#  Electron-only reconnection and inverse magnetic-energy transfer at sub-ion
scales

Zhuo Liu  Plasma Science and Fusion Center, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Caio Silva  Plasma Science and Fusion
Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Lucio
M. Milanese  Proxima Fusion GmbH, Munich, 81671, Germany  Muni Zhou  School of
Natural Sciences, Institute for Advanced Study, Princeton, NJ 08544, USA  Noah
R. Mandell  Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA
Nuno F. Loureiro  Plasma Science and Fusion Center, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA

###### Abstract

We derive, and validate numerically, an analytical model for electron-only
magnetic reconnection applicable to strongly magnetized (low-beta) plasmas.
Our model predicts sub-ion-scale reconnection rates significantly higher than
those pertaining to large-scale reconnection, aligning with recent
observations and simulations. We apply this reconnection model to the problem
of inverse magnetic-energy transfer at sub-ion scales. We derive time-
dependent scaling laws for the magnetic energy decay and the typical magnetic
structure dimensions that differ from those previously found in the MHD
regime. These scaling laws are validated via two- and three-dimensional
simulations, demonstrating that sub-ion scale magnetic fields can reach large,
system-size scales via successive coalescence.

## Introduction.

Decaying turbulence in plasmas is sometimes accompanied by the transfer of
energy from small to large scales; a so-called inverse transfer. Though poorly
understood, this phenomenon is thought to be crucial to magnetic field
evolution at cosmological scales [banerjee2004evolution,
durrer2013cosmological, hosking2023cosmic] as well as in various astrophysical
systems such as the solar wind [chen2011anisotropy] and the heliosphere
[khabarova2015small, drake2012power]. The standard explanation for inverse
energy transfer appeals to the conservation of net magnetic helicity
[pouquet1978two, matthaeus1986turbulent, christensson2001inverse]. However,
recent studies have shown that this process can also occur in non-helical
turbulent systems [brandenburg2015nonhelical, reppin2017nonhelical]. It has
been proposed that this may be due to the successive merger of magnetic
structures via magnetic reconnection, thus transferring energy to
progressively larger scales [zhou2019magnetic, zhou2020multi, bhat2021inverse,
zhou2021statistical, hosking2021reconnection, milanese2023].

So far, investigations of this topic have mostly been restricted to the
resistive MHD framework. This implies they cannot be directly applied to the
(mostly) collisionless environments often encountered in astrophysical
contexts. For example, recent in-situ spacecraft measurements have provided
substantial evidence for the existence of numerous rope-like magnetic
structures at ion scales, with substructures within these flux ropes observed
at electron scales (e.g., [stawarz2018intense, zhong2020direct,
huang2021electron, hasegawa2023ion]). As such, the existing understanding of
inverse energy cascade needs to be extended to account for (i) kinetic
modifications to turbulent dynamics; and (ii) changes to magnetic reconnection
— as compared to resistive MHD systems.

The latter is of particular importance given the observation that the physics
of reconnection between sub-ion-scale structures is expected to be
substantially different: when the reconnection region is sufficiently small,
the ions can become unresponsive, and a transition to so-called “electron-only
reconnection” is expected. This variant of magnetic reconnection has been
recently observed in the Earth’s magnetosphere [phan2018electron,
stawarz2019properties], and is numerically found to exhibit substantially
higher reconnection rates compared to conventional scenarios
[jain2012electron, sharma2019transition, califano2020electron,
arro2020statistical, vega2020electron, greess2022kinetic,
guan2023reconnection].

The goal of this Letter is thus to generalize the current resistive-MHD-based
understanding of inverse energy transfer via magnetic reconnection to kinetic,
collisionless plasma environments.

## Equations.

We employ a two-field isothermal fluid model valid for low-beta plasmas [e.g.,
schekochihin2009astrophysical, zocco2011reduced, passot2018gyrofluid,
meyrand2021violation, zhou2023spectrum]. The dynamics of this system are
described by equations for the perturbed electron density $\delta n_{e}$ and
the parallel (to the guide-field $B_{z}$) vector potential $A_{z}$, which read
111These equations are the isothermal limit of the Kinetic Reduced Electron
Heating Model (KREHM), a rigorous analytical framework derived from
gyrokinetics under the condition of low electron plasma beta ($\beta_{e}\sim
m_{e}/m_{i}$, where $\beta_{e}$ is the ratio of the electron to the magnetic
pressure, and $m_{e}$ and $m_{i}$ are the electron and proton masses)
[zocco2011reduced]. See Supplemental Material for more details and results in
the non-isothermal limit.:

$\displaystyle\frac{1}{n_{0e}}\frac{d\,\delta
n_{e}}{dt}=\hat{\mathbf{b}}\cdot\nabla\frac{e}{cm_{e}}d_{e}^{2}\nabla_{\perp}^{2}A_{z},$
(1)
$\displaystyle\frac{d}{dt}\left(A_{z}-d_{e}^{2}\nabla_{\perp}^{2}A_{z}\right)=-c\frac{\partial\varphi}{\partial
z}+\hat{\mathbf{b}}\cdot\nabla\frac{\delta n_{e}}{n_{0e}}.$ (2)

Here,
$\hat{\mathbf{b}}\cdot\nabla\equiv\partial/\partial_{z}-1/B_{z}\\{A_{z},...\\}$
is the parallel (to the total field) gradient operator, and
$d/dt\equiv\partial/\partial t+c/B_{z}\\{\varphi,...\\}$ denotes the
convective time derivative, with $\varphi$ the electrostatic potential; both
operators feature a Poisson bracket, defined as
$\\{P,Q\\}\equiv\partial_{x}P\partial_{y}Q-\partial_{y}P\partial_{x}Q$. These
equations are closed via the gyrokinetic Poisson law [krommes2002fundamental],

$\frac{\delta
n_{e}}{n_{0e}}=\frac{T_{0e}}{T_{0i}}(\hat{\Gamma}_{0}-1)\frac{e\varphi}{T_{0e}},$
(3)

where $\hat{\Gamma}_{0}$ denotes the inverse Fourier transform of
$\Gamma_{0}(\alpha)=I_{0}(\alpha)e^{-\alpha}$, with $I_{0}$ the zeroth-order
modified Bessel function of the first kind and
$\alpha=k_{\perp}^{2}\rho_{i}^{2}/2$, where $\rho_{i}=v_{{\rm
th}i}/\Omega_{i}$ is the ion Larmor radius, with $v_{{\rm
th}i}=\sqrt{2T_{0i}/m_{i}}$ the ion thermal velocity and
$\Omega_{i}=eB_{z}/m_{i}c$ the ion Larmor frequency. Lastly,
$d_{e}=c/\omega_{pe}$ is the electron skin depth, with $\omega_{pe}=\sqrt{4\pi
n_{0e}e^{2}/m_{e}}$ the electron plasma frequency. In the following, we refer
to “perpendicular” as the direction perpendicular to the guide field $B_{z}$,
i.e., the $xy$ plane.

## Electron-only Reconnection.

We now use these equations to derive a two-dimensional reconnection model
valid at scales below the ion Larmor radius. We consider two identical
magnetic islands with typical magnetic field $B_{\perp}$ and radius $R$ such
that $d_{e}\ll R\ll\min(\rho_{i},\rho_{s})$, merging in the $x$ direction,
where $\rho_{s}=\rho_{i}/\sqrt{2T_{0i}/T_{0e}}$ is the ion sound Larmor
radius. The total magnetic flux to be reconnected is
$A_{z,\rm{total}}=B_{\perp}R$, and the time for this process to occur
(reconnection time) is denoted as $\tau_{\rm rec}$. The width of the current
sheet between the islands during reconnection is denoted as $\delta_{x}$,
assumed comparable to $d_{e}$ (the scale at which the frozen flux condition is
broken in our model), and the length of the current sheet is denoted as
$\delta_{y}$. In the current sheet, $A_{z}=A_{z,\rm{sheet}}\sim
B_{\perp}\delta_{x}$, consistent with an upstream reconnecting field
$B_{y}=\partial_{x}A_{z}\sim B_{\perp}$. The out-of-plane current in the sheet
is then $-(4\pi/c)J_{z}=\nabla_{\perp}^{2}A_{z}\sim B_{\perp}/\delta_{x}$.

