# ALMA CN Zeeman Observations of AS 209: Limits on Magnetic Field Strength and
Magnetically Driven Accretion Rate

Rachel E. Harrison Department of Astronomy, University of Illinois, Urbana, IL
61801, USA  Leslie W. Looney Department of Astronomy, University of Illinois,
Urbana, IL 61801, USA  Ian W. Stephens Harvard-Smithsonian Center for
Astrophysics, Cambridge, MA 02138, USA  Department of Earth, Environment and
Physics, Worcester State University, Worcester, MA 01602, USA  Zhi-Yun Li
Department of Astronomy, University of Virginia, Charlottesville, VA 22903,
USA  Richard Teague Department of Astronomy, University of Michigan, Ann
Arbor, MI 48109, USA  Richard M. Crutcher  Department of Astronomy, University
of Illinois, Urbana, IL 61801, USA  Haifeng Yang Institute for Advanced Study,
Tsinghua University, Beijing, 100084, China  Erin Cox  Center for
Interdisciplinary Exploration and Research in Astrophysics (CIERA) and
Department of Physics andAstronomy, Northwestern University, Evanston, IL
60208, USA  Manuel Fernández-López Instituto Argentino de Radioastronomía
(CCT-La Plata, CONICET; CICPBA),Buenos Aires, Argentina  Hiroko Shinnaga
Department of Physics and Astronomy, Graduate School of Science and
Engineering, Kagoshima University, Amanogawa Galaxy Astronomy Research Center
(AGARC), Kagoshima 890-0065, Japan Rachel Harrison<EMAIL_ADDRESS>
(Received 24 June 2020; Revised 15 September 2020; Accepted 5 January 2021)

###### Abstract

While magnetic fields likely play an important role in driving the evolution
of protoplanetary disks through angular momentum transport, observational
evidence of magnetic fields has only been found in a small number of disks.
Although dust continuum linear polarization has been detected in an increasing
number of disks, its pattern is more consistent with that from dust scattering
than from magnetically aligned grains in the vast majority of cases. Continuum
linear polarization can reveal information about the magnetic field’s
direction, but not its strength. On the other hand, observations of circular
polarization in molecular lines produced by Zeeman splitting offer a direct
measure of the line-of-sight magnetic field strength in disks. We present
upper limits on the net toroidal and vertical magnetic field strengths in the
protoplanetary disk AS 209 derived from Zeeman splitting observations of the
CN 2-1 line. The 3$\sigma$ upper limit on the net line-of-sight magnetic field
strength in AS 209 is 5.0 mG on the redshifted side of the disk and 4.2 mG on
the blueshifted side of the disk. Given the disk’s inclination angle, we set a
3$\sigma$ upper limit on the net toroidal magnetic field strength of 8.7 and
7.3 mG for the red and blue sides of the disk, respectively, and 6.2 and 5.2
mG on the net vertical magnetic field on the red and blue sides of the disk.
If magnetic disk winds are a significant mechanism of angular momentum
transport in the disk, magnetic fields of a strength close to the upper limits
would be sufficient to drive accretion at the rate previously inferred for
regions near the protostar.

Magnetic fields; protoplanetary disks; accretion, accretion disks

††journal: ApJ

##  1 Introduction

Magnetic fields are thought to play an essential role in the evolution of
protoplanetary disks by providing a means of angular momentum transport
through the magnetorotational instability (MRI) (Balbus & Hawley, 1998) or
magnetic disk winds (Blandford & Payne, 1982). The MRI mechanism requires that
material have a high enough ionization fraction to be well-coupled to the
magnetic field. In protoplanetary disks, the ionization fraction is only
expected to be high enough to produce magnetorotational instability within
$\sim$0.1 au of the central protostar, where the temperature is high enough to
produce thermal ionization, and in the surface layers of the disk, where non-
thermal ionization sources such as cosmic rays, The poloidal component of the
disk magnetic field gives rise to the magnetic disk wind. The disk wind system
modeled in Suriano et al. (2018) predicts that the disk magnetic field will
have both toroidal and poloidal components, with the toroidal component being
stronger than the poloidal component in regions above and below the disk
midplane (see Suriano et al. 2018 Figures 4 and 5).

