in a manner that is meaningful to end users,
(3) Resilience in how AI systems deliver accurate results and respond to
change in inputs, (4) Accountability for the system’s outcomes, and (5)
Safeguarding the confidentiality of training data through privacy enhancing
measures. However, providing each aspect of RDS has its own challenges from
contextualizing the aspect to implementing it in data science and AI systems.
In [6], the authors describe the challenges regarding the confidentiality
aspect for process mining which combines process and data science. In the
following, we provide the challenges regarding the fairness aspect.

###  4.1 Contextualizing Fairness in AI Systems: Challenges

The idea of fairness is somewhat amorphous. At its highest level of
abstraction, fairness is a normative concept that comes from our conscience.
Dator defines a fair system as follows: “What is fairness then? We all have
desires and we want people to treat us according to those desires. We also
know that people around us have similar desires and want to be treated
accordingly. Fairness is closely related to fair play so it seems logical to
conclude that a fair system is a system where everybody is treated in a
similar way” [4]. There are a number of challenges associated with
contextualizing and applying such a high level of abstraction to a more
concrete algorithmic AI fairness framework.

First, fairness may be influenced by cultural, sociological, economic, and
legal considerations. What may be considered as fair in one culture may be
perceived as unfair in another. Unequal distribution of opportunity may
require the application of distributive fairness that levels the playing
field. For example, in the context of credit applications, there ought to be
an equal probability of loan eligibility by ensuring that AI algorithmic
outcomes do not discriminate against members of protected groups [3]. There
are other instances where the application of corrective fairness may be
necessary, for example, to remedy adverse impacts in the administration of
justice, housing, education, and employment.

Second, equality does not necessarily result in the fairness of outcomes.
While under Human Rights legislations disparate treatment on the basis of
race, gender, nationality, disability, and sexual orientation is prohibited
there may still be instances of adverse outcomes, based on other facially-
neutral variables that cause a disparate impact, i.e., unintentional
discrimination [5]. Consider Amazon’s free same day delivery service based on
an AI algorithm that included attributes, such as distance to the nearest
fulfillment center, local demand in designated zip code areas, and frequency
distribution of prime members to determine profitable locations for free Same-
Day Delivery. The algorithm was found to be biased against minorities even
though race was deemed not to be a factor in the determination of same day
delivery, and minority residents in the selected zip codes were about half as
likely to be eligible as white residents.555https://eu.usatoday.com/

The third challenge is balancing algorithmic fairness with fairness outcomes
[10]. In this context, fairness encompasses policy and legal considerations,
and leads us to ask: what ought to be fair? For example, in the context of
hiring practices, what ought to be a fair percentage of women in management
positions that AI algorithms should incorporate as thresholds to promote
gender parity?

The fourth challenge relates to trade-off in balancing demographic parity with
the utility of outcomes. For example, if AI algorithms remove disparate impact
in the incarceration of minorities, how would that impact broader policy
considerations such as the administration of justice?

Finally, fairness implicates issues of power. Before we can decide what is
fair, we need to decide who gets to decide that. The conundrum we must
confront is that the minority groups who are so typically the victims of
algorithmic bias are rarely given a seat at the table when it is time to
define what is fair. The unfortunate result is that far too often, the
definition of fairness is simply what those already in power need it to be to
maintain that power.

###  4.2 Implementing Fairness: Challenges for Data Scientists

Fairness constraints need to be considered in the context of specific use
cases and for desired outcomes. Bias may be introduced at various levels
within an AI system. Training data may introduce proxies that discriminate.
Historical bias may unconsciously result in adverse outcomes, for example
through word embeddings [4]. Representation bias through under or, over
representation of training data may produce disparate impacts. The algorithms
may not sufficiently adjust for fairness constraints. Inadequate testing for
disparate treatment and impact may have adverse consequences for protected
groups. While some argue that AI algorithms in fact minimize bias there is
compelling evidence that they can and often amplify biases. Examples span
facial recognition, criminal justice, hiring practices, and loan approvals
[9].

Regardless of any contextualization, any definition, and any implementation
approach of the fairness which is the cornerstone for Trustworthy AI, what is
essential is to gain visibility to and remediate potential gaps in Trustworthy
AI compliance processes. In the next section, we demonstrate how process
mining could play a role in fulfilling such requirements.

##  5 Process Mining for Promoting Trustworthy AI

Compliance with the proposed EU AIA requires an understanding of process
execution and interactions between multiple internal and external
stakeholders, risk assessment of diverse systems of record that incorporate AI
systems, and cooperation with various regulatory bodies and standards
organizations.

The proposed AI regulation operationalizes and codifies trustworthy AI
principles with prescribed mandates to institute appropriate data governance
and management practices. The governance mechanism is complex and requires
human and systems-based interactions between diverse internal and external
stakeholders and EU and national regulators. Monitoring conformance with AIA
is delegated to national supervisory authorities, they are empowered to order
companies to take corrective actions, access all information, documentation,
and data required to enforce compliance with the proposed regulation.

Given the complexity and variability of interactions implicit in achieving
compliance with the proposed regulation it is our contention that process
mining can be a valuable tool to help organizations gain visibility to various
dimensions of prescribed process flows stipulated by the regulation,
accelerate the analysis of how information flows, surface process bottlenecks,
visualize interactions generated by event logs from disparate systems of
record that may reveal areas of compliance and reputational risks. Process
mining bridges the gap between data science and process science using event
data captured from different types of information systems [1]. It is a data-
driven approach that enables organizations to gain insight into interactions
between people, systems, and organizations based on “as-is” visualization of
process execution.

There are many techniques and activities in the context of process mining.
However, the three main types of activities in process mining are process
discovery, conformance checking, and enhancement. Process discovery techniques
take an event log and discover a process model without using any other
information. Conformance checking techniques take a process model and an event
log of the same process to check whether reality, as recorded in the event
log, conforms to the model and vice versa. Enhancement techniques are used to
extend or improve a given process model using the information about the
process recorded in some event logs [1].

Process Mining can facilitate compliance with AIA by many functionalities such
as: (1) Surfacing AI regulatory compliance process gaps and uncertainties, (2)
Capturing user interactions performing compliance tasks, (3) Comparing process
execution variations, (4) Highlighting compliance task outliers and errors,
(5) Identifying potential root causes for improper execution, (6) Real-time
monitoring of processes to ensure conformance to prescribed process execution
paths, and (7) Triggering alerts in the event of non-compliant process tasks
or changes in conditions. Furthermore, the AIA proposed regulation is
inherently collaborative in nature wherein process execution spans across
different organizations.

As discussed in [11], in collaborative processes where different organizations
execute different parts of a shared process, the internal activities carried
out by each organization are beyond the control of the other collaborators
resulting in uncertainty regarding process execution. Whenever there is
uncertainty in a process, there is a need for trust. Hence, collaborative
business processes are especially trust-intensive. In such trust-intensive
environments, process mining can be used to clarify the flow of activity
execution among several organizations.

Figure 2: Process mining cadence to meet AIA prescriptive compliance
obligations.

Compliance with AIA constitutes a number of interdependent steps. Performing
these steps may involve variabilities in process execution paths and hand off
between different stakeholders and prescribed conformance obligations to meet
Articles 16-23 and Annex VII of the AIA:

  * •

Step 1: R&D teams develop and bring to market AI systems in accordance with
the risk classification system defined by the proposed regulation. If it is a
high-risk AI system then a priori conformance assessment must be undertaken
and a declaration of conformity must be submitted to the appropriate National
Supervisory Authority. Then the AI system may be placed on the market.

  * •

Step 2: Legal and Compliance teams must institute compliance measures in
accordance with Chapter 2 of the proposed regulation that ensures adherence to
data governance, accountability, transparency, accuracy, robustness, and
cybersecurity provisions.

  * •

Step 3: Data Science teams must undertake continuous monitoring of AI systems,
collect data on the system’s operation and take corrective action if needed.
The post-market monitoring system must actively and systematically collect,
document, and analyze relevant data provided by users.

  * •

Step 4: Customer-facing functions such as Sales, Marketing, and Support, are
responsible for providing clarity and certainty as to the expected AI system
inputs and outputs in a way that users are informed that they are interacting
with an AI system, augmented with human oversight who monitor their operation
and be able to decide, to override or reverse the output of the high-risk AI
system.

  * •

Step 5: Implementation of a Quality Management System with auditable and
traceable documentation relating to the techniques, procedures for the design,
of the high-risk AI systems, including procedures for data management, data
analysis, data labeling, data storage, data aggregation, data retention and
report serious incidents that may result in adverse outcomes.

Figure 2 further maps the compliance steps, the obligation provisions of the
AIA, and process mining functionality to support Trustworthy AI. The figure
illustrates how process mining techniques can facilitate AIA obligations. The
FACT challenges of RDS are also taken into consideration in process mining as
a subdiscipline called Responsible Process Mining (RPM) which is recently
receiving increasing attention [15, 14, 7, 8].

##  6 Conclusion

Trustworthy AI engenders a climate of trust essential for achieving
sustainable competitive advantages in an intensely competitive environment
where the application of AI is a disruptive force. The proposed EU regulation
of AI is a comprehensive prescriptive measure which imposes onerous
obligations, redress mechanisms on AI developers and businesses deploying AI
systems. To mitigate compliance, reputational, and business risks process
mining is poised to provide a data-driven approach to discover how existing
Trustworthy AI compliance processes work, surface and remediate process
bottlenecks, visualize different pathways of process execution and identify
and remediate variations from prescribed protocols. Process mining can be a
useful toolbox for ensuring that certain AI systems are designed and developed
in accordance with common necessary requirements before they are put on the
market and operationalized through harmonized technical standards.

## Acknowledgments

Funded under the Excellence Strategy of the Federal Government and the Länder.
We also thank the Alexander von Humboldt Stiftung for supporting our research.

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