In the limit $k_{\perp}\rho_{i}\gg 1$, Eq. (3) reduces to $\delta
n_{e}/n_{0e}=-e\varphi/T_{0i}$. Thus the advection of $\delta n_{e}$ term in
Eq. (1) vanishes. The advection of $A_{z}$ term in Eq. (2), i.e.,
$\\{\varphi,A_{z}\\}$, becomes $-(cT_{0i}/eB_{z})\\{\delta
n_{e}/n_{0e},A_{z}\\}$, which can be combined with the R.H.S. of Eq. (2).
Taking $\partial_{x}\sim 1/\delta_{x}$ and $\partial_{y}\sim 1/\delta_{y}$ and
balancing both sides of Eq. (1) within the sheet, we find $(\delta
n_{e}/n_{0e})/\tau\sim(ed_{e}^{2}/B_{z}cm_{e})(B_{\perp}^{2}/\delta_{x}\delta_{y})$,
where $\tau$ is the time it takes to reconnect the magnetic flux
$A_{z,\rm{sheet}}$. A similar term-balancing procedure for Eq. (2) yields
$A_{z}/\tau\sim(c(T_{0e}+T_{0i})/eB_{z}n_{e0})(B_{\perp}\delta
n_{e}/\delta_{y})$. Note that the electron inertia term does not affect this
balance since
$d_{e}^{2}\nabla_{\perp}^{2}A_{z}\sim(d_{e}/\delta_{x})^{2}A_{z}\sim A_{z}$.

Combining these results, we find
$\tau\sim(1/\sqrt{1+T_{0i}/T_{0e}})(\delta_{y}/\rho_{s})(d_{e}/v_{A,\perp})$,
where $v_{A,\perp}=B_{\perp}/\sqrt{4\pi n_{0}m_{i}}$ is the (ion) Alfvén
speed. The last step is to relate the reconnection time, $\tau_{\rm rec}$, and
$\tau$ via $\tau_{\rm rec}/\tau\sim A_{z,\rm{total}}/A_{z,\rm{sheet}}\sim
R/d_{e}$. Writing the reconnection time in the form of $\tau_{\rm
rec}\sim\mathcal{R}_{\rm rec}^{-1}R/v_{A,\perp}$, we obtain the normalized
reconnection rate:

$\mathcal{R}_{\rm
rec}\sim\sqrt{1+\frac{T_{0i}}{T_{0e}}}\frac{\rho_{s}}{\delta_{y}}\sim\sqrt{1+\frac{T_{0i}}{T_{0e}}}\frac{\rho_{s}}{R},$
(4)

where we have assumed $\delta_{y}\sim R$ (see Supplemental Material for
validation). When $\rho_{s}>R$, the (normalized) reconnection rate can become
much larger than the value of $\sim 0.1$ typically associated with
collisionless reconnection [birn2001GEM, cassak2017review, liu2017does] (but
may not exceed $\sim\rho_{s}/d_{e}$).

The outflow velocity, $u_{\rm out}$, can be calculated as $u_{\rm
out}\sim(c/B_{x})E_{z}\sim cR/(d_{e}B_{\perp})(B_{\perp}R/c\tau_{\rm
rec})\sim\sqrt{1+T_{0i}/T_{0e}}(\rho_{s}/d_{e})v_{A,\perp}\sim\sqrt{1+T_{0i}/T_{0e}}\sqrt{\beta_{e}}v_{Ae,\perp}$,
with $v_{Ae,\perp}\equiv B_{\perp}/\sqrt{4\pi n_{0e}m_{e}}$ the electron
Alfvén speed based on perpendicular magnetic field. The conclusion that the
outflow velocity is proportional to electron Alfvén speed is consistent with
previous laboratory observations [shi2022laboratory]. Therefore, if instead we
normalize the reconnection time to the outflow time, i.e., $\tau_{\rm
rec}\sim\mathcal{R}_{\rm rec,e}^{-1}R/u_{\rm out}$, we obtain, unsurprisingly,

$\mathcal{R}_{\rm rec,e}\sim{d_{e}}/{R};$ (5)

i.e., the aspect ratio of the reconnecting current sheet.

We now proceed to validate this model via direct numerical simulations. Eqs.
(1,2,3) are solved numerically using the pseudo-spectral code GX
[mandell2018laguerre, mandell2022gx]. Hyper-diffusive terms of the form
$\nu_{H}\nabla_{\perp}^{6}$ are added to the R.H.S. of Eqs. (1,2), with
$\nu_{H}=1/\Delta t(\Delta x/\pi)^{6}$; this value ensures that these terms
are only significant at scales comparable to the grid scale. These details
apply to all the simulations we include in this Letter.

In what follows, all quantities are presented in dimensionless units. The
simulations are performed in a doubly periodic box of dimensions $L_{x}\times
L_{y}$, where $L_{x}=L_{y}=2\pi$. The initial condition consists of two
identical magnetic islands of Gaussian shape, specified as
$A_{z}=A_{0}\exp(-(2\pi a(x-0.25L_{x})/L_{x})^{2}-(2\pi
a(y-0.5L_{y})/L_{y})^{2})+A_{0}\exp(-(2\pi a(x-0.75L_{x})/L_{x})^{2}-(2\pi
a(y-0.5L_{y})/L_{y})^{2})$ 222This initial configuration is approximately an
equilibrium but it is unstable to coalescence instability
[finn1976coalescence]. We choose this configuration instead of the typical
Harris sheet [harris1962plasma] for consistency with the study of the inverse
magnetic energy transfer problem discussed below. However, we do not expect
the results to be sensitive to the specific configuration chosen.. We set
$a=1.5$ so that the root-mean-squared width of each island is about $1.1$,
i.e., $R\approx 0.55$. Multiple simulations are performed by varying the
values of $\rho_{i}$ (we set $T_{i0}/T_{e0}=1$ in all the simulations so
$\rho_{s}=\rho_{i}/\sqrt{2}$) and $d_{e}$ to encompass both the regime where
$R\gtrsim\rho_{i}>d_{e}$ (traditional ion-coupled reconnection) and
$\rho_{i}>R\gg d_{e}$ (electron-only reconnection). The numerical resolution
is adjusted across simulations to resolve the $d_{e}$ scale adequately (see
the Supplemental Material for more information).

Figure 1:  Dimensionless peak reconnection rates, $\mathcal{R}_{\rm rec}$,
from simulations with different values of $\rho_{i}/R$. In the electron-only
regime ($d_{e}<R\leq\rho_{i}$), red, orange, and yellow symbols refer to
simulations with $d_{e}/R=0.05,0.1,0.2$, respectively. In the ion-coupled
regime ($d_{e}<\rho_{i}<R$), we set $d_{e}/\rho_{i}=0.1$ (except for the
leftmost data-point, where $d_{e}/\rho_{i}=0.2$). The dashed line is Eq. (4)
(with $T_{i0}/T_{e0}=1$).

For each simulation, we calculate the reconnection rate as the time derivative
of the magnetic flux between the X- and the O-points. The peak reconnection
rates are plotted in Fig. 1 against the parameter $\rho_{i}/R$. Along the
horizontal axis, the progression from left to right signifies a transition
from MHD to sub-$\rho_{i}$ scales in terms of the initial island size. At MHD
scales, we observe that the reconnection rate converges to a constant value
$\sim 0.2$, somewhat higher than the often quoted value of $0.1$ [e.g.,
birn2001GEM, Isobe2005MeasurementOT, Zenitani2008, Drake2008, Bessho_2012,
cassak2017review, Chen2017, liu2017does] but consistent with previous results
for collisionless reconnection in the strong guide-field limit
[loureiro2013fast, numata2015ion]. However, upon transition to sub-ion scales
(i.e., $R<\rho_{i}$), the reconnection rates exhibit a substantial increase,
aligning almost perfectly with our model’s prediction, Eq. (4), represented by
the dashed line in Fig. 1. Consistently, at fixed $\rho_{i}/R$, the
reconnection rate depends only weakly on the value of $d_{e}$ (and such
dependence weakens as $d_{e}$ becomes smaller). We also note that the
reconnection rate starts to significantly deviate from the large-scale (ion
coupled) value ($\sim 0.2$) at around $R\sim 10\rho_{i}$. This observation is
consistent with the analysis in [mallet2020onset] and with simulation results
[sharma2019transition], where it is predicted that ions start to decouple from
the reconnecting magnetic field at scales of around 10 ion-scales. However, it
is not until $R\sim\rho_{i}$ that the reconnection rate perfectly aligns with
Eq. (4).

Lastly, we note that although our focus here is on low-beta plasmas, it is
relatively straightforward to extend our analytical model to the
$\beta_{e}\sim 1$ case; see Supplemental Material for details.

## Magnetic Flux Tube Coalescence.

We now apply the electron-only reconnection model to the problem of inverse
transfer of magnetic energy at sub-ion scales within the framework of Eqs.
(1-3). We first address two-dimensional dynamics in the perpendicular plane.
Consider two identical circular magnetic islands of radius $R$ such that
$\rho_{i}>R\gg d_{e}$. Each carries the magnetic flux $RB_{\perp}$, which is
assumed to be conserved by their merger [zhou2019magnetic, zhou2020multi]. The
lifetime of these two islands is determined by the reconnection time,
$\tau_{{\rm rec}}\sim(R/\rho_{s})(R/v_{A,\perp})$, as shown above.