Observationally determining magnetic field strengths and morphologies in
protoplanetary disks to constrain the various angular momentum transport
mechanisms has proven difficult. The original motivation behind many
millimeter and sub-millimeter continuum polarization observations of
protoplanetary disks was to find evidence of dust grain alignment to disk
magnetic fields (e.g., Stephens et al., 2014; Rao et al., 2014; Segura-Cox et
al., 2015). However, the polarized emission seen in many disks at these
wavelengths is better explained by dust scattering of thermal emission (e.g.,
Kataoka et al., 2015; Yang et al., 2016, 2017). While possible evidence of
grain aligment to disk magnetic fields has been found in the circumbinary
disks BHB07-11 (Alves et al., 2018) and VLA 1623 (Sadavoy et al., 2018; Harris
et al., 2018), and in the disk of HD 142527 (Ohashi et al., 2018), these
observations provide information about the direction and morphology of the
magnetic field lines, not the magnetic field strength.

The Zeeman effect offers a direct means of constraining the line-of-sight
magnetic field strength without contamination from continuum dust scattering;
however, it is possible to produce circular polarization in molecular lines
through resonant scattering (Houde et al., 2013). In the presence of a
magnetic field, spectral lines split apart in frequency to a degree that
depends on the magnetic field strength as $\nu=\nu_{0}\pm\frac{eB}{4\pi
m_{e}c}$ (e.g., Crutcher & Kemball, 2019), where $\nu_{0}$ is the line
frequency in the absence of a magnetic field and $B$ is the magnetic field
strength. In astronomical sources with intrinsically weak magnetic fields
compared to those in, for example, stellar photospheres, the observable of
Zeeman splitting is circular polarization of the spectral line, which measures
the line of sight magnetic field strength. Paramagnetic species, such as the
CN radical, are particularly sensitive to the Zeeman effect. CN is also one of
only three species in which the Zeeman effect has been unambiguously detected
in extended molecular gas in star forming regions (Crutcher, 2012). However,
the Zeeman effect has yet to be detected in a protoplanetary disk, despite a
recent attempt with ALMA (Vlemmings et al., 2019).

AS 209 is a protoplanetary disk located in the Ophiuchus star forming region
at a distance of 126 pc from the Sun (Gaia Collaboration et al., 2016). The
protostar has a mass of 1.25 $M_{\odot}$ (Teague et al., 2018) and a
luminosity of 1.5 $L_{\odot}$ (Tazzari et al., 2016). The disk is known to
have two rings at $r$ = 75 au and $r$ = 130 au and two gaps at $r$ = 62 au and
$r$ = 103 au (Fedele et al., 2018). This source was selected as a target
source based on its inclination angle, the presence of a bright CN line (Öberg
et al., 2011), and high accretion rate of nearly $10^{-7}$ $M_{\odot}$ yr-1
(Johns-Krull et al., 2000). The disk’s inclination angle ($i$) of 35.3∘
(Fedele et al., 2018), where $i=0^{\circ}$ represents a face-on disk, means
that both toroidal and vertical magnetic field lines would have a component
along the line of sight, with the toroidal component $B_{\phi}=B_{LOS}/\sin i$
along the disk’s major axis and the vertical component $B_{z}=B_{LOS}/\cos i$.
Finally, a strong magnetic field would likely be needed to drive its high
accretion rate.

In this paper, we present Zeeman observations toward AS 209 using CN 2-1. In
Section 2, we present the observations, including the continuum polarization.
Without an obvious Zeeman detection, we then use two analysis approaches in
Section 3. In Section 4, we place limits on the magnetic strength, and in
Section 5, we summarize the results.

##  2 Observations

These observations were taken with the Atacama Large Millimeter/Sub-millimeter
Array (ALMA). All data discussed in this paper were taken as part of ALMA
project 2018.1.01030.S (PI: Rachel Harrison). The observations were taken in
four execution blocs between 6 March and 7 March 2019 while the array was in
configuration C43-1. The total observing time was 6.1 hours, of which 2.7
hours were spent on AS 209. The average sampling time was $\sim$6 seconds.
J1733-1304 was the phase calibrator, J1751+0939 was the polarization
calibrator, and J1427-4206 was the bandpass calibrator. The dataset consists
of two spectral line windows with a channel width of 122.070 kHz and a total
bandwidth of 117.1875 MHz each, and two continuum windows with a channel width
of 976.562 kHz and a total bandwidth of 937.5000 MHz each. The spectral line
windows are centered on 226.64013 GHz and 226.88081 GHz. One spectral line
window covered lines 1-4 listed in Table 1, and the other covered lines 5-9.
The Zeeman factors for all of the lines observed are from Shinnaga and
Yamamoto (in preparation). Shinnaga and Yamamoto calculated under the
framework of the first-order perturbation, as the interstellar magnetic field
is quite weak, as weak as 100 $\mu$G or less. The authors employ the Hund’s
case (b) for the coupling scheme of the angular momenta for the calculation .
We quote Shinnaga and Yamamoto’s Zeeman factor values to the second decimal
place, and they are in agreement with those reported in Vlemmings et al.
(2019).

Table 1: CN N=$2\rightarrow 1$ Hyperfine Lines Line $J,F\rightarrow
J^{\prime},F^{\prime}$ $\nu$ (GHz) Z (Hz/$\mu$G) 1
$\frac{3}{2},\frac{3}{2}\rightarrow\frac{1}{2},\frac{3}{2}$ 226.63217 $-0.72$
2 $\frac{3}{2},\frac{5}{2}\rightarrow\frac{1}{2},\frac{3}{2}$ 226.65956
$-0.71$ 3 $\frac{3}{2},\frac{1}{2}\rightarrow\frac{1}{2},\frac{1}{2}$
226.66369 $-0.62$ 4
$\frac{3}{2},\frac{3}{2}\rightarrow\frac{1}{2},\frac{1}{2}$ 226.67931 $-1.18$
5 $\frac{5}{2},\frac{5}{2}\rightarrow\frac{3}{2},\frac{3}{2}$ 226.87419 +0.71
6 $\frac{5}{2},\frac{7}{2}\rightarrow\frac{3}{2},\frac{5}{2}$ 226.87478 +0.40
7 $\frac{5}{2},\frac{3}{2}\rightarrow\frac{3}{2},\frac{1}{2}$ 226.87590 +1.18
8 $\frac{5}{2},\frac{3}{2}\rightarrow\frac{3}{2},\frac{3}{2}$ 226.88742 +1.47
9 $\frac{5}{2},\frac{5}{2}\rightarrow\frac{3}{2},\frac{5}{2}$ 226.89213 +1.06
Frequencies and Zeeman splitting factors (Z) calculated from theory for the
hyperfine lines (Shinnaga and Yamamoto in preparation) covered in these
observations. The Zeeman splitting factor is a measure of how much a line will
be split in frequency by a given magnetic field strength. A higher absolute
value of $Z$ means that a line is more sensitive to the line of sight magnetic
field.

The data were calibrated by data analysts at the NAASC using a script
developed for calibrating ALMA polarization observations. All data reduction
was performed using the Common Astronomy Software Applications (CASA) version
5.4.0. The data were cleaned using Briggs weighting with a robust parameter of
0.5 to create image cubes and continuum images for all four Stokes parameters.
We performed one round of phase-only self calibration on the continuum $I$
data, with the solution interval set to the scan length. Before making the
line image cubes, we subtracted the continuum emission from the spectral line
windows. The image cubes were created with a spectral resolution of 0.25 km/s.
The images have a beam size of 1.40$\arcsec$ $\times$ 1.27$\arcsec$. The
linear polarized intensity map was debiased using the average noise value
determined from the $Q$ and $U$ maps, an estimator used by e.g. Wardle &
Kronberg (1974) and Vidal et al. (2016):

$P=\begin{cases}\sqrt{Q^{2}+U^{2}-\sigma^{2}}&\text{if
$\sqrt{Q^{2}+U^{2}}\geq\sigma$}\\\ 0&\text{otherwise}\end{cases}$ (1)

We estimate that the uncertainty on the amplitude calibration is $\pm$10%, and
from here on, we only give statistical uncertainties. The rms values of the
image cubes are $\sigma_{(I,Q,U,V)}$ = (0.91, 0.95, 0.96, 0.95) mJy beam-1 per
0.25 km/s channel in the spectral window containing lines 1-4 and
$\sigma_{(I,Q,U,V)}$ = (0.95, 0.88, 0.90, 0.89) mJy beam-1 per 0.25 km/s
channel in the spectral window containing lines 5-9. The continuum I rms value
was 0.60 mJy beam-1 before self calibration and 0.33 mJy beam-1 after self
calibration. The rms values for the $Q$, $U$, and $V$ continuum images are
$\sigma_{(Q,U,V)}$ = (0.015, 0.012, 0.013) mJy beam-1. The higher noise value
in Stokes $I$ compared to $Q$, $U$, and $V$ is due to dynamic range limits.

##  3 Results

In the dust continuum total intensity (Stokes $I$), the disk has a peak flux
of 127.4 mJy/beam and a total intensity of 215.0 $\pm$ 1.8 mJy. All nine of
the hyperfine components that we targeted were detected in Stokes $I$. The
integrated line intensity for the hyperfine components of lines 5-7 in Table 1
is 3.874 $\pm$ 0.015 Jy km/s, which is consistent with the value of 3.32 $\pm$
0.14 Jy km/s from (Öberg et al., 2011), given that the absolute flux
calibration uncertainty is $\sim$10% for ALMA and $\sim$10-15% for the SMA.
The velocity map of the source shows the pattern expected from a rotating disk
(see Figure 1b).

Figure 1: Integrated intensity (moment 0) map for hyperfine components 1-4
(a), and velocity (moment 1) map for hyperfine 1 (b). The moment maps are
created from line emission with intensity $\geq 5\sigma_{Iline}$. Contours
represent total intensity (Stokes $I$) of 3, 10, 50, 100, 200, 325 and
500$\sigma$ levels.

###  3.1 Line circular polarization

The Stokes $V$ polarization produced by the Zeeman effect is related to the
strength of the line of sight B-field $B_{LOS}$, the derivative of the line
Stokes $I$ flux with respect to frequency $dI/d\nu$, and the Zeeman splitting
factor $Z$ by $V=\frac{1}{2}ZB_{LOS}\frac{dI}{d\nu}$ in the case where the
frequency splitting is small compared to the Stokes $I$ line width. We made
maps of all of the CN hyperfine lines in Table 1 looking for detections in
Stokes $V$. Stokes $V$ was not obviously detected in the channel maps,
spectra, or moment 0 maps. However, the signal could be contaminated by
instrumental terms. To solve for and eliminate instrumental effects that could
mimic Zeeman splitting, we use the technique developed by Crutcher et al.
(1996). This technique fits the observed Stokes $V$ profile of each hyperfine
component to the expression

$V_{j}(\nu)=C_{1}I_{j}(\nu)+C_{2}\frac{dI_{j}(\nu)}{d\nu}+\frac{1}{2}BZ_{j}\frac{dI_{j}(\nu)}{d\nu}$
(2)

where $j$ refers to each hyperfine component. $C_{1}$ absorbs any gain
difference between left and right polarization that was not calibrated out
previously, as well as any linearly polarized line signal. C2 absorbs any
instrumental polarization effects such as beam squint that produce pseudo-
Zeeman splitting, which shows up as the same splitting in each hyperfine line.
$B$ will be non-zero only if there is circular polarization due to the Zeeman
effect. We used this method to simultaneously fit all the unblended portions
of the lines integrated over the circumstellar disk, but we were still unable
to find a detection of Zeeman splitting.

Figure 2:

However, unlike the case for the clouds studied by Crutcher et al. (1996), the
disk of AS 209 is rotating, which broadens the line emission and would impact
any disk spatial averaging. To account for this, we used the package gofish
(Teague, 2019) to shift the velocities in the Stokes $I$ and Stokes $V$ cubes
by the amounts appropriate to correct for Keplerian motion. Based on the
disk’s inclination angle, position angle, distance, and stellar mass, gofish
calculates the expected Keplerian velocity for each pixel in the image cube.
This allows us to shift the line emission for each Stokes parameter onto a
common centroid velocity range based on the disk’s deprojected Keplerian
rotation profile.

Our angular resolution is sufficient to resolve the redshifted and blueshifted
sides of the disk. This is advantageous for investigating any toroidal
component of the magnetic field because we expect a toroidal magnetic field to
have opposite signs on the red and blue sides of the disk. We used two
different methods to extract information about the magnetic field in the disk.
First, we created average Stokes $I$ and average Stokes $V$ profiles for the
redshifted and blueshifted sides of the disk and fit these data using the
technique described earlier in this section. The Stokes $I$ and $V$ spectra
for all of the hyperfine components created using gofish are shown in Figure
3. To calculate the uncertainties in each velocity bin, we calculated the per-
channel rms in a region outside the disk from spectral line cubes whose
velocities had been Kepler-corrected in gofish.

Figure 3:  Stokes $V$ and Stokes $I$ profiles created using gofish for
hyperfine components 1-9, averaged across the redshifted and blueshifted sides
of AS 209. Velocities are LSRK velocities with respect to the velocity of
hyperfine component 2. The red line represents the redshifted side of the
disk, and the blue line represents the blueshifted side of the disk.

After using gofish to account for line broadening due to Keplerian motion,
lines 5, 6, and 7 were still blended. We therefore used only the lower-
frequency portion of the Kepler-corrected line 5 data and the higher-frequency
portion of the Kepler-corrected line 7 data in the Crutcher et al. (1996) fit.
The magnetic field strength that produced the best fit to these data was 1.9
$\pm$ 1.7 mG on the redshifted side of the disk and 1.0 $\pm$ 1.4 mG on the
blueshifted side of the disk. Using the uncertainties on the magnetic field
strengths as the 1$\sigma$ value, the limit on the net line-of-sight magnetic
field strength is 5.0 mG on the redshifted side of the disk and 4.2 mG on the
blueshifted side of the disk. Given the disk’s inclination angle of 35.3∘,
this places 3$\sigma$ upper limits of 8.7 mG (red) and 7.3 mG (blue) on the
net toroidal magnetic field and 6.3 mG (red) and 5.1 mG (blue) on the net
vertical magnetic field. The Stokes $I$ and $V$ averaged spectra from the red
and blue sides of each line are shown in Figure 3.

Second, we stacked the Stokes $I$ and Stokes $V$ spectra from both the red and
blue sides of the disk for only the unblended lines. The Stokes $I$ spectra
were scaled by their relative intensities, and the Stokes $V$ spectra were
scaled by the relative Stokes $I$ intensities and the Zeeman splitting
factors. If the disk field is toroidal, we would expect the Stokes $V$
spectrum from one side of the disk to be the same shape as the spectrum from
the other side of the disk, but mirrored across the velocity axis. This is
because the line-of-sight component of a toroidal field would have opposite
signs on the redshifted and blueshifted sides of the disk, and $V(v)$ is
proportional to the line-of-sight component of $B_{z}$, including the sign.
Therefore, stacking $V(v)$ from the red side of the disk with $-V(v)$ from the
blue side of the disk should increase the SNR of any Stokes $V$ emission that
comes from a toroidal field. However, this method of stacking would destroy
any Stokes $V$ signal from a vertical magnetic field, as we expect the
direction of the vertical component of the field to remain the same across the
disk. Nonetheless, stacking the lines in this way did not lead to a detection
of circularly polarized emission. We fit the weighted sum of the line profiles
$V_{tot}=\frac{1}{2I_{0}}B_{los}\sum_{i}Z_{i}\frac{dI_{i}}{d\nu_{i}}w_{I}w_{Z}$
to the stacked line data, where the $i$’s are lines 1, 2, 3, 4, 8, and 9;
$I_{0}$ is the maximum Stokes $I$ intensity, and $w_{I}$ and $w_{Z}$ are
weighting factors that account for the relative intensities of the lines and
the relative strengths of their Zeeman splitting factors, respectively. The
stacked spectra are shown in Figure 4. The best fit values for $B_{los}$ from
this stacking technique were -4.2 $\pm$ 6.9 mG for the redshifted side of the
disk, 2.0 $\pm$ 4.6 mG for the blueshifted side of the disk, and 0.4 $\pm$ 4.9
mG for the full disk. Because this method does not account for circular
polarization from instrumental effects, we report the results of using the
first fitting technique described above as our final upper limits on the
magnetic field strength.

Figure 4: Stacked Stokes $I$ and $V$ spectra using gofish for the redshifted
(a) and blueshifted (b) sides of the disk, and for the full disk (c). The
dashed horizontal lines in the plots of $V/I_{0}$ represent $\pm$ 1.8% of the
$I/I_{0}$ peak.

###  3.2 Continuum linear polarization

The continuum polarization pattern in AS 209 at 1.3 mm is shown in Figure 5.
Our observations are similar to those observed at 870 $\mu$m by Mori et al.
(2019), with the direction of polarization oriented parallel to the disk’s
minor axis in the inner part of the disk and oriented azimuthally in the outer
part of the disk. This pattern closely matches the polarization from
scattering in a disk with a similar inclination to AS 209’s predicted by the
model in Yang et al. (2016). We plan to explore the possible mechanisms behind
this polarization pattern in a future paper.

Figure 5: 1.3 mm continuum linear polarization in AS 209. The black contours
in (a) represent total intensity (Stokes $I$) of 3, 10, 50, 100, 200, 325 and
500$\sigma$ levels. The solid contours in (b) represent the locations of
rings, and the dashed contours represent the locations of gaps from (Fedele et
al., 2018). The colormap represents debiased polarized intensity with the
scale on the right of each source. The length of the polarization vectors
corresponds to the polarization fraction. The vectors are plotted with $\sim$
5 segments per beam. Vectors are only plotted where the polarized intensity
$P\geq 3\sigma_{P}$ and $P/I\geq 0.1$.

##  4 Discussion

We did not detect circular polarization in any of the individual CN 2-1 lines
or in the stacked lines, so we have calculated 3-$\sigma$ upper limits on the
net toroidal and vertical magnetic field strengths. The minimum detectable
degree of circular polarization with ALMA is 1.8% of the peak Stokes $I$ flux
according the ALMA Cycle 7 Technical Handbook. The Stokes $V$ flux does not
reach this 1.8% threshold in any channel of any individual hyperfine line or
the stacked lines in our observations. Because the fitting technique described
in Section 3.1 removes instrumental effects that could produce spurious Stokes
$V$ signal, and because the Stokes $V$ spectrum is noise-like, the
uncertainities on the magnetic field strengths calculated using the fitting
technique can be used to calculate upper limits on the disk’s magnetic field
strength. The average rms values on the Stokes $V$ spectra shown in Figure 3
were $\sim$0.9% of the Stokes $I$ peak of the brightest line, which means that
a Stokes $V$ signal $\geq$ 0.02 Jy beam-1 would be detectable in the brightest
line. Here, we discuss the implications of our field strength upper limits for
the disk’s mass accretion rate, as well as possible reasons for the non-
detection.

Our non-detection of Zeeman splitting in the CN 2-1 line allows us to put
constraints on the mass accretion rate that the magnetic field can drive in
the disk on the 10’s of au scale. The magnetically driven accretion rate
${\dot{M}}_{\rm mag}$ is related to the magnetic stresses through, e.g.,
equation 18 of Wang et al. (2019). Making the simplifying assumptions that
$|B_{R}|$, $|B_{z}|$, and $|B_{\phi}|$ are constant, and that $B_{\phi}$ has
opposite signs above and below the midplane, this equation can be recast into
a rough order-of-magnitude estimate of the magnetically driven accretion rate:

$\begin{split}{\dot{M}}_{\rm
mag}\approx\frac{2R}{\Omega}|B_{z}B_{\phi}|+\frac{2h}{\Omega}|B_{R}B_{\phi}|\\\
=\frac{2R|B_{z}B_{\phi}|}{\Omega}\left(1+\frac{h}{R}\frac{|B_{R}|}{|B_{z}|}\right),\end{split}$
(3)

where $B_{R}$, $B_{z}$ and $B_{\phi}$ are, respectively, the (cylindrically)
radial, vertical and azimuthal component of the magnetic field, $h$ the disk
scale height, and $\Omega=(GM_{*}/R^{3})^{1/2}$ is the angular Keplerian speed
at radius $R$. The first term on the right hand side of the equation is mass
accretion driven by a magnetized disk wind and the second term is that from
magnetic stresses internal to the disk. They are consistent with the estimates
from Bai & Goodman 2009, see their equations 6 and 15 respectively. Since the
disk is geometrically thin, with $h/R\ll 1$, the magnetic disk wind tends to
remove angular momentum more efficiently than the internal magnetic stresses
for comparable $B_{z}$ and $B_{R}$. In this case, we have

$\begin{split}{\dot{M}}_{\rm
mag}\approx\frac{2R|B_{z}B_{\phi}|}{\Omega}=2.1\times
10^{-6}\left(\frac{|B_{z}|}{6.2\ {\rm mG}}\right)\left(\frac{|B_{\phi}|}{8.7\
{\rm mG}}\right)\\\ \left(\frac{M_{*}}{1.25\
M_{\odot}}\right)^{-1/2}\left(\frac{R}{50\ {\rm au}}\right)^{5/2}\
\frac{M_{\odot}}{\rm yr},\end{split}$ (4)

where we have normalized the vertical and toroidal components of the magnetic
field by their respective $3\sigma$ upper limits and the stellar mass $M_{*}$
by the value inferred in Teague et al. (2018). The $\sim 10^{-6}$ $M_{\odot}$
yr${}^{-}1$ value we estimate from our magnetic field strength upper limits is
to be compared with the mass accretion rate from the disk onto the central
star of $10^{-7}$ $M_{\odot}$/yr estimated by Johns-Krull et al. (2000) based
on the luminosity of ultraviolet lines (especially CIV). This accretion rate
is on the high side for classical T Tauri stars and needs to be checked
through independent methods.

It is possible that the absolute value of the disk magnetic field strength
$|B|$ is substantially higher than our upper limits on the net line-of-sight
magnetic field strength. This is particularly true in the case where the
magnetic field in the disk is dominated by the toroidal component $B_{\phi}$
and $B_{\phi}$ reverses polarity across the disk midplane. Such a field
reversal would be naturally produced if there is a net magnetic flux threading
the disk (as is likely given that the disk forms out of magnetized dense cores
that appear to have fairly regular magnetic fields as traced by dust
polarization; for a recent review, see Hull & Zhang 2019). The differential
rotation between the disk midplane and the atmosphere and/or disk wind
naturally twists the polodial field into a toroidal field that reverses
direction around the midplane (for an illustration, see Suriano et al. 2018).
In this case, if the disk is not too optically thick, an individual line of
sight will contain CN emission from above and below the disk midplane and thus
sample gas with reversed toroidal magnetic fields and thus Zeeman signals of
opposite sign. The optical depths of the bright rings of AS 209 are estimated
to be 0.46 and 0.52 at ALMA Band 6 (Dullemond et al., 2018) and should be much
lower in the gaps. It is likely that the bulk of the CN emitting materials
both above and below the disk midplane contribute to the observed signals,
which leads to a cancellation of the Zeeman signal. In this case, the Zeeman
measurement can significantly underestimate the absolute strength of the
magnetic field, as illustrated quantitatively by Mazzei et al. (2020, MNRAS,
submitted).

##  5 Conclusions

We present the first Zeeman observations toward the circumstellar disk of AS
209, using 9 hyperfine components of CN 2-1 in ALMA Band 6. Although we easily
detect the dust polarization of the disk, which is consistent with previous
Band 7 observations, we do not detect any polarized emission in the CN lines.
After correcting for the Keplerian rotation of the disk using gofish, we used
two approaches to derive upper limits on the magnetic field strengths: fitting
the equation described in Equation 2 to each hyperfine component as described
in Crutcher et al. (1996) and fitting an equation with the form of the Zeeman
splitting profile to the scaled sum of all of the un-blended components. We
derived these limits for the redshifted and blueshifted sides of the disk, as
well as the entire disk. We present 3$\sigma$ upper limits based on the
stacking technique described in Crutcher et al. (1996) because this technique
allows us to remove circular polarization from instrumental effects from the
Stokes $V$ spectra. In that case, we have calculated 3$\sigma$ upper limits on
the net toroidal and vertical magnetic field strengths of $B_{\phi}<8.7$ mG
and $B_{z}<6.1$ mG. A change in polarity of the toroidal magnetic field across
the disk midplane may cause us to underestimate the absolute strength of the
toroidal component of the magnetic field, and therefore the true toroidal
magnetic field strength could be $>$ 8.7 mG. Our constraints on the magnetic
field strength provide an upper limit on the magnetically-driven mass
accretion rate on the 50 au scale of order $10^{-6}$ M${}_{\odot}yr^{-1}$ or
smaller, which is consistent with the mass accretion rate onto the star
previously inferred for this object.

##  6 Acknowledgements

This paper makes use of the following ALMA data: ALMA #2018.1.01030.S. ALMA is
a partnership of ESO (representing its member states), NSF (USA) and NINS
(Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. REH and
LWL acknowledge support from NSF AST-1910364. ZYL is supported in part by NASA
80NSSC18K1095 and 80NSSC20K0533 and NSF AST-1716259 and AST-1910106. RMC is
supported in part by NSF-1815987. We would like the North American ALMA
Science Center data analysts for performing the initial data calibration and
imaging. We would also like to thank Ryan Loomis and Mark Lacy for their
support during our data reduction visit to the NAASC.

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11institutetext:  Leipzig University, Faculty of Physics and Earth Sciences,
Institute for Theoretical Physics, Brüderstraße 16, 04081 Leipzig 11email:
<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>
Charles University, Faculty of Mathematics and Physics, Department of
Macromolecular Physics, V Holešovičkách 2, CZ-180 00 Praha 11email:
<EMAIL_ADDRESS>

# Force-Free and Autonomous Active Brownian Ratchets

Constantin Rein 11 Martin Kolář 22 Klaus Kroy 11 Viktor Holubec 221122

###### Abstract

Autonomous active Brownian ratchets rectify active Brownian particle motion
solely by means of a spatially modulated but stationary activity, without
external forces. We argue that such ratcheting requires at least a two-
dimensional geometry. The underlying principle is similar to the ratcheting
induced by steric obstacles in microswimmer baths: suitably polarized swimmers
get channeled, while the others get trapped in low-activity regions until they
loose direction. The maximum current is generally reached in the limit of
large propulsion speeds, in which the rectification efficiency vanishes.
Maximum efficiency is attained at intermediate activities and numerically
found to be on the order of a few percent, for ratchets with simple wedge-
shaped low-activity regions.

##  1 Introduction

Brownian ratchets are subtle microscale transport devices [1, 2]. They combine
two effects that individually do not promote directed transport, namely
isotropic Brownian motion and asymmetric environments, such that a net
directed particle current is produced [3, 4, 5]. Conventional designs with
passive particles usually break the spatial symmetry by imposing an asymmetric
potential. They also introduce a non-equilibrