# Non-ideal fields solve the injection problem in relativistic reconnection

Lorenzo Sironi<EMAIL_ADDRESS>Department of Astronomy and
Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA

###### Abstract

Magnetic reconnection in relativistic plasmas is well established as a fast
and efficient particle accelerator, capable of explaining the most dramatic
astrophysical flares. With particle-in-cell simulations, we demonstrate the
importance of non-ideal fields for the early stages (“injection”) of particle
acceleration. Most of the particles ending up with high energies (near or
above the mean magnetic energy per particle) must have passed through non-
ideal regions where the assumptions of ideal magnetohydrodynamics are broken
(i.e., regions with $E>B$ or nonzero $E_{\parallel}={\bf E}\cdot{\bf B}/B$),
whereas particles that do not experience non-ideal fields end up with Lorentz
factors of order unity. Thus, injection by non-ideal fields is a necessary
prerequisite for further acceleration. Our results have important implications
for the origin of nonthermal particles in high-energy astrophysical sources.

Magnetic reconnection in the relativistic regime [1, 2, 3], where the magnetic
energy is larger than the particle rest-mass energy (equivalently, the mean
magnetic energy per particle is $\sim\sigma mc^{2}\gg mc^{2}$, with $\sigma$
the magnetization), has been invoked to explain the most dramatic flaring
events in astrophysical high-energy sources [e.g. 4, 5, 6, 7, 8, 9, 10, 11].
Our understanding of the physics of relativistic reconnection has greatly
advanced thanks to fully-kinetic particle-in-cell (PIC) simulations, which
have established reconnection as an efficient particle accelerator [e.g. 12,
13, 14, 15, 16]. It is widely accepted that most of the energy gain of ultra-
relativistic particles comes from ideal fields [e.g. 13, 17]. It was then
argued that the spectrum of high-energy particles would remain unchanged, if
non-ideal fields were to be ignored [17].

In this Letter, we demonstrate that, instead, non-ideal fields have a key role
in the acceleration process. They are essential for solving the “injection
problem,” so that non-relativistic particles can be promoted to relativistic
($\sim\sigma mc^{2}\gg mc^{2}$) energies. Injection by non-ideal fields is a
necessary prerequisite to access further acceleration channels, primarily
governed by ideal fields [13, 18, 17, 19, 20, 16]. High-energy particles
receive most of their energy by ideal fields (in 2D, via Fermi-like
acceleration with the “slingshot” mechanism [17] or via magnetic moment
conservation in the increasing field of compressing plasmoids [19, 20]; in 3D,
via grad-B-drift acceleration while their orbits sample both sides of the
layer [16]). However, we find that the particles must be pre-energized in non-
ideal regions, before being further accelerated by ideal fields. We find that
if non-ideal fields were artificially excluded, all particles would end up
with low energies.

Setup—We perform 2D and 3D PIC simulations with TRISTAN-MP [21, 22]. We
initialize a force-free field of strength $B_{0}$, whose direction rotates
from $+\hat{x}$ to $-\hat{x}$ across a current sheet at $y=0$. We consider a
cold electron-positron plasma with rest-frame density $n_{0}$ of 16 particles
per cell in 2D and 4 in 3D (Suppl. Mat. for convergence studies). Particles
initially in the current sheet are excluded from our analysis, so to obtain
results independent from specific choices at initialization. The field
strength $B_{0}$ is parameterized by the magnetization $\sigma=B_{0}^{2}/4\pi
n_{0}mc^{2}=\left(\omega_{\rm c}/\omega_{\rm p}\right)^{2}$, where
$\omega_{\rm c}=eB_{0}/mc$ is the Larmor frequency and $\omega_{\rm
p}=\sqrt{4\pi n_{0}e^{2}/m}$ the plasma frequency. We vary $\sigma$ in the
range $3\lesssim\sigma\lesssim 200$, with $\sigma=50$ as our reference case.
Most of our runs assume a vanishing guide field, but we also present results
with guide field $B_{g}=B_{0}$ initialized along $z$ as in [23].

Along the $y$-direction of inflows, two injectors continuously introduce fresh
plasma and magnetic flux into the domain, see [24]. We employ periodic
boundary conditions in $z$. Most of our runs have periodic boundaries also in
$x$ (but our conclusions also hold for outflow $x$-boundaries, Suppl. Mat.).
Each simulation is evolved for $\sim 2.5\,L_{x}/c$ (the periodic
$x$-boundaries would artificially choke reconnection at longer times). We
usually let reconnection start spontaneously from numerical noise, but we also
show similar results when reconnection is “triggered” by hand at the initial
time. We resolve the plasma skin depth with $\,c/\omega_{\mathrm{p}}=5$ cells,
and employ large domains up to $L_{x}\simeq 6400\,c/\omega_{\mathrm{p}}$ in 2D
(our reference is $L_{x}\simeq 1600\,c/\omega_{\mathrm{p}}$), and
$L_{x}=2L_{z}\simeq 800\,c/\omega_{\mathrm{p}}$ in 3D.

Figure 1: Evolution of the layer for our reference simulation
($\sigma=50,\;B_{g}=0,\;L_{x}=1600\,c/\omega_{\mathrm{p}}$). (a) Density in
the midplane $y=0$, normalized to $n_{0}$; (b)
$\chi=(B^{2}-E^{2})/(B^{2}+E^{2})$ at $y=0$, to identify regions of magnetic
($\chi>0$) or electric ($\chi<0$) dominance; the particle $x$-locations at
their first $E>B$ encounter are shown in (c) and (d), where we distinguish
between particles that experience $E>B$ with $E_{z}>0$ (c) vs $E_{z}<0$ (d)
(see text).