Now considering a “sea” of (volume-filling) magnetic islands with similar
sizes in the sub-$\rho_{i}$ regime and assuming the merging process to occur
sequentially between island pairs, the evolution of total magnetic energy can
be described by

$\frac{d\mathcal{E}}{dt}\propto\frac{dB_{\perp}^{2}}{dt}\sim-\frac{B_{\perp}^{2}}{\tau_{\rm
rec}}\propto-B_{\perp}^{5}.$ (6)

We thus find $\mathcal{E}=[a(t-t_{0})+\mathcal{E}_{0}^{-3/2}]^{-2/3}$, where
$t_{0}$ is the lifetime of the first island generation, $a$ is a coefficient,
and $\mathcal{E}_{0}$ is the initial magnetic energy. The perpendicular
coherence scale and magnetic field strength can be derived in a similar way.
The resulting long-term ($\tilde{t}\equiv t/t_{0}\gg 1$) evolution scaling
laws are

$k_{\perp}\propto\tilde{t}^{-1/3},\quad
B_{\perp}\propto\tilde{t}^{-1/3},\quad\mathcal{E}\propto\tilde{t}^{-2/3}.$ (7)

When extending to the three-dimensional case (with a strong guide field), a
magnetic island of radius $R$ and magnetic field $B_{\perp}$ becomes a
magnetic flux tube with radius $R$, poloidal magnetic field $B_{\perp}$, and
length $l$ in the direction of the local mean field. The arguments above
leading to Eq. (7) should remain valid (which we will confirm via direct
numerical simulation). The length of the flux tube $l$ can be estimated from
the assumption that the inverse linear time scale,
$\gamma_{l}\sim\omega_{KAW}\sim k_{\perp}\rho_{s}k_{\|}v_{A}$ (with
$k_{\perp}\sim 1/R$ and $l\sim 1/k_{\|}$), should be comparable to the inverse
nonlinear time scale, $\gamma_{nl}^{-1}$ (critical balance
[goldstein1995properties]), where $\gamma_{nl}^{-1}$ should be the smallest of
the reconnection time, $\tau_{\rm rec}$, and the perpendicular advection time,
$R/u_{\perp}$, with $u_{\perp}\sim(c/B_{z})\varphi/R$ being the in-plane
($\mathbf{E}\times\mathbf{B}$) velocity that pertains to Eqs. (1-2). By
assuming scale-by-scale equipartition between the density and magnetic energy
fluctuations, $(\delta
n_{e}/n_{0e})^{2}n_{0e}T_{0e}\sim|\nabla_{\perp}A_{z}|^{2}/8\pi$
[zhou2023spectrum] (see Supplemental Material for validation of this
hypothesis), together with Eq. (3) in the limit $k_{\perp}\rho_{i}\gg 1$, we
find $\varphi\sim\sqrt{T_{0i}/T_{0e}}(\rho_{i}v_{A}/c)B_{\perp}$ and, thus,
$R/u_{\perp}\sim(R/\rho_{s})(R/v_{A,\perp})\sim\tau_{\rm rec}$. Therefore, we
conclude that the nonlinear advection time and reconnection time are
comparable within this regime, i.e., $\gamma_{nl}\sim\tau_{\rm rec}^{-1}\sim
k_{\perp}\rho_{s}k_{\perp}v_{A,\perp}$. This conclusion differs from that in
the MHD regime, where reconnection is always slower [zhou2020multi]. Finally,
balancing $\gamma_{l}$ with $\gamma_{nl}$ yields

$l\sim{B_{z}}/({k_{\perp}B_{\perp}})\propto\tilde{t}^{2/3}.$ (8)

To verify these predictions, we conducted a three-dimensional simulation in a
periodic domain with dimensions $L_{x}\times L_{y}\times L_{z}$, where
$L_{x}=L_{y}=2\pi$ and $L_{z}=4L_{x}$ (in normalized units). We specify the
initial equilibrium as $A_{z}(x,y,z,t=0)=A_{z0}\cos(k_{0}x)\cos(k_{0}y)$ with
$k_{0}=10$ with $A_{z0}=0.1$, yielding a $2k_{0}\times 2k_{0}$ static array of
magnetic flux tubes with sequentially opposite polarities. We set
$\rho_{i}=2.0$ (and $T_{i0}=T_{e0}$) and $d_{e}=0.04$. These choices ensure
that all flux tubes satisfy $d_{e}\ll R<\rho_{i}$, and thus Eqs. (7-8) should
be valid after an initial transient.

Figure 2:  Three-dimensional simulation results. (a) Temporal evolution of the
wavenumber corresponding to the peak of the magnetic spectrum (top panel) and
total magnetic energy (bottom panel). $a,b,c,d$ are fitting parameters. We use
the data to the right of the gray vertical dashed lines to avoid the transient
stage. (b) Parallel coherence length $l$ at different times. (c) The
perpendicular magnetic spectra at progressively later times (from blue to
green). The vertical dashed line denotes the $d_{e}$ scale.

Fig. 2 presents the temporal evolution of some key quantities. The top panel
of Fig. 2 (a) shows the time evolution of the wavenumber at which the magnetic
energy spectrum peaks, $k_{\perp,\rm{max}}$; whereas the bottom panel shows
the box-averaged magnetic energy, $\mathcal{E}$. Both quantities conform to
the theoretical predictions after $t\approx 0.3\tau_{A}$, with the scaling
laws of Eq. (7) becoming applicable for $t\gtrsim 1.0\,\tau_{A}$ 333The fit is
slightly worse for $k_{\perp,\rm{max}}$ than for $\mathcal{E}$ due to the
discrete nature of this diagnostic and the limited change in the value of
$k_{\perp,\rm{max}}$, given the constrained size of the simulation domain..
Likewise, the parallel coherence length $l$, estimated using the five-point
second-order structure function [cho2009simulations, cerri2019kinetic,
zhou2023spectrum] (see the Supplemental Material for further details), aligns
with Eq. (8) after approximately $t\approx 2.0\,\tau_{A}$, as evident from
Fig. 2 (b) 444Before the system settles into a natural state of decaying
turbulence described by our theory, it undergoes a transient stage
characterized by the development of strong turbulence. An essential indicator
of strong turbulence is the establishment of critical balance, where flux
tubes initially confined to box length break in the $z$-direction. This
reduction in the parallel coherence length $l$ is captured by the first two
data points in Fig. 2 (b)..

Fig. 2 (c) shows the evolution of the perpendicular magnetic-energy spectrum.
A reasonable agreement with a $k_{\perp}^{-8/3}$ [boldyrev2012spectrum,
zhou2023spectrum] inertial range power-law behavior is observed, suggesting
that, in addition to the inverse cascade arising from the flux-tube merger,
there is also a direct cascade driven by the nonlinear interaction of kinetic
Alfvén wave packets. This observation is consistent with measurements of
turbulence in the Earth’s magnetosheath in instances where electron-only
reconnection is prevalent [stawarz2019properties].

Lastly, we note that although the above results are obtained within the
isothermal limit, we verify (see Supplemental Material) that the temporal
power laws still hold in the non-isothermal case.

## Transition to MHD scales.

As the flux tube (or island) merging proceeds, structures get progressively
larger and eventually transition to MHD scales, i.e., $R\gg\rho_{i}$. In this
regime, electron-only reconnection no longer applies; instead, we expect the
(normalized) reconnection rate to be a constant, independent of either the
island size or the kinetic scales (Fig. 1 indicates $\mathcal{R}_{\rm
rec}\approx 0.2$). This observation, together with flux conservation, is all
that is required to conclude that, at this stage, the evolution scalings of
resistive MHD [zhou2019magnetic, zhou2020multi, zhou2021statistical] should
apply here as well. Namely, we expect

$k_{\perp}\propto\tilde{t}^{-1/2},\quad
B\propto\tilde{t}^{-1/2},\quad\mathcal{E}\propto\tilde{t}^{-1}.$ (9)

To directly investigate these predictions, we conducted a two-dimensional
simulation for which the initial condition is
$\hat{A_{z}}(\mathbf{k_{\perp}})=A_{z0}(k_{\perp}^{4}/k_{c}^{4})\exp(i\phi(\mathbf{k_{\perp}}))$
for $k_{\perp}\leq k_{c}$ and
$\hat{A_{z}}(\mathbf{k_{\perp}})=A_{z0}(k_{\perp}^{4}/k_{c}^{4})\exp(1-k_{\perp}^{4}/k_{c}^{4})\exp(i\phi(\mathbf{k_{\perp}}))$
for $k_{\perp}>k_{c}$, where $\phi(\mathbf{k_{\perp}})$ is a random phase,
$A_{z0}=0.001$ and $k_{c}=160$. The domain is a doubly periodic box of
dimensions $L_{x}\times L_{y}$, with $L_{x}=L_{y}=2\pi$, and the spatial grid
size is $8192^{2}$. We set $\rho_{i}=0.025$ ($\rho_{s}\approx 0.0177$) and
$d_{e}=0.002$, so that the initial condition satisfies
$\rho_{i}^{-1}<k_{c}<d_{e}^{-1}$; thus, this simulation should first capture
merging at kinetic scales (as in the 3D simulation discussed earlier),
followed by a transition to MHD scales.

Figure 3:  Temporal evolution of magnetic energy $\mathcal{E}$ (blue) and
characteristic island radius $R$ (red) from the two-dimensional simulation.
The horizontal red dashed line denotes $R=\rho_{i}$.

Time traces of magnetic energy $\mathcal{E}$ and characteristic radius $R$ of
the magnetic structures, defined as $R\equiv 2\pi/4k_{\perp,{\rm max}}$, are
presented in Fig. 3. Initially, these quantities are found to evolve according
to the power laws $\tilde{t}^{-2/3}$ and $\tilde{t}^{1/3}$, respectively, in
accordance with Eq. (7). As $R$ exceeds $\rho_{i}$ (around $t\approx
0.2\tau_{A}$) the evolution of these quantities starts to deviate from those
power laws. This threshold is consistent with the departure from the electron-
only reconnection regime (i.e., Eq. (4)) as $\rho_{i}/R$ decreases identified
in Fig. 1. Around $t\approx 10.0\tau_{A}$, $R$ reaches $10\rho_{i}$, the scale
at which the reconnection rate converges to the constant large-scale value and
ions are fully coupled (see Fig. 1). Consistently, transitions to
$\mathcal{E}\sim\tilde{t}^{-1}$ and $R\sim\tilde{t}^{1/2}$ power-law scalings
are observed, in agreement with the predictions of Eq. (9).

## Conclusions.

We present an analytical model for electron-only strong-guide-field
reconnection in low-$\beta$ plasmas. Unlike large-scale collisionless
reconnection, where the (normalized) reconnection rate is conjectured to be a
constant of order $0.1$ [cassak2017review, liu2017does], we find that this
regime is characterized by reconnection rates that scale as the ratio between
the ion gyroradius and the size of the reconnecting structure and which can
significantly exceed 0.1. This finding provides a theoretical explanation for
previous observational and numerical studies [e.g., sharma2019transition,
greess2022kinetic, guan2023reconnection]. The validity of our model is
confirmed through direct numerical simulations, which also reveal that
deviation from the fully ion-coupled regime begins when $R\approx 10\rho_{i}$,
consistent with previous numerical studies [sharma2019transition] in a
different plasma regime. Although our focus in this study is low beta plasmas,
we also propose an extension of our derivation to $\beta_{e}\sim 1$ (see
Supplemental Material) which appears to be consistent with existing numerical
data.

Using this reconnection model, we derive time-dependent scaling laws for
inverse magnetic-energy transfer in the sub-$\rho_{i}$ range, which we
validate via a three-dimensional direct numerical simulation. Finally, through
a two-dimensional simulation starting with random Gaussian magnetic fields, we
demonstrate the ability of magnetic-island mergers to transport magnetic
energy from kinetic to MHD scales, accompanied by predicted changes in the
time-dependent scaling laws. In addition to its relevance to the understanding
of the self-organization of systems dominated by flux-rope-like structures
[e.g., eastwood2016ion, sun2019mms, huang2021electron], this conclusion lends
credence to recently proposed ideas for the viability of electron-scale seeds
as the origin of large-scale magnetic fields [zhou2020multi,
zhou2022spontaneous, zhou2023magnetogenesis].

## Acknowledgements.

This work was supported by DOE Awards DE-SC0022012, DE-FG02-91-ER54109, and by
the Schmidt Futures at the Institute for Advanced Study. This work used
resources of the Satori cluster at the MGHPCC facility funded by DOE award No.
DE-FG02-91-ER54109, and the National Energy Research Scientific Computing
Center, a DOE Office of Science User Facility supported by the Office of
Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231
using NERSC award FES-ERCAP0020063.

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# Average cluster size inside sediment left after droplet desiccation

P A Zolotarev1,2 K S Kolegov1,2,3 1Astrakhan State University, 20A Tatishchev
St., Astrakhan, 414056, Russia  2Landau Institute for Theoretical Physics
Russian Academy of Sciences, 1-A Academician Semenov Ave., Chernogolovka,
142432, Russia  3Volga State University of Water Transport, Caspian Institute
of Maritime and River Transport, 6 Nikolskaya St., Astrakhan, 414000, Russia
<EMAIL_ADDRESS>
###### Abstract

In this work, we continue to study the formation of particle chains (clusters)
inside the annular sediment during the drying of a colloidal droplet on a
substrate. The average value of the cluster size was determined after
processing experimental data from other authors. We performed a series of
calculations and found the value of the model parameter allowed to get
numerical results agreed with the experiment. Also, a modification of the
previously proposed algorithm is analyzed here.

##  1 Introduction

Evaporation-induced self-assembly (EISA) methods are useful in applications
such as inkjet printing, production of photonic crystals, application of
transparent conductive coatings, development of biosensors, and others [1].
Modeling the formation of colloidal particle precipitation structures is
important. It allows us to understand the main mechanisms of such processes,
identify key parameters, and study how to influence the system to obtain the
required patterns. For example, the model [2] describes the self-assembly of
particles on the free surface of a droplet. The capillary interaction of
particles and their transport by surface flow resulting from evaporation is
considered in this paper. The authors [2] performed calculations for spherical
and triangle droplets using the lattice Boltzmann method (LBM). Numerical
results showed the dependence of the sediment on the curvature of the free
surface. Clusters of particles formed at the liquid-vapor interface appear on
the substrate after complete evaporation of the liquid. Another mechanism for
the formation of particle clusters as a result of their capillary attraction
is their self-assembly on the substrate where the thickness of the liquid
layer is less than the particle size [3]. The calculations were performed
using the modification of LBM. The feature of the algorithm is the ability to
simulate the dynamics of soft particles [3]. The mass transfer of hard
particles in a drying film was studied in [4]. The LBM-based model takes into
account the hydrodynamic interaction of particles and their diffusion. The
authors [4] performed calculations for different values of the Peclet number
associated with different evaporation rates. At high values of the Peclet
number, a layer of particles is formed near the free surface of the film due
to the rapid movement of the two-phase boundary as a result of intense
evaporation. A uniform distribution of particles is observed at a small value
of the Peclet number. It is explained by diffusion transfer. The periodic
predominance of capillary flow or solutal Marangoni flow sometimes leads to
the formation of concentric precipitation of colloidal particles [5]. The
authors [5] used the lattice model and the Monte Carlo method to study the
effect of the surfactant concentration on this process. The magnetic
interaction of particles in a liquid and the formation of chains was studied
in [6]. The model is based on the Soft Sphere Discrete Element Method. At each
time step, Newton’s equations were solved to predict the motion of particles.
Calculations were performed for small viscosity values [6]. The model [7]
takes into account the diffusion and sedimentation of particles. These
particles have no volume, but they have mass. The model is based on the phase
field method and the Monte Carlo method. Two shapes of the droplet surface
were considered: a spherical segment and an asymmetric shape from the
experiment [8]. In both cases, a large number of deposited particles per unit
area near the periphery is shown (the coffee stain effect). The model predicts
a significant accumulation of particles in areas of low curvature for an
asymmetric droplet shape. In the experiment [8], the particles accumulated
mainly in areas of high curvature. The reason for this is the capillary flow,
which is not taken into account in the model [7]. The formation of particle
chains (clusters) inside the annular sediment during the evaporation of a
colloidal droplet on a substrate was described in [9]. The model takes into
account diffusion, advection, and capillary attraction of particles. Capillary
interaction of particles occurs near the boundary (fixation radius $R_{f}$)
where the thickness of the liquid layer is comparable to the particle size.
This process was mimicked as follows. In one time step, the particle was
shifted to the nearest neighboring particle in the small neighborhood $R_{n}$
at the presence of the last one. Calculations were performed for only one
parameter value $R_{n}$. In the current work, we consider the set of parameter
$R_{n}$ values and determine the value which leads the average cluster size is
similar to the experiment [10]. Also, here we analyze the modification of the
algorithm associated with multiple attempts to shift those moving particles
whose location has not changed in a time step as a result of a collision with
a neighboring particle.

##  2 Methods

###  2.1 Problem statement and model description

Here we consider an axisymmetric sessile droplet on a substrate in the mode of
a pinned three-phase boundary. Colloidal particles are suspended in the
liquid. They are subject to transfer by flow, diffusion, and capillary
attraction. At the initial time, the contact angle $\theta_{0}=\pi/18$. The
contact radius of the droplet with the substrate $R=$ 60 $\mu$m. The radius of
spherical particles $r_{p}=$ 0.35 $\mu$m. As the liquid dries, these particles
precipitate. The deposit is a monolayer of particles since the droplet is thin
(its height $h\ll R$). Time of full droplet evaporation $t_{\mathrm{max}}=$ 10
s. A more detailed description of the problem statement is given in [9].

The mathematical model [9] is semi-discrete. Each particle is considered
separately, but the liquid is described with a continuum approach. It is
assumed that the particle velocity is equal to the fluid flow velocity, which
is calculated by the formula

$\bar{v}_{r}(r,t)=\frac{R}{4r/R(t_{\mathrm{max}}-t)}\left[\frac{1}{\sqrt{1-(r/R)^{2}}}-\left(1-(r/R)^{2}\right)\right].$

It was derived from the mass conservation law (references and formula
derivation are given in [9]). The model considers the radial flow averaged
over the drop height. Here $r$ and $t$ is the radial coordinate of the
particle center and the current time of the process, respectively. Coordinate
of the particle at the next time step as a result of displacement by the flow

$r_{\tau+1}=\begin{cases}r_{\tau}+\bar{v}_{r}\delta t,\;r_{\tau+1}\eqslantless
R_{f}\\\ R_{f},\text{ otherwise,}\end{cases}$

where $\delta t$ is the time step and $\tau$ is the time step number.

Capillary forces act on the particles in the region of the fixation boundary
$R_{f}(t)=\sqrt{R^{2}-4r_{p}R/\theta(t)}$. The time dependence of the contact
angle is written as $\theta(t)=\theta_{0}\left(1-t/t_{\mathrm{max}}\right)$.
We mimic the capillary attraction of particles as follows. If there are
particles in the vicinity of the current particle that have precipitated or
are also subject to capillary attraction, then it moves close to the nearest
neighboring particle. By neighborhood, we mean a circle with a radius $R_{n}$.
Some particles precipitate if $r>R_{f}+r_{p}$. In this case, the local
thickness of the liquid layer is less than the particle size, $h<2r_{p}$.

The diffusion motion of particles is modeled by the Monte Carlo method. A
random offset angle $\alpha\in[-\pi;\pi)$ is generated at each time step
$\tau$. The new position of the particle is calculated using the formulas
$x_{\tau+1}=x_{\tau}+\cos(\alpha)\sqrt{2D\delta t}$ and
$y_{\tau+1}=y_{\tau}+\sin(\alpha)\sqrt{2D\delta t}$, where $x$ and $y$ are the
Cartesian coordinates of the particle center. The diffusion coefficient is
calculated using the Einstein formula $D=kT/(6\pi\eta r_{p})$, where $k$ is
the Boltzmann constant. The values of the temperature $T$ and the viscosity
$\eta$ of the liquid are taken for water under normal room conditions
($D\approx 6\cdot 10^{-13}$ m2/s). The value of the time step $\delta t=$ 0.1
ms was selected based on a series of computational experiments in such a way
that the Einstein relation for the mean square displacement $\langle
l^{2}\rangle=2Dt_{max}$ was fulfilled. Here

$\langle
l^{2}\rangle=\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}\left(\left(x_{i}(t_{\mathrm{max}})-x_{i}(0)\right)^{2}+\left(y_{i}(t_{\mathrm{max}})-y_{i}(0)\right)^{2}\right)$

and $N_{p}$ is the number of particles.

###  2.2 Algorithm

For each particle, we store coordinates and status in memory. Let’s denote
movable particles with green and black colors. Here green particles are
subject to convective and diffusive transfer. We do not use a separate status
for particles diffusing near $R_{f}$ to simplify the algorithm [9]. Capillary
forces act on black particles. Unmovable particles that have precipitated are
marked in red. The particle status changes according to the rules in Figure 1
(right).

Figure 1:  Scheme to the algorithm description (left) and rules for changing
the particle status (right).

For more details about the algorithm, see Figure 1 (left) and pseudocode 1. At
the beginning of the process, all particles are green by default. If a green
particle touches the fixation radius, it is repainted black. The green
particle cannot go beyond the fixation radius, unlike the black one. After
all, the movement in that area is mainly due to capillary forces. But due to
the movement of the fixation radius itself, the green particle may be behind
it (after that, it will be repainted red). A black particle becomes red after
it is beyond the moving boundary $R_{f}$ as a result of the motion of the last
one or after it shifts due to capillary forces.

Algorithm 1 Particle dynamics algorithm.

1:Problem parameters definition: $R_{n}$, $r_{p}$, $R$, $N_{p}$, and
$t_{\mathrm{max}}$.

2:Generation random coordinates of the particles $x_{i}$ and $y_{i}$
($i\in[1;N_{p}]$).

3:By default, all particles are marked green.

4:for $\tau\leftarrow 1,t_{\mathrm{max}}/\delta t$ do

5: calculate $R_{f}$.

6: for $i\leftarrow 1,N_{p}$ do

7: changing the particle status if necessary.

8: end for

9: while (not all particles are displaced) $\And$ (there is a shift of at
least one particle) do

10: shuffle the array of particle numbers.

11: for $i\leftarrow 1,N_{p}$ do

12: red particles are skipped.

13: if (green particle) then

14: calculate the new particle coordinates due to diffusion.

15: if (no collision) then

16: move the particle.

17: end if

18: Calculate the new particle coordinates due to advection.

19: if (no collision) then

20: move the particle.

21: end if

22: end if

23: if (the particle is black) $\And$ (there is a black or red particle within
$R_{n}$ ) then

24: Calculate the new particle coordinates due to the capillary shift.

25: if (no collision) then

26: move the particle close to the nearest one.

27: end if

28: end if

29: end for

30: end while

31: Write the particle coordinates and colors to a file for the current time
step.

32:end for

At each time step, we calculate the new value of $R_{f}$. Then we go through
all the non-red particles and check the conditions for changing their status.
Next, we shuffle the numbers of particles so that their subsequent search is
random. After that, we repeatedly iterate over the moving particles that have
not yet moved at the current time step and try to move each of them in turn.
Failure to shift can be caused by a collision (neighbor particle overlapping)
or the exit of a green particle beyond the $R_{f}$ boundary as a result of a
diffusive motion. Repeated iterations are completed if all the moving
particles were displaced at the current time step or there were no
displacements at all during a repeat of the loop. To check that there is no
collision of a particle with any other, it is advisable to consider the
distance only to the nearest neighbors. To do this, we have to store
additionally and periodically update information about subdomains containing
particles (numbers of particles included in the subdomain).

###  2.3 Determination of cluster sizes

To determine the value of the average cluster size, an experimental picture
was taken from [10] for further processing. The data we are interested in was
extracted as follows. The image was imported into the bitmap editor GIMP. A
transparent layer was applied on top of it, which we worked with later. The
brush size was set to match the size of the particles in the image [10]. All
particles not included in the annular sediment were drawn on the transparent
layer manually on top of the experimental image. At the same time, we tried to
choose different colors for each cluster for the convenience of further
calculation.

Algorithm 2 Algorithm for finding clusters.

1:Read the coordinates of the particles $x_{i}$, $y_{i}$ and their cluster
numbers $c_{i}\leftarrow 0$ ($i\in[1;N_{p}]$).

2:for $i\leftarrow 1,N_{p}$ do

3: $r\leftarrow\sqrt{x_{i}^{2}+y_{i}^{2}}$

4: if $r>r_{*}$ then

5: remove this particle from the array.

6: end if

7:end for

8:Cluster counter $C_{n}\leftarrow 1$.

9:for $i\leftarrow 1,N_{p}$ do

10: for $j\leftarrow 1,N_{p}$ do

11: Calculate the distance $s_{ij}$ between particles $i$ and $j$.

12: if $s_{ij}=d_{p}$ $\And$ $i\neq j$ then

13: if $c_{i}=0$ $\And$ $c_{j}=0$ then

14: $c_{i}\leftarrow c_{j}\leftarrow C_{n}$

15: $C_{n}\leftarrow C_{n}+1$

16: end if

17: if $c_{i}\neq 0$ $\And$ $c_{j}=0$ then

18: $c_{j}\leftarrow c_{i}$

19: end if

20: if $c_{i}=0$ $\And$ $c_{j}\neq 0$ then

21: $c_{i}\leftarrow c_{j}$

22: end if

23: end if

24: end for

25:end for

26:while there are changes to cluster numbers do

27: for $i\leftarrow 1,N_{p}$ do

28: for $j\leftarrow 1,N_{p}$ do

29: skip the zero cluster particles.

30: $s_{ij}=\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}}$

31: if $s_{ij}=d_{p}$ $\And$ $i\neq j$ then

32: if particles $i$ and $j$ are not from one cluster then

33: We assign a value $\max(c_{i},c_{j})$ to the particle with
$\min(c_{i},c_{j})$.

34: end if

35: end if

36: end for

37: end for

38:end while

Sometimes it was difficult to visually determine whether a particular particle
belongs to a particular cluster or not. In the process, we were guided by the
following rule. If a particle is located close to another particle in the
cluster, or the distance between these particles is much smaller than the
particle size, then we assume that the current particle also belongs to this
cluster. As a result, the layer with the original image was removed. Based on
the received image, we manually calculated the average cluster size.

Processing of numerical simulation results was performed using the algorithm 2
for finding clusters. To determine the contact of two particles, the condition
$s=d_{p}\pm\varepsilon d_{p}$ was used, where $s$ is the distance between two
particles, the diameter of a spherical particle $d_{p}=2r_{p}$, and
$\varepsilon$ is the constant ($\varepsilon=$ 0.01 is used here). This nuance
was also taken into account when detecting a collision of two particles
($s<d_{p}-\varepsilon d_{p}$). The calculation of the average cluster size
based on the results of ten repeated tests was performed using a script
written in Python.

##  3 Results and discussions

A series of calculations were performed for different values of $R_{n}$ in the
range from $3r_{p}$ to $22r_{p}$. Then the numerical result was approximated
using the least-squares method (Figure 2). The plot in Figure 2 resembles a
sigmoid. The value of the average cluster size determined by us based on
experimental data [10] is $\langle N_{c}\rangle\approx 8.2$ (average number of
particles in the chain). The model predicts this value for $R_{n}\approx
7.9r_{p}$.

Figure 2: Dependence of the average cluster size on the parameter $R_{n}$ (the
standard deviation was used to estimate the error).

For comparison, the Figure 3 shows the final sediment structures based on
experimental data [10] and our simulation results. Only the particles located
inside the annular sediment are shown. The subset of particles belonging to
the ring was subtracted from their set using the condition $r>r_{*}$. The
inner radius of the ring $r_{*}\approx 0.75R$ was determined based on
calculation data [9] ($r_{*}$ corresponds to the place of a high gradient of
the number of particles per unit area). The internal structure of such rings
was discussed in detail earlier [9].

Here we observe that for the given parameters, the model predicts the absence
of single particles (Figure 3). The visualization of experimental data shows a
very small number of such particles. In both cases, there are clusters of
different sizes (from two to several dozen particles in a cluster). Some
clusters are similar in shape. It should be noted that based on the results of
modeling, tree-shaped clusters are expected to appear. Their root element is
located closer to the periphery. These “trees” grow towards the center of the
area. It is explained by the nature of movement and the shape of the fixation
boundary $R_{f}$. Such branching structures are not observed in the
experiment. This distinctive feature of the numerical results is most likely
related to the approximate description of the domain geometry [9]. After all,
the shape of the drop at the final stage of evaporation is more like a film,
in which local ruptures can occur [1]. Thus, the action of capillary forces
occurs in a larger area. The model boundary $R_{f}$ and its neighborhood are
only a rough approximation of such a region. Besides, the mimic of the
capillary interaction of particles is used here. In a good way, it is
necessary to solve the n-body problem numerically. Only in this problem,
instead of gravity, there are capillary forces.

Figure 3:  Visualisation of clusters from experiment [10] (left) and
simulation results for $R_{n}=8r_{p}$ (right).

Analysis of the algorithm 1 showed a weak nonlinearity of the dependence of
the simulation time on the number of particles $N_{p}$ (Figure 4). Also, it
was found that the average (over time steps) rate of movable particle
displacement failures for the modified algorithm (with multiple attempts to
particle displacement) is 1–2% according to the results of calculations. For
comparison, we tested the original algorithm (with a single attempt to shift
the particles) and got a failure rate of 3-4%. Five tests were run
simultaneously for each type of calculation (the error is less than the marker
size on the Figure 4). Thus, for a sufficiently small value of the time step,
the error associated with the failure of the particle shift is small. Repeated
attempts to shift particles can slow down the program and not give a
significant difference in the result.

Figure 4: Dependence of the simulation time on the particle number (the
program was written in Python, Intel (R) Core (TM) i9-9900K CPU 3.6 GHz was
used).

##  4 Conclusion

Numerical results [9] are qualitatively agreed with the experiment [10]. As
the droplet dries, chains of particles (clusters) are formed inside the
annular sediment. In the current work, we have processed experimental data
[10], found out the average cluster size, and defined the value of a model
parameter $R_{n}$ to predict a quantitatively agreed value $\langle
N_{c}\rangle$. But it is also worth mentioning the qualitative difference
between the simulation and experimental results, related to the shape of some
clusters. The simple model under consideration often predicts tree-shaped
clusters, which is not typical for the experiment [10]. Further study is
needed in this direction to better understand the processes that are taking
place. Some questions can be answered by using the high-speed shooting of the
formation of such clusters. Additional experimental data are also needed, for
example, on the shape of the free surface of the liquid at the time stage when
these clusters begin to appear. Further creation of more complex and accurate
models that can predict the internal structure of such precipitation is
important.

## Acknowledgment

This work is supported by the grant 18-71-10061 from the Russian Science
Foundation. We thank Svetlana Kolegova for her help in processing experimental
data and Yuri Tarasevich for useful comments.

## References

## References

  * [1] Kolegov K S and Barash L Y 2020 Advances in Colloid and Interface Science 285 102271 ISSN 0001-8686
  * [2] Zhao M, Luo W and Yong X 2019 Journal of Colloid and Interface Science 540 602–611
  * [3] Wouters M, Aouane O, Sega M and Harting J (Preprint 2007.15405v2)
  * [4] Chun B, Yoo T and Jung H W 2020 Soft Matter 16 523–533
  * [5] Seo H W, Jung N and Yoo C S 2020 Journal of Mechanical Science and Technology 34 801–808
  * [6] Darras A, Opsomer E, Vandewalle N and Lumay G 2017 Scientific Reports 7
  * [7] Yang J, Kim H, Lee C, Kim S, Wang J, Yoon S, Park J and Kim J 2020 Theoretical and Computational Fluid Dynamics 34 679–692
  * [8] Deegan R D, Bakajin O, Dupont T F, Huber G, Nagel S R and Witten T A 1997 Nature 389 827–829
  * [9] Kolegov K S and Barash L Y 2019 Physical Review E 100
  * [10] Park J and Moon J 2006 Langmuir 22 3506–3513


Pedestrian detection with high-resolution event camera P. Wzorek, T. Kryjak

# Pedestrian detection with
high-resolution event camera

Piotr Wzorek  Tomasz Kryjak  Embedded Vision Systems Group, Computer Vision
Laboratory
Department of Automatic Control and Robotics
AGH University of Krakow, Poland
<EMAIL_ADDRESS>
###### Abstract

Despite the dynamic development of computer vision algorithms, the
implementation of perception and control systems for autonomous vehicles such
as drones and self-driving cars still poses many challenges. A video stream
captured by traditional cameras is often prone to problems such as motion blur
or degraded image quality due to challenging lighting conditions. In addition,
the frame rate – typically 30 or 60 frames per second – can be a limiting
factor in certain scenarios. Event cameras (DVS – Dynamic Vision Sensor) are a
potentially interesting technology to address the above mentioned problems. In
this paper, we compare two methods of processing event data by means of deep
learning for the task of pedestrian detection. We used a representation in the
form of video frames, convolutional neural networks and asynchronous sparse
convolutional neural networks. The results obtained illustrate the potential
of event cameras and allow the evaluation of the accuracy and efficiency of
the methods used for high-resolution (1280 x 720 pixels) footage.

###### keywords:

pedestrian detection, event camera, convolutional neural networks, sparse
convolutional neural networks

##  1 Introduction

Event cameras are neuromorphic vision sensors inspired by the structure and
behaviour of the human eye [1]. They are increasingly being used in computer
vision. They respond to changes in the brightness of the observed scene
independently for each pixel. Each so-called “event” is generated when the
change in the logarithm of the brightness sensed by a given pixel reaches a
certain threshold. A single event is described by four values:
$e=\\{t,x,y,p\\}$, where: $t$ is the timestamp of the event (in microseconds),
$x$ and $y$ are the coordinates of the pixel registering the event, and $p$ is
its polarity in the form of a value of 1 (positive change in brightness) or -1
(negative change).

The growing popularity of event cameras is the result of their ability to be
used in conditions of rapid movement of the object relative to the sensor
(temporal resolution of microseconds) and unfavourable lighting conditions
(dynamic range of 120 dB, relatively good performance in low light). The event
camera only records changes in the scene. This means that redundant
information is largely eliminated and the number of events generated depends
on the dynamics of the scene, camera ergo-motion or lighting changes. As a
result, the power consumption of the sensor is low (in typical situations) and
the processing of the most important information can be efficient.

However, there are significant challenges in applying known computer vision
algorithms to event data. These type of data differ significantly from
traditional video frames in that they form a sparse spatio-temporal cloud.
Meanwhile, state-of-the-art object detection solutions use traditional video
frames as input to deep convolutional neural networks.

In this work, we have compared different methods for applying deep learning
algorithms to event data. We considered using the representation of event data
in the form of frames generated by accumulating them over a defined time
window and using as input to a CNN, as well as the use of asynchronous sparse
neural networks to optimise energy consumption by reducing the number of
operations performed. We focused on the problem of pedestrian detection as a
key issue for different types of autonomous vehicles – both drones and self-
driving cars. The main contribution of this paper is the evaluation of
selected methods available in the literature for a high resolution dataset and
their comparison in terms of accuracy and efficiency.

The remainder of this paper is organised as follows. Section 2 describes the
different methods considered in the literature for applying deep learning with
event data to the task of object detection. Section 3 describes the research
we have conducted. A summary and considerations for further development plans
for the implemented systems conclude the paper.

##  2 Object detection with event cameras

State-of-the-art object detection algorithms use deep convolutional neural
networks adapted to process data represented as two- or three-dimensional
matrices. The unusual nature of event data requires both the use of so-called
event data representations and the introduction of modifications to the
algorithms used. Many approaches to this problem have been proposed in the
literature.

The simplest solution is to accumulate the event data in a matrix, analogous
to the frames recorded by classical cameras, and then use deep convolutional
neural networks. The paper [2] proposes a method in which each pixel is
assigned the polarity value of the last event recorded. An extension of this
idea is the exponentially decaying time surface, where the time of occurrence
of an event is also taken into account. Also popular are methods that take
into account the frequency of occurrence of an event for a given pixel [3], or
the so-called leaky surface [4], where the memory of previous events is
retained.

An extension of the idea of using deep convolutional neural networks to
process event data is the use of asynchronous sparse convolutional neural
networks (ASCNN). The authors of [5] exploit the sparsity of event data to
reduce the computational complexity and energy consumption of the detection
system, using networks in which only the convolution results for changing
input values are updated. The use of ASCNNs allows to perform detection
efficiently and asynchronously (updating the necessary values for each
incoming pixel).

The literature also considers methods based on the fusion of event data and
traditional RGB frames [6], the use of transformer-based architectures [7],
recurrent architectures [8], Graph Neural Networks [9] or Spiking Neural
Networks [10].

##  3 The analysed pedestrian detection methods

For our research, we used the Prophesee 1 Megapixel automotive detection
dataset [8] containing sequences recorded with a 1280x720 pixel resolution
event camera in road conditions with labelled objects such as pedestrians,
traffic signs, traffic lights or cars. The dataset was filtered to select only
fragments containing pedestrians. This yielded more than one million 10ms
sequences on which pedestrians were labelled. For some experiments, a subset
of 100,000 10ms sequences was used to speed up the neural network training
process.

The first method considered was to accumulate event data in time windows of
10ms and use them as input to a deep convolutional neural network. To maximise
the amount of information in the representation, we used a fusion of
representations using different characteristics of the data – polarity,
temporal resolution and frequency of occurrence, analogous to [11]. As a
detector, we used the YOLOv7 architecture of [12], which represents the state
of the art in terms of execution time and accuracy. The detector trained on
the described dataset using the transfer learning mechanism achieved a
detection accuracy of<EMAIL_ADDRESS>(38%mAP@.5:.95). The YOLOv7 model is
characterised by a value of 104.7 GFLOPs (floating point operations). We use
this metric to evaluate and compare the computational efficiency between
multiple models.

Figure 1: Example of correct detection on the fused representations

As an alternative to the detection system described above, we also decided to
test ASCNNs. Analogous to [5], we used the event histogram as event data
representation, the VGG-16 network as feature extraction and the YOLO output
layer. This model has a reported computational efficiency of 205 MFLOPs, which
is a significant reduction compared to the YOLOv7-based detector.

However, we were unable to achieve satisfactory accuracy scores for a ASCNNs
after numerous experiments on a reduced data set (100,000 samples). The
maximum result was 0.04%mAP. To increase the network input, we also tried to
extend the model by two convolution layers and a max-pooling layer. For the
transformed model, we obtained a maximum result of 0.1%mAP. As a comparison,
for the same subset of dataset we obtained<EMAIL_ADDRESS>(19.8%mAP@.5:.95) for
YOLOv7 model.

##  4 Summary

In this work, we investigated the accuracy and computational efficiency of
event camera-based detection systems with deep convolutional network YOLOv7
and ASCNNs. The accuracy score achieved for the YOLOv7 detector (67.7%
<EMAIL_ADDRESS>38%mAP@.5:.95) are satisfactory and comparable to the RED network
accuracy (43%mAP) reported in [8], which is the highest result achieved for
this dataset. The accumulation of data in a 10ms window provides enough
information to perform detection with a large YOLOv7 model and a diverse and
efficient representation. However, the event histogram representations
generated in the same time windows and the use of a relatively small network
model (VGG16 with YOLO output) do not allow for satisfactory performance
results for ASCNNs.

The conducted research makes it possible to plan the direction of further work
on detection systems for high-resolution event data, taking into account their
accuracy, computational complexity and the possibility of hardware
implementation using SoC FPGA or eGPU platforms. We plan to further explore
ASCNNs architectures to improve the accuracy of detection systems. We will
conduct further experiments considering other event data accumulation times
(to increase the amount of information) and larger sparse network models. The
significant reduction in computational complexity in sparse models is a
motivation for further work on this type of network. Another approach that we
also want to test is the use of binary networks. A relatively simplified event
representation should work well with this type of model and allow for improved
computational efficiency. We are also considering various methods to
accelerate the mentioned networks using the FPGA platform. Another type of
network that we plan to evaluate and study are spiking neural networks (SNN),
which allow direct processing of events. The use of Dynamic Vision Sensors and
neuromorphic platforms such as BrainChip or Loihi would enable the task of
pedestrian detection to be realised effectively and efficiently.

## Acknowledgements

The work presented in this paper was supported by the programme “Excellence
initiative – research university” for the AGH University of Krakow. We
gratefully acknowledge Poland’s high-performance computing infrastructure
PLGrid (HPC Centers: ACK Cyfronet AGH) for providing computer facilities and
support within computational grant no. PLG/2023/016130.

## References

  * [1] Gallego, G., Delbrück, T., Orchard, G., Bartolozzi, C., Taba, B., Censi, A., Leutenegger, S., Davison, A. J., Conradt, J., Daniilidis, K., et al.  Event-based vision: A survey.  _IEEE transactions on pattern analysis and machine intelligence_ , 44(1):154–180, 2020.
  * [2] Afshar, S., Ralph, N., Xu, Y., Tapson, J., Schaik, A. v., and Cohen, G.  Event-based feature extraction using adaptive selection thresholds.  _Sensors_ , 20(6):1600, 2020.
  * [3] Chen, N. F.  Pseudo-labels for supervised learning on dynamic vision sensor data, applied to object detection under ego-motion.  In _Proceedings of the IEEE conference on computer vision and pattern recognition workshops_ , pages 644–653. 2018.
  * [4] Cannici, M., Ciccone, M., Romanoni, A., and Matteucci, M.  Asynchronous convolutional networks for object detection in neuromorphic cameras.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops_ , pages 0–0. 2019.
  * [5] Messikommer, N., Gehrig, D., Loquercio, A., and Scaramuzza, D.  Event-based asynchronous sparse convolutional networks.  In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part VIII 16_ , pages 415–431. Springer, 2020.
  * [6] Jiang, Z., Xia, P., Huang, K., Stechele, W., Chen, G., Bing, Z., and Knoll, A.  Mixed frame-/event-driven fast pedestrian detection.  In _2019 International Conference on Robotics and Automation (ICRA)_ , pages 8332–8338. IEEE, 2019.
  * [7] Gehrig, M. and Scaramuzza, D.  Recurrent vision transformers for object detection with event cameras.  _arXiv preprint arXiv:2212.05598_ , 2022.
  * [8] Perot, E., De Tournemire, P., Nitti, D., Masci, J., and Sironi, A.  Learning to detect objects with a 1 megapixel event camera.  _Advances in Neural Information Processing Systems_ , 33:16639–16652, 2020.
  * [9] Schaefer, S., Gehrig, D., and Scaramuzza, D.  Aegnn: Asynchronous event-based graph neural networks.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 12371–12381. 2022.
  * [10] Cordone, L., Miramond, B., and Thierion, P.  Object detection with spiking neural networks on automotive event data.  In _2022 International Joint Conference on Neural Networks (IJCNN)_ , pages 1–8. IEEE, 2022.
  * [11] Wzorek, P. and Kryjak, T.  Traffic sign detection with event cameras and dcnn.  In _2022 Signal Processing: Algorithms, Architectures, Arrangements, and Applications (SPA)_ , pages 86–91. IEEE, 2022.
  * [12] Wang, C.-Y., Bochkovskiy, A., and Liao, H.-Y. M.  Yolov7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors.  _arXiv preprint arXiv:2207.02696_ , 2022.


Overview of searches for single production of vector-like top and bottom
quarks with the ATLAS experiment at 13 TeV

Johannes Erdmann
on behalf of the ATLAS Collaboration111Copyright 2018 CERN for the benefit of
the ATLAS Collaboration. Reproduction of this article or parts of it is
allowed as specified in the CC-BY-4.0 license.

Technische Universität Dortmund, Experimentelle Physik 4, Otto-Hahn-Str. 4a,
D-44227 Dortmund, Germany

> Vector-like quarks (VLQ) that couple preferentially to third-generation
> Standard Model (SM) quarks are a well-motivated extension of the SM, which
> could solve in particular the hierarchy problem. While VLQs can be produced
> in pairs via the strong interaction at the LHC, the cross section for single
> production may be larger than the pair production cross section for VLQ
> masses above the current exclusion limits. An overview of the searches for
> the single production of VLQs at the ATLAS experiment that use data taken at
> a centre-of-mass energy of 13 TeV, corresponding to up to 79.8 fb-1, is
> presented. Vector-like top quarks are searched for in the decay channel
> $T\rightarrow Wb$ and in the decay channel $T\rightarrow Zt$ with the $Z$
> boson decaying to two charged leptons. Vector-like bottom quarks are
> searched for in the decay channel $B\rightarrow Hb$ with the Higgs boson
> decaying to two photons. No significant excess over the SM prediction is
> found and 95% CL exclusion limits are set on the single production of VLQs
> as a function of their mass and coupling to SM quarks.

> PRESENTED AT
>
>
>
>
> $11^{\mathrm{th}}$ International Workshop on Top Quark Physics
> Bad Neuenahr, Germany, September 16–21, 2018

##  1 Introduction

Vector-like quarks (VLQ) are predicted in many extensions of the Standard
Model (SM). Particular examples are models that predict a new strong
interaction and solve the hierarchy problem by introducing the Higgs boson as
the pseudo-Nambu–Goldstone boson of the explicitly broken symmetry of that
interaction. While VLQ can be pair-produced at the LHC via the strong
interaction, at large VLQ masses the cross sections for single-VLQ production
may be larger than for pair production. However, the single-production cross
section depends on the coupling of the VLQ to SM particles. This coupling $C$
[2] can be expressed in terms of the branching ratios (BR) of VLQ and a
generalized coupling $\kappa$ [3], or in terms of the mixing angle with SM
quarks [4]. In the searches discussed here, only mixing with third-generation
SM quarks is assumed, allowing the decays of vector-like $T$ ($B$) quarks to
$Wb$ ($Wt$), $Zt$ ($Zb$) or $Ht$ ($Hb$). The BRs are predicted in certain VLQ
multiplet representations. In the ($T$) singlet, these are approximately
50:25:25, and in the ($B$ $Y$) doublet, they are approximately 50:50:0.
Searches for single-$T$ and single-$B$ production with 13 TeV data taken by
the ATLAS experiment [1] at the LHC are presented.

##  2 Search for $B\rightarrow Hb$

A search for the $B$ quark is performed in the decay mode to $Hb$, using the
$H\rightarrow\gamma\gamma$ decay in 79.8 fb-1 of data [5]. In the event
selection, a Higgs-like diphoton system is required, as well as at least one
forward jet, characteristic of single-VLQ production, and at least one
$b$-tagged jet. Novel $b$-tagging is used that applies a fixed cut on the
$b$-tagging discriminant value up to a jet $p_{\text{T}}$ of 250 GeV, and
above 250 GeV, the cut on the discriminant value is loosened as a function of
$p_{\text{T}}$, so that a constant $b$-tagging efficiency is maintained. No
reconstructed electrons or muons are allowed in the events, and the invariant
mass of the diphoton system and the highest-$p_{\text{T}}$ $b$-tagged jet,
$m_{\gamma\gamma b}$, is required to be larger than 300 GeV. In order to
further separate the signal from the background,
$|y^{*}|=0.5\cdot|y^{\gamma\gamma}-y^{b}|$ is required to be smaller than 1.1,
where $y^{\gamma\gamma}$ ($y^{b}$) is the rapidity of the diphoton system (the
highest-$p_{\text{T}}$ $b$-tagged jet).

A resonance is searched for in the $m_{\gamma\gamma b}$ spectrum, where the
main background process is continuum diphoton production, and a smaller
contribution comes from events with a $H\rightarrow\gamma\gamma$ decay. The
shape of the continuum background is estimated from data using events with an
invariant diphoton mass in the range 105–120 GeV or 130–160 GeV. Its
normalization is determined from the data in the search region. No significant
excess above the SM background is found, and 95% confidence level (CL) limits
are set on the cross section times BR to $Hb$ as a function of the mass of the
$B$ quark (Figure 1) for a benchmark coupling $\kappa_{B}=0.5$ [3]. In the
($B$ $Y$) doublet model, $B$ quarks with masses smaller than 1210 GeV are
excluded.

Figure 1: Limits at 95% CL on the cross section times BR to $Hb$ for the
single production of a $B$ quark as a function of its mass, assuming a
benchmark coupling value of $\kappa_{B}=0.5$ [3]. Also shown is the expected
cross section for the ($B$ $Y$) doublet model, which predicts a BR to $Hb$ of
50% [5].

##  3 Search for $T\rightarrow Wb$

A search for the $T$ quark is performed in the decay mode to $Wb$, using the
$W\rightarrow\ell\nu$ decay in 3.2 fb-1 of data [6]. In the event selection,
exactly one electron or muon, missing transverse momentum larger than 120 GeV,
and a $b$-tagged jet with $p_{\text{T}}>350~{}\text{Ge\kern-1.00006ptV}$ are
required. Events are rejected if they contain additional high-$p_{\text{T}}$
jets that are either close to the highest-$p_{\text{T}}$ $b$-tagged jet or far
away from it ($\Delta R<1.2$ or $>2.7$). The distance in azimuth between the
lepton and the highest-$p_{\text{T}}$ $b$-tagged jet is required to be larger
than 2.5 and at least one forward jet is required to be present.

The mass of the $T$ quark is reconstructed from the lepton, the missing
transverse momentum, and the highest-$p_{\text{T}}$ $b$-tagged jet, and a
resonance is searched for in this spectrum, where the main background
processes are $W$+jets, $t\overline{t}$ and single-top production. For
$W$+jets and $t\overline{t}$ production, control regions (CR) are defined to
validate and improve the modeling of these backgrounds. No significant excess
above the SM background is found, and 95% confidence level (CL) limits are set
on the coupling and mixing angle of the $T$ quark as a function of its mass
(Figure 2). As the coupling value does not only enter the production cross
section but also the decay, the intrinsic VLQ width depends on the coupling
value. When setting limits on the coupling or mixing angle, the change in
width is taken into account by reweighting the MC samples, which were
generated at a benchmark value of the coupling, using MC samples at truth
level with a variable coupling value.

(a)  (b)

Figure 2: Limits at 95% CL on (a) the left-handed coupling of the $T$ quark to
the $W$ boson and the $b$-quark, and (b) the left-handed mixing angle of the
$T$ quark and the top quark as a function of the mass of the $T$ quark in the
singlet model [6].

##  4 Search for $T\rightarrow Zt$

A search for the $T$ quark is performed in the decay mode to $Zt$, using the
$Z\rightarrow\ell\ell$ decay in 36.1 fb-1 of data [7]. Events are selected
with two electrons or muons of opposite electric charge and with a mass close
to the mass of the $Z$ boson ($Z$-boson candidate). Two channels are defined,
which use events with exactly two charged leptons and events with at least
three charged leptons, respectively. In both channels, at least one $b$-tagged
jet is required in the event selection, as well as a forward jet. The
$p_{\text{T}}$ of the $Z$-boson candidate is required to be larger than 200
GeV (150 GeV) in the dilepton (trilepton) channel. In the dilepton channel, at
least one top-tagged jet with a large radius parameter is required. In the
trilepton channel, the $p_{\text{T}}$ of the highest-$p_{\text{T}}$ electron
or muon that is not part of the $Z$-boson candidate must be larger than 200
GeV. In both channels, a requirement is made in the event selection that
suppresses the contribution from VLQ pair production, so that the focus of the
analysis is solely on the single-production process.

A resonance is searched for in the spectrum of the invariant mass of the
highest-$p_{\text{T}}$ top-tagged large-radius jet and the $Z$-boson candidate
(the scalar sum of the transverse momenta of all jets and electrons and muons)
in the dilepton (trilepton) channel. The main background processes are
$Z$+jets (diboson and $t\overline{t}X$, where $X$ is a $Z$ boson in a large
fraction of the events) production in the dilepton (trilepton) channel. For
these processes, CRs are defined to validate and improve the modeling of the
backgrounds. No significant excess above the SM background is found, and 95%
confidence level (CL) limits are set on the coupling and mixing angle of the
$T$ quark as a function of its mass (Figure 3). As in the $T\rightarrow Wb$
analysis, the MC samples were reweighted to set limits as a function of the
coupling or mixing angle.

Figure 3: Limits at 95% CL on (a) the coupling of the $T$ quark to the $W$
boson and the $b$-quark, and (b) the left-handed mixing angle of $T$ quark and
the top quark as a function of the mass of the $T$ quark in the singlet model
[7].

##  5 Conclusions

Single production of vector-like quarks was searched for in proton–proton
collision data taken at the LHC with the ATLAS detector at a center-of-mass
energy of 13 TeV, using data corresponding to integrated luminosities of
3.2–79.8 fb-1. No significant excess above the Standard Model background
predictions was found, and 95% confidence level limits were set on the mass of
the $B$ quark, and on the coupling and mixing angles of $T$ quarks in
benchmark models.

ACKNOWLEDGEMENTS

The author would like to acknowledge the support by BMBF, Germany (FSP-103).

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