Results—Near X-points the reconnected field scales as $B_{y}\sim
B_{0}\,x/\Delta$ over some length $\Delta$, whereas the electric field is
$E_{z}\sim\eta_{\rm rec}B_{0}$, with $\eta_{\rm rec}\sim 0.1$ the reconnection
rate [e.g., 24]. For $B_{g}=0$, magnetic dominance is broken (i.e., $E>B$) at
$|x|\lesssim\eta_{\rm rec}\Delta$. For $B_{g}=0$ the parallel electric field
$E_{\parallel}=\boldsymbol{E}\cdot\boldsymbol{B}/B$ necessarily vanishes, so
non-ideal effects are captured by $E>B$, rather than by $E_{\parallel}$. In
contrast, in the presence of a nonzero $B_{g}$, the region of electric
dominance disappears if $B_{g}/B_{0}\gtrsim\eta_{\rm rec}$. Here,
$E_{\parallel}\sim\eta_{\rm rec}B_{0}$ near the X-point, i.e., the electric
field is entirely accounted for by $E_{\parallel}$. Thus, non-ideal effects
are captured by $E>B$ for $B_{g}/B_{0}\lesssim\eta_{\rm rec}$, and by
$E_{\parallel}$ for stronger guide fields. We verified this argument with a
sweep of $B_{g}/B_{0}$ values, but here we focus only on $B_{g}=0$ (our main
case) and $B_{g}/B_{0}=1$.

The layer evolution for our reference simulation
($\sigma=50,\;B_{g}=0,\;L_{x}=1600\,c/\omega_{\mathrm{p}}$) is shown in Fig.
1(a), where we present the spatio-temporal structure of density in the
midplane ($y=0$). At early times, the layer breaks into a series of primary
plasmoids. Over time, they grow and coalesce, and new secondary plasmoids
appear in the under-dense regions in-between primary plasmoids (e.g., top
right in Fig. 1(a)). Regions with $E>B$ are rather ubiquitous in between
plasmoids (see the green and blue areas in Fig. 1(b), where we plot
$\chi=(B^{2}-E^{2})/(B^{2}+E^{2})$ at $y=0$). For each simulation particle, we
detect the first time (if any) it experiences $E>B$, and record its position
at this time. The particle $x$-locations at their first $E>B$ encounter are
shown in panels (c) and (d), where we distinguish between particles that
experience $E>B$ with $E_{z}>0$ (c) vs $E_{z}<0$ (d). The former ($E_{z}>0$)
is expected for X-points of the main layer, whereas $E_{z}<0$ in between
merging plasmoids, as demonstrated by comparing (c) and (d) with (a).

Figure 2: (a) Particle spectra at $t\simeq 1.8\,L_{x}/c$ (solid lines), for
simulations with $\sigma=50,\;B_{g}=0$ and different box sizes:
$L_{x}=L_{0}=1600\,c/\omega_{\mathrm{p}}$ (blue),
$L_{x}=2L_{0}=3200\,c/\omega_{\mathrm{p}}$ (green), and
$L_{x}=4L_{0}=6400\,c/\omega_{\mathrm{p}}$ (red). Dashed lines show the
spectrum $E>B$ particles (see text), dotted lines of $E<B$ particles. Inset:
comparison of 2D and 3D triggered simulations with
$\sigma=50,\;B_{g}=0,\;L_{x}=400\,c/\omega_{\mathrm{p}}$ at $t\simeq
1,000\,\omega_{\rm p}^{-1}\simeq 2.5L_{x}/c$. Dashed and solid lines as in the
main panel. (b) Spectra at $t\simeq 4000\,\omega_{\rm p}^{-1}\simeq
2.5L_{x}/c$ for simulations with $L_{x}=1600\,c/\omega_{\mathrm{p}},\;B_{g}=0$
and varying $\sigma$, see the legend; line style as in (a). The inset presents
the spectra of $E>B$ particles, but $\gamma-1$ on the horizontal axis is
rescaled by $(50/\sigma)$. (c) Contribution of $E>B$ particles to the total
census in the reconnection region (blue), and to the number of particles with
$\gamma>\sigma/4$ (green) and $\gamma>\sigma$ (red), as a function of
$\sigma$. The black points show the fraction of length along $y=0$ occupied by
$E>B$ regions.

The high-energy part of the spectrum extracted from the reconnection region
111We define the reconnection region such that it contains a mixture of
particles starting from $y>0$ and $y<0$ [41], with both populations
contributing at least 10% (our results do not significantly depend on this
fraction). Unless otherwise noted, we only show the spectrum of particles
belonging to the reconnection region. is dominated by particles that
experienced $E>B$ at some point in their history. In Fig. 2, we plot the
overall spectrum with solid lines, the spectrum of particles that experienced
$E>B$ (hereafter, “$E>B$ particles”) with dashed lines, and the spectrum of
particles that never experienced $E>B$ (hereafter, “$E<B$ particles”) with
dotted lines. “$E>B$ particles” are particles that at some point experienced
$E>B$, so they are not only those currently in $E>B$ regions.

For high magnetizations, the high-energy tail of the spectrum is mostly
populated by $E>B$ particles, regardless of system size (Fig. 2(a) 222The
shift of the spectral cutoff to higher energies with increasing system size
has been extensively characterized, in both 2D [15, 19, 20] and 3D [16].) and
dimensionality (inset in Fig. 2(a), showing a comparison between 2D and 3D).
With increasing $\sigma$ (Fig. 2(b)), the spectrum of $E>B$ particles shifts
to higher energies as $\propto\sigma$ (see inset in Fig. 2(b), where
$\gamma-1$ on the horizontal axis is rescaled by $(50/\sigma)$), and it
increases in normalization. In contrast, the spectrum of $E<B$ particles peaks
at $\gamma-1\sim{\rm few}$ for all magnetizations, and at high energies it
drops much steeper than the $E>B$ spectrum. Thus, the overall spectrum can be
described as a combination of two populations: a low-energy peak at trans-
relativistic energies contributed by $E<B$ particles, and a high-energy bump
with mean Lorentz factor $\propto\sigma$ populated by $E>B$ particles.

A robust result of PIC simulations is that higher magnetizations display
harder spectra, with $p=-d\log N/d\log\gamma\rightarrow 1$ for $\sigma\gg 1$
[e.g. 12, 13, 14, 15]. For domain sizes within the reach of current PIC
simulations, the spectrum does not extend much beyond the post-injection
spectrum (the cutoff is at $\lesssim 10\,\sigma mc^{2}$ [19, 20, 16]; indeed,
hard power-laws with $p<2$ could not extend to much higher energies without
running into an energy crisis [19]). The fact that higher magnetizations
display harder spectra has a simple explanation. While the peak of the $E<B$
population is nearly $\sigma$-independent, the $E>B$ component shifts to
higher energies ($\sim\sigma mc^{2}$) and higher normalizations with
increasing $\sigma$, thus hardening the overall spectrum (Fig. 2(b)).

In the asymptotic limit $\sigma\gg 1$, $E>B$ particles contribute a fraction
$\gtrsim 90\%$ at $\gamma>\sigma$ (red points in Fig. 2(c)) and $\gtrsim 70\%$
at $\gamma>\sigma/4$ (green). These fractions are nearly constant in time for
$t\gtrsim L_{x}/c$ (Suppl. Mat.), and independent of the domain size (Fig.
2(a)). For $\sigma\gtrsim 50,\;E>B$ particles account for $\sim 20\%$ of the
overall census in the reconnection region (Fig. 2(c), blue). This can be
related to the fraction of length along the $y=0$ line (of area in the $y=0$
plane, for 3D) occupied by non-ideal regions, as we now explain.

The black points in Fig. 2(c) denote the “occupation fraction” of $E>B$
regions along $y=0$, averaged over $1\lesssim ct/L_{x}\lesssim 2$. This
fraction increases with $\sigma$ 333The increase with $\sigma$ has two
reasons: first, fragmentation of the layer by the secondary plasmoid
instability [42] is more pronounced for higher $\sigma$, increasing the number
of non-ideal regions [24]; second, a given $E>B$ region is more extended at
higher $\sigma$, since $\eta_{\rm rec}$ increases with magnetization [43].
Both effects reach an asymptotic limit at $\sigma\gtrsim 50$, and in the limit
$\sigma\gg 1$ approaches $\sim 10\%$ (black points in Fig. 2(c)), which is
about half of the fraction of $E>B$ particles (blue).

This factor of two has a simple explanation. The dashed blue line in Fig. 3
shows the spectrum measured, for each particle, at its first $E>B$ encounter
with $E_{z}>0$, as appropriate for X-points in the main layer. It contains
$\sim 10\%$ of post-reconnection particles, i.e., exactly equal to the $E>B$
occupation fraction. The dotted blue line in Fig. 3, instead, shows the
spectrum of particles experiencing $E>B$ with $E_{z}<0$, i.e., in between
merging plasmoids. It also contains $\sim 10\%$ of particles. Thus, for
$\sigma\gtrsim 50$, $\sim 10\%$ of particles encounter $E>B$ fields when
entering the reconnection region, and an additional $\sim 10\%$ in secondary
layers between merging plasmoids. The latter extend along $y$, so their
X-points are not accounted for by the black markers in Fig. 2(c). This
justifies why for $\sigma\gtrsim 50$ the fraction of $E>B$ particles (blue in
Fig. 2(c)) is twice larger than the $E>B$ occupation fraction (black).

Figure 3: Evidence of fast energization near non-ideal regions, for
$\sigma=50,\;B_{g}=0$ and $L_{x}=1600\,c/\omega_{\mathrm{p}}$. The spectrum of
particles at their first $E>B$ encounter is shown by the dashed blue line if
$E_{z}>0$, and dotted blue if $E_{z}<0$. Their sum is the solid blue. The
series of spectra from light to dark red are measured, for those same
particles, respectively $\sim 9,\,27,\,90,\,270\;\omega_{\rm p}^{-1}$ after
their first $E>B$ encounter. For comparison, the black line shows the overall
spectrum at the final time $\omega_{\rm p}t\simeq 4000$.

In Fig. 3, we provide evidence of fast particle acceleration near non-ideal
regions. The spectrum of particles at their first $E>B$ encounter is shown by
the solid blue line, demonstrating that at this point the particles still have
low energies. The series of spectra from light to dark red are measured, for
those same particles, respectively $\sim 9,\,27,\,90,\,270\,\omega_{\rm
p}^{-1}$ after their first $E>B$ encounter. The spectral peak quickly shifts
up to $\gamma-1\sim 5$ (first red line; at this time, most of the particles
are still in $E>B$ regions), yielding a mean acceleration rate $d\gamma/dt\sim
0.5\,\omega_{\rm p}^{-1}$, comparable to the maximal rate $\sim\eta_{\rm
rec}|\beta_{z}|\sqrt{\sigma}\omega_{\rm p}^{-1}$ [16] assuming a $z$-velocity
$|\beta_{z}|\simeq 0.7$ (particles accelerated at X-points also have some
$\beta_{x}$ along the outflow). Rapid acceleration continues up to
$\gamma-1\simeq 20\sim\sigma/2$ (third red line) 444As we show in Suppl. Mat.,
this is also the peak of the spectrum of particles currently residing in $E>B$
regions.. Beyond this stage, the spectrum still shifts up in energy [19, 20],
but at a slower rate (compare the two darkest red lines). The later stages are
governed by ideal fields [17].

Figure 4: Experiments with test-particles that do not experience non-ideal
fields, for $B_{g}=0$ (top) and $B_{g}=B_{0}$ (bottom). Spectra are computed
at $\omega_{\rm p}t=4000$ for simulations with
$L_{x}=1600\,c/\omega_{\mathrm{p}}$ and varying $\sigma$ (see legend). Top:
solid and dotted lines as in Fig. 2(b), whereas dash-dotted lines present the
final spectrum of test-particles. They are initialized as regular particles,
but when they pass through $E>B$ regions, we artificially fix their Lorentz
factor at $\gamma-1\sim{\rm few}$ (more specifically, to the same peak as the
dotted lines). Bottom: solid lines for regular particles, while dash-dotted
lines for test-particles evolved without the contribution by $E_{\parallel}$.

To corroborate our conclusions on the importance of non-ideal fields for
particle injection, we present in Fig. 4 two additional experiments. For
$3\lesssim\sigma\lesssim 200$, we analyze the cases of vanishing ($B_{g}=0$,
top) and strong ($B_{g}=B_{0}$, bottom) guide fields. The final spectra are
indicated by solid lines. In the top panel, dotted lines show the spectra of
$E<B$ particles, whereas dash-dotted lines the spectra of “test-particles” —
not contributing to the electric currents in the simulation, but otherwise
initialized and evolved as regular particles. When test-particles pass through
$E>B$ regions, we artificially fix their Lorentz factor at $\gamma-1\sim{\rm
few}$. The remarkable agreement between dash-dotted and dotted lines
demonstrates that if we do not allow test-particles to gain energy while in
$E>B$ regions, they display similar spectra as particles that never had $E>B$
encounters (we have confirmed this also for the 3D simulation of Fig. 2(a),
inset). Equivalently, energization in non-ideal regions plays a key role in
shaping the high-energy end of the particle spectrum.

Fig. 4(b) refers instead to $B_{g}=B_{0}$. As we discussed, here non-ideal
effects are well captured by $E_{\parallel}$. Comparison with Fig. 4(a) shows
that spectra are softer for larger $B_{g}$. The fraction of injected particles
decreases with increasing $B_{g}/B_{0}$ because the layer is less prone to
fragmentation into plasmoids [29, 30], and so to formation of non-ideal
regions. In Fig. 4(b), dash-dotted lines present the spectrum of test-
particles evolved without $E_{\parallel}$, so in response to
$\boldsymbol{E}-E_{\parallel}(\boldsymbol{B}/B)$. When inhibiting energization
by $E_{\parallel}$, the test-particles stay at non-relativistic energies.

Conclusions—We investigate the injection physics of particle acceleration in
relativistic reconnection. In contrast to earlier claims [17], we find that
energization by non-ideal fields is a necessary prerequisite for further
acceleration (in Suppl. Mat. we compare to [17]). Particles that are
artificially evolved without non-ideal fields do not even reach relativistic
energies. While it is true that high-energy particles receive most of their
energy via ideal fields [17], this can only happen after an injection phase
necessarily governed by non-ideal fields. We then argue that studies of
reconnection-powered acceleration that employ test-particles in
magnetohydrodynamics simulations need to properly include non-ideal fields.

The spectral component of particles that encountered non-ideal regions shifts
to greater energies ($\sim\sigma mc^{2}$) and higher normalizations with
increasing magnetization, whereas particles that do not experience non-ideal
fields always end up with Lorentz factors near unity. The overall spectrum
then gets harder for higher $\sigma$, which explains the $\sigma$-dependent
spectral hardness reported in PIC simulations [e.g., 13, 14, 15]. This
statement applies to the range $1\lesssim\gamma\lesssim 10\,\sigma$ of the
post-injection spectrum. At higher energies, an additional power-law tail will
emerge, whose slope is set by the dominant acceleration mechanism, which is
different between 2D [31, 20] and 3D [16].

Finally, we remark that, even though we have assumed an electron-positron
plasma, it is well known that $\sigma\gg 1$ reconnection behaves similarly in
electron-positron, electron-proton [32, 33, 34] and electron-positron-proton
[35] plasmas, so our results should apply regardless of the plasma
composition. The importance of non-ideal fields for particle injection has
also been emphasized in studies of non-relativistic low-beta turbulence and
reconnection [36, 37, 38, 39] and magnetically-dominated turbulence [40].

xx

###### Acknowledgements.

We thank L. Comisso, D. Groselj, A. Spitkovsky and N. Sridhar for comments
that greatly improved the clarity of the paper. We thank F. Guo for
discussions on this topic. L.S. acknowledges support from the Cottrell
Scholars Award, NASA 80NSSC20K1556, NSF PHY-1903412, DoE DE-SC0021254 and NSF
AST-2108201. This project made use of the following computational resources:
NASA Pleiades supercomputer, Habanero and Terremoto HPC clusters at Columbia
University. This work is dedicated to the memory of my father.

##  I Supplementary Material

###  I.1 Secondary Particle Energization

In Fig. 5(a) we present with the orange line a representative spectrum of
particles that currently reside in $E>B$ regions. The spectrum is measured at
$\omega_{\rm p}t\simeq 900$ for our reference simulation with $\sigma=50$,
$B_{g}=0$ and $L_{x}=1600\,c/\omega_{\mathrm{p}}$. The red line, instead, is
obtained by integrating over time (until the final time $\omega_{\rm p}t\simeq
4000$) the instantaneous spectra from $E>B$ regions, assuming that particles
spend there an average time of $\sim 10\,\omega_{\rm p}^{-1}$. The red
spectrum has a broad peak at $\gamma-1\sim 20$, i.e., the characteristic
Lorentz factor of particles in $E>B$ regions is $\gamma-1\sim 20\sim\sigma/2$.
For comparison, the black line shows the overall spectrum in the reconnection
region at the final time.

The blue spectrum is computed as follows: for each of the 450 output snapshots
of our simulation, we consider the instantaneous (at time $t_{i}$) spectrum of
particles located in $E>B$ regions, and we extrapolate their energy to the
final time $t_{f}\simeq 4000\,\omega_{\rm p}^{-1}$ as

$\gamma_{f}-1=(\gamma_{i}-1)\left(1+\frac{t_{f}-t_{i}}{\Delta t}\right)^{1/2}$
(1)

where $\gamma_{i}$ is the Lorentz factor at the injection time $t_{i}$ and
$\gamma_{f}$ the one at $t_{f}$, and we have chosen $\Delta
t=0.2\,L_{x}/c\simeq 300\,\omega_{\rm p}^{-1}$. The blue line is then obtained
by summing the contributions of the individual snapshots $t_{i}$, still
assuming that particles spend in $E>B$ regions an average time of $t\sim
10\,\omega_{\rm p}^{-1}$ (as we assumed for the red line).

The scaling in Eq. 1 is motivated by the dominant mechanism of secondary
particle energization in 2D relativistic reconnection [19, 20] — beyond the
primary acceleration/injection discussed in this paper. 2D PIC simulations
have shown that the Lorentz factor of high-energy particles scales as
$\gamma\propto t^{1/2}$. This is driven by a linear increase in the field
strength felt by the particles trapped in plasmoids, coupled with the
conservation of their first adiabatic invariant [19, 20]. The blue line is in
good agreement — as regard to both shape and normalization — with the high-
energy part of the overall spectrum (black). This demonstrates that primary
acceleration/injection at/near $E>B$ regions, followed by secondary
acceleration in compressing plasmoids (as prescribed by Eq. 1, and appropriate
for 2D), provides an excellent description of the history of high-energy
particles. In particular, even though at any given time the fraction of
particles residing in $E>B$ regions is small (orange line), their time-
integrated contribution (blue line) fully accounts for the high-energy part
($\gamma\gtrsim\sigma$) of the final spectrum (black line).

Figure 5: (a) In orange, we show a representative spectrum of particles that
currently reside in $E>B$ regions, measured at $\omega_{\rm p}t\simeq 900$ for
our reference simulation with $\sigma=50$ and
$L_{x}=1600\,c/\omega_{\mathrm{p}}$. The red line is obtained by integrating
over time (until the final time $\omega_{\rm p}t\simeq 4000$) the
instantaneous spectra of $E>B$ regions, assuming that particles spend there an
average time of $t\sim 10\,\omega_{\rm p}^{-1}$. The blue line is obtained
from the $E>B$ spectra measured at 450 different time snapshots (one every
$9\,\omega_{\rm p}^{-1}$), by extrapolating the particle energies to later
times as discussed in the text (see Eq. 1). For comparison, the black line
shows the overall spectrum in the reconnection region at the final time
($\omega_{\rm p}t\simeq 4000$). (b) The solid black line is the total particle
spectrum at the final time ($\omega_{\rm p}t=4000$) for our fiducial
simulation with $\sigma=50$ and $L_{x}=1600\,c/\omega_{\mathrm{p}}$. The
dotted black line shows the spectrum of $E<B$ particles at the same time.
Dash-dotted lines present the final spectrum of test-particles, which are
initialized as regular particles, but when they pass through $E>B$ regions we
artificially fix their $\gamma\beta$ to the value indicated in the legend. (c)
Convergence with respect to the number of particles per cell, which we vary
from 16 (green and red) to 64 (blue and orange), for two magnetizations, as
indicated in the legend: $\sigma=3$ (green and blue) and $\sigma=50$ (red and
orange). The simulations have $L_{x}=1600\,c/\omega_{\mathrm{p}}$. (d)
Comparison with [17] (see the text for details), showing the particle spectra
at three different times, as indicated in the legend. In both panels (c) and
(d), we show the total particle spectrum (solid), as well as the spectrum of
$E>B$ particles (dashed) and of $E<B$ particles (dotted). All the spectra in
this figure refer to $B_{g}=0$ simulations.

##  II Test-particles in $B_{g}=0$ simulations

In Fig. 5(b), we show with the solid black line the total particle spectrum at
the final time ($\omega_{\rm p}t=4000$) for our fiducial simulation with
$\sigma=50$, $B_{g}=0$ and $L_{x}=1600\,c/\omega_{\mathrm{p}}$. The dotted
black line shows the spectrum of $E<B$ particles at the same time, whereas
dash-dotted lines present the final spectrum of test-particles. They are
initialized as regular particles, but when they pass through $E>B$ regions, we
artificially fix their $\gamma\beta=\sqrt{\gamma^{2}-1}$ to the value
indicated in the legend. The dash-dotted lines for $\gamma\beta=1$ and $3$ are
nearly identical, demonstrating that our conclusions are robust as long as
test-particles are constrained to stay at trans-relativistic energies while in
$E>B$ regions. Both cases are similar to the dotted black line, i.e., test-
particles that are forced to keep a low $\gamma\beta$ while in $E>B$ regions
display similar spectra as particles that never had $E>B$ encounters.

Higher values of $\gamma\beta$ bring the high-energy end of the test-particle
spectrum closer to the one of regular particles (black solid line). In
particular, the spectrum of $\gamma\beta=20$ test-particles nearly overlaps
with the spectrum of regular particles. Once again, this suggests that $E>B$
regions lead to acceleration up to $\gamma\sim 20\sim\sigma/2$, as
demonstrated by Fig. 3 in the main paper. As long as $E>B$ regions are allowed
(or constrained, as we do here with test-particles) to accelerate particles up
to $\gamma\sim\sigma/2$, the resulting test-particle spectrum is consistent
with the spectrum of regular particles.

###  II.1 Convergence studies

In Fig. 5(c), we show that the total particle spectra (solid lines), as well
as the spectra of $E>B$ particles (dashed lines) and of $E<B$ particles
(dotted lines), are identical when employing 16 particles per cell (our
fiducial value) or 64 particles per cell. This conclusion holds for both low
($\sigma=3$) and high ($\sigma=50$) magnetizations.

###  II.2 Comparison with Guo et al. (2019)

In Fig. 5(d), we directly compare our results to the work by [17], since their
conclusions are in contradiction with our findings. In particular, they
reported evidence that most high-energy particles never had $E>B$ encounters,
and they argued that the spectrum of high-energy particles would remain the
same if non-ideal fields were to be excluded.

To ensure a fair comparison, we initialize the plasma with the same
temperature as in [17] ($kT=0.36\,mc^{2}$) and we initiate reconnection by
hand (“triggered” setup, where we reduce the pressure near the center
$(x,y)=(0,0)$ of the layer at the initial time). Our spatial resolution
$\,c/\omega_{\mathrm{p}}=5$ is larger than in [17], and our box length
$L_{x}=1600\,c/\omega_{\mathrm{p}}$ is also greater. The “cold” magnetization
(normalized to the plasma rest mass energy density) is $\sigma=50$,
corresponding to an effective magnetization (normalized to the enthalpy
density) of $\sigma_{\rm eff}\simeq 24$. The simulation has $B_{g}=0$.

In all the spectra showed so far, we only accounted for particles belonging to
the reconnection region (defined such that it contains a mixture of particles
starting from $y>0$ and $y<0$ [41], with both populations contributing at
least 10%). Instead, in order to compare directly with Fig. 3(c) by [17], Fig.
5(d) includes all particles within $\sim 0.25L_{x}$ from the midplane $y=0$
(upstream particles populate the low-energy bump at $\gamma-1\lesssim 3$). We
present the total particle spectrum (solid), as well as the spectrum of $E>B$
particles (dashed) and of $E<B$ particles (dotted), at three different times,
as indicated by the legend. While at early times the high-energy end is mostly
populated by $E<B$ particles, at later times — when the spectrum has finally
developed a significant high-energy component — the situation is reversed,
with $E>B$ particles clearly dominating the high-energy spectrum. At late
times, the total particle spectrum (solid) is significantly harder than the
spectrum of $E<B$ particles (dotted). We therefore disagree with the
conclusions by [17], who claimed that most high-energy particles never had an
$E>B$ encounter, and that the spectrum of high-energy particles would remain
the same if non-ideal fields were to be ignored.

Note that the trend we observe, with $E>B$ particles becoming dominant at
later times, can also be seen in the time sequence displayed by Fig. 3(c) of
[17], yet this was apparently overlooked when deriving their conclusions.

###  II.3 Steady state

In Fig. 6, we show the time dependence of the fractions reported in Fig. 2(c)
of the main paper (same color coding). We display the contribution of $E>B$
particles to the total census in the reconnection region (blue), and to the
number of particles with $\gamma>\sigma/4$ (green) and $\gamma>\sigma$ (red),
as a function of time. The black line shows the fraction of length along $y=0$
occupied by $E>B$ regions. To ensure that the system can achieve a statistical
steady state, here we employ a triggered setup and outflow boundary conditions
in the $x$ direction. Otherwise, the numerical and physical parameters are the
same as in our fiducial run, with $\sigma=50$, $B_{g}=0$ and
$L_{x}=1600\,c/\omega_{\mathrm{p}}$. In contrast to the case of periodic
boundaries in $x$, which artificially choke reconnection after a few
$L_{x}/c$, with outflow boundaries one can follow the quasi-steady evolution
of the system as long as computational resources allow [24, 16].

Fig. 6 yields two important conclusions. First, a quasi-steady state is
established at $\omega_{\rm p}t\gtrsim 1200\sim 0.75L_{x}/c$. This is the time
when the two reconnection fronts that were generated at $t=0$ near the center
of the domain advect out of the boundaries (for details, see [24]). Both the
fractions of $E>B$ particles and the occupation fraction of $E>B$ regions
along $y=0$ are nearly constant in time. Second, these fractions are
remarkably the same for the outflow simulation presented in Fig. 6 and for the
periodic run in Fig. 2(c) (see the data points at $\sigma=50$). In particular,
it still holds true that $E>B$ particles contribute a fraction $\sim 20\%$ of
the particle census in the reconnection region, which is twice larger than the
occupation fraction of $E>B$ regions. See the main text for the explanation of
this factor of two.

Figure 6: Time evolution of the fraction of $E>B$ particles in the
reconnection region (blue), and the fraction of $E>B$ particles with
$\gamma>\sigma/4$ (green) and $\gamma>\sigma$ (red). The black line shows the
fraction of length along $y=0$ occupied by $E>B$ regions. We adopt the same
physical parameters as our fiducial run ($\sigma=50$, $B_{g}=0$ and
$L_{x}=1600\,c/\omega_{\mathrm{p}}$), but we employ outflow boundaries in $x$
and a triggered setup. The vertical dashed line indicates the time when the
two reconnection fronts that were generated at $t=0$ near the center of the
domain advect out of the boundaries. This illustrates that the fractions of
$E>B$ particles and the occupation fraction of $E>B$ regions attain a quasi-
steady state at $\omega_{\rm p}t\gtrsim 1200$.

###  II.4 Particle trajectories

We present a representative set of particle trajectories in the animation
particletrack.mov. We employ our fiducial case with $\sigma=50$, $B_{g}=0$ and
$L_{x}=1600\,c/\omega_{\mathrm{p}}$. The top panel shows the trajectory of
1500 simulation particles (yellow points), selected randomly to start around
$y\simeq 170\,c/\omega_{\mathrm{p}}$ at $\omega_{\rm p}t\simeq 650$. We follow
these particles over time, superimposed over the 2D plot of density (grey
scale). When a particle interacts for the first time with an $E>B$ region
(i.e., the particle is injected), it is depited as an open green circle (for
each particle, the green circle is shown only at the time of injection). The
title of the second panel tracks the fraction of particles (from the set of
yellow points) that over time get injected at $E>B$ regions. The fraction of
injected particles rises up to 7% after their first interaction with the
layer, and then further increases up to 18%, primarily as a result of the
merger of two large plasmoids. This is in line with what we describe in the
main paper when commenting on Fig. 2(c).

The bottom panel of the animation displays the energy history of three
particles, whose spatial trajectory is shown in the top panel by the filled
circles with the same color. The particles are selected randomly for each
decade in Lorentz factor ($\gamma-1$ from $1$ to 10, from $10$ to 100, from
100 to 1000). The low-energy particle (yellow) belongs to the $E<B$
population. This is expected, since most particles ending up with low energies
do not go through non-ideal regions. Instead, both the orange and the red
particles belong to the $E>B$ population; again, this is expected, since $E>B$
particles dominate the high-energy end of the spectrum. For the orange and red
particles, their injection time (i.e., the first time they experience $E>B$)
is denoted by the vertical dashed line having the same color.

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# Two-color differential dynamic microscopy for capturing fast dynamics

R. You  R. McGorty<EMAIL_ADDRESS>Department of Physics and
Biophysics, University of San Diego, San Diego, CA 92110, USA

###### Abstract

Differential dynamic microscopy (DDM) is increasingly used in the fields of
soft matter physics and biophysics to extract the dynamics of microscopic
objects across a range of wavevectors using optical microscopy. Standard DDM
is limited to detecting dynamics no faster than the camera frame rate. We
report on an extension to DDM where we sequentially illuminate the sample with
spectrally-distinct light and image with a color camera. By pulsing blue and
then red light separated by a lag time much smaller than the camera’s exposure
time we are able to use this two-color DDM method to measure dynamics
occurring much faster than the camera frame rate. The following article has
been accepted by Review of Scientific Instruments. After it is published, it
will be found at https://aip.scitation.org/journal/rsi.

††preprint: AIP/123-QED

A number of optical techniques allow users to quantify the dynamics of small
particles, molecules, intracellular bodies or whole cells. These include
single-particle trackingCrocker and Grier (1996), image correlation
spectroscopyPetersen  _et al._ (1993), fluorescence correlation
spectroscopyMagde, Elson, and Webb (1972), and dynamic light scatteringBerne
and Pecora (2000). Researchers were given another option to add to this list
in 2008: differential dynamic microscopy (DDM)Cerbino and Trappe (2008). Its
advantages include being able to extract particle dynamics from time series of
images even when the contrast is too low or the particle concentration is too
high to allow for accurate particle localizations and being compatible with a
number of optical microscopy modalities including bright-field, dark-
fieldBayles, Squires, and Helgeson (2016), wide-field fluorescence, confocalLu
_et al._ (2012), and light-sheet microscopiesWulstein  _et al._ (2016).

As DDM can be used with nearly any optical microscope with a digital camera
and yields data analogous to what one would obtain with dynamic light
scattering, it has been applied to numerous system. To provide just a partial
list, DDM has been used to measure the dynamics of: colloidal particles or
nanoparticlesHe  _et al._ (2012); anisotropic particlesReufer  _et al._
(2012); swimming bacteriaWilson  _et al._ (2011); biomacromolecules or
particles in biomimetic environmentsRegan  _et al._ (2019); Burla  _et al._
(2020); colloidal gelsCho, Cerbino, and Bischofberger (2020); probe particles
for microrheological determinations of viscoelasticityBayles, Squires, and
Helgeson (2017); particles in crowded environmentsSentjabrskaja  _et al._
(2016); and foamsGiavazzi, Trappe, and Cerbino (2020).

In this paper, we present a modification to DDM that allows dynamics faster
than the camera frame rate to be measured. With standard DDM, one is unable to
quantify dynamics occurring over times shorter than the time interval between
camera frames. This makes studying fast dynamics problematic without access to
high-speed cameras. We show that by using a color camera and illuminating the
sample with spectrally-separated pulses of light one can uncover dynamics
occurring faster than the camera frame rate. Inspiration for this method came
from two-color particle velocimetry which has been used to measure fast
dynamicsAdrian (1986); Goss  _et al._ (1991).

To use standard DDM, one acquires a time series of images taken using any
spatially invariant microscopy method and analyzes the difference between
images separated by a given lag time in Fourier space. One can then find the
time it takes for intensity fluctuations to decay as a function of the spatial
frequency or wavevector, $\bm{q}$. In practice, one first calculates the
difference between images separated by a lag time $\Delta t$,
$d(\bm{x},t,\Delta t)=I(\bm{x},t+\Delta t)-I(\bm{x},t)$ where $\bm{x}$ is the
pixel position. One next obtains the image structure function by taking two
dimensional Fourier transforms of these differences and averaging over all
times $t$: $D(\bm{q},\Delta t)=\langle|\hat{d}(\bm{x},t,\Delta
t)|^{2}\rangle_{t}$. Isotropic samples will result in a radially symmetric
image structure function which can then be azimuthally averaged to yield
$D(q,\Delta t)$ where $q=\sqrt{q_{x}^{2}+q_{y}^{2}}$. This function can then
be fit to

$\displaystyle D(q,\Delta t)=A(q)\big{(}1-f(q,\Delta t)\big{)}+B(q),$ (1)

where the amplitude, $A(q)$, depends on the scattering properties of the
sample and the optical properties of the microscope, the background, $B(q)$,
depends on the noise in the image, and $f(q,\Delta t)$ is the intermediate
scattering function which accounts for the sample dynamics. For diffusive
dynamics $f(q,\Delta t)=\exp(-\Delta t/\tau(q))$ where $\tau=(Dq^{2})^{-1}$
and $D$ is the diffusion coefficient.

Given that DDM analyzes images acquired with a digital camera on an optical
microscope, the range of accessible spatial scales span from the diffraction
limit or pixel size (whichever is greater) to the size of the field of view.
In typical studies of thermally-diffusing colloids or moving bacteria, this
range is often from the submicron to 100 $\mu$m. Over what time scales will
dynamics over this spatial range occur? For diffusive dynamics, the time for a
density fluctuation to decay is proportional to the length scale squared. That
is $\tau\propto q^{-2}$, where $\tau$ is the decay time and $q$ is the
wavevector. Therefore, to probe diffusive dynamics across two orders of
magnitude in space requires data over four orders of magnitude in time.

How can one acquire data over such a span of timescales? In several DDM
studies, image sequences are acquired at multiple frame rates. As one example,
Germain et al. investigate the dynamics of micron-sized colloids and bacteria
and acquire images at both 400 Hz and 4 HzGermain, Leocmach, and Gibaud
(2016). This allows the authors to cover time scales from 2.5 ms to 1000 s.
More recently, Arko and Petelin developed a dual-camera method to acquire DDM
data over about 6 orders of magnitude in timeArko and Petelin (2019). Using a
beam splitter they imaged their sample onto two separate cameras, each
recording frames at 200 Hz. By triggering the cameras at offset times and
comparing frames between the two cameras, they could measure time lags much
smaller than with a single camera.

In this paper, we describe a new method which likewise provides access to
dynamics faster than the frame rate of our camera. Using a single color camera
and illuminating the sample with pulses of blue and red light, we can compare
the blue and red channels of a single image to observe changes within the
sample over times much smaller than the exposure time of a single frame.

We perform two-color DDM measurements with the setup shown in Figure 1a. The
light from red and blue LEDs (M625L2 and M455L3, Thorlabs) are combined with a
longpass dichroic mirror (DMLP550R, Thorlabs) and illuminate the sample. A 40×
objective (0.65 NA, Olympus) and $f$ = 200 mm tube lens (AC254-200-ML-A,
Thorlabs) image the sample onto a color CMOS camera (DFK 37BUX287, The Imaging
Source). Triggering the two LEDs and the camera is a Digilent Analog Discovery
2 (National Instruments).

Figure 1:  (a) Schematic of our optical microscope shows two LED light sources
(one with a peak wavelength of 625 nm, the other 455 nm) used to illuminate a
sample. A 40× objective and tube lens image the sample onto a color CMOS
camera. (b) Within a single exposure time of the camera, we pulse each LED.
The time interval between pulses is varied and, for the data shown, ranges
from 3 to 91 ms.

The trigger sequencing is indicated in Figure 1b. We trigger the camera to
start an exposure every 100 ms, resulting in a frame rate of 10 Hz. Within a
single exposure time, both the blue (455 nm) and red (625 nm) LEDs each turn
on for 1 ms. The spacing between the blue and red pulses is staggered. We use
a sequence of 10 pulse delays in our experiments: 3, 5, 7, 10, 14, 20, 29, 43,
62 and 91 ms. We repeat this sequence of different pulse delays 800 times.
Therefore, we acquire 8000 color images acquired at 10 Hz. However, the short
delay times between the blue and red illumination pulses allow us to measure
dynamics occurring much faster than 10 Hz.

The analysis of this data follows closely to what was previously described for
standard DDM. However, we now first separate the recorded time series of
images into blue and red channels. We then calculate the image structure
function using the differences between data in the blue and red channels:
$d_{two-color}(\bm{x},t,\Delta t)=I_{red}(\bm{x},t+\Delta
t)-I_{blue}(\bm{x},t)$. Here, $\Delta t$ no longer must be some integer
multiple of the time between adjacent frames (as in standard DDM) but can be
as small as the time separating the blue and red illumination pulses.

We evaluate this method using a sample of 180-nm-diameter polystyrene
particles (FluoSpheres F8811, Molecular Probes) suspended in water. These
particles are diluted by a factor of 100 and a few microliters are sealed
between a glass slide and coverslip. This sample was first investigated with
standard DDM using a camera running at 80 Hz and constant bright-field
illumination. We found a diffusion coefficient of $2.48\pm 0.03\mathrm{\mu
m^{2}/s}$. We then acquired data using the two-color DDM method as described
where we acquired 8000 frames of 256$\times$256 pixels with the camera running
at 10 Hz and the blue and red illumination being pulsed with the times between
pulses ranging from 3 to 91 ms.

Figure 2:  (a) For five different values of the wavevector $q$ we plot
$f(q,\Delta t)$. In blue and red split circles, the values of $f(q,\Delta t)$
are shown for 10 lag times corresponding to the time intervals between the
blue and red illumination pulses that occur within a single camera frame.
These lag times range from 3 to 91 ms. In violet diamonds, the values of
$f(q,\Delta t)$ are shown using standard DDM methods on just the blue channel
of the acquired image sequence. Since the camera was running at 10 Hz, for
standard DDM methods the minimum time lag corresponds to 100 ms. The solid
lines represent the theoretical intermediate scattering function for diffusive
dynamics, $f(q,\Delta t)=\exp(-Dq^{2}\Delta t)$. (b) The determined decay
rate, $\tau^{-1}$, is plotted versus $q^{2}$. For diffusive dynamics,
$\tau^{-1}=Dq^{2}$ and the solid line here shows
$D=2.48\mu\mathrm{m}^{2}\textfractionsolidus\mathrm{s}$. Data from the
standard DDM method (violet diamonds) starts deviating from this straight line
just after the expected decay rate exceeds 10 s-1, as anticipated given the
images were acquired at 10 frames per second. Data from the two-color DDM
method follows the expected linear relationship up to around 70 s-1. The inset
focuses on the low-$q$ region. The two-color method does not accurately detect
data fluctuations slower than about 10 s-1.

After splitting this time series of images into blue and red channels, we find
the two-color image structure function, $D_{two-color}(q,\Delta t)$. This two-
color image structure function is found for the 10 time lags, $\Delta t$, that
correspond to the delay times between the blue and red illumination pulses
that occur within a single camera frame. With the same sequence of 8000 images
we also find $D_{standard}(q,\Delta t)$ using only data from the blue channel
of each frame. For this, the values of $\Delta t$ range from 100 ms (the time
between frames) to 10 s.

For both $D_{two-color}(q,\Delta t)$ and $D_{standard}(q,\Delta t)$, we
proceed with typical DDM analysis where for each $q$ we fit the image
structure function to Equation 1 and find $A$, $B$ and $\tau$. In Fig. 2a we
show $f(q,\Delta t)$, and the relationship between $\tau$ and $q$ is shown in
Fig. 2b. We observe that our data follows the expected exponential decay of
$f(q,\Delta t)$ whether using the two-color or standard method. However, for
the two-color method, we are able to observe the dynamics at larger
wavevectors. Our data is consistently fit for wavevectors with decay rates of
up to about 70 s-1. Such fast dynamics are inaccessible using standard DDM on
data acquired at 10 Hz. In Fig. 2b, we observe, as expected, that $\tau^{-1}$
depends linearly on $q^{2}$. Decay rates determined through
$D_{standard}(q,\Delta t)$ start to deviate from this linear relationship
beyond about 10 s-1. However, with $D_{two-color}(q,\Delta t)$ we find decay
rates that match the expected linear relationship up to several times faster
than the rate at which images were acquired.

The decay rates, $\tau^{-1}$, determined from the two-color DDM method appear
noisier than those from the standard method. This is likely the result of less
data going into the two-color DDM method. When fitting our data to Equation 1,
for each value of $q$ we have ten data points (corresponding to the 10 time
intervals between blue and red illumination pulses) going into $D_{two-
color}(q,\Delta t)$ and 30 data points contributing to $D_{standard}(q,\Delta
t)$ (corresponding to lag times logarithmically spaced from 100 ms to 10 s).
Furthermore, for each lag time, $\Delta t$, we average together 800 Fourier
transformed image differences for $D_{two-color}(q,\Delta t)$. Whereas with
the standard DDM method applied to the same data set, we have 7999 Fourier
transformed image differences to average for $\Delta t$ of one frame (100 ms).

We note that modifications to this two-color method could be made. For
example, an image splitter system could be used to separate the two colors
onto distinct regions of the camera sensor which would allow for the use of
monochromatic cameras. Furthermore, while we applied this method to bright-
field imaging, one could use similar principles using fluorescence with
spectrally distinct fluorophores.

In summary, we have devised a method to allow differential dynamic microscopy
to be used to study dynamics faster than the camera frame rate. This obviates
the need for high speed cameras to measure fast dynamics if one can instead
illuminate the sample with spectrally-separated pulses offset in time and then
split the different color signals on the image sensor. We show that this two-
color DDM method allows us to extract the dynamics of colloidal particles that
occur up to several times faster than the camera frame rate.

R.M. acknowledges support from the Research Corporation for Science
Advancement through the Cottrell Scholars program.

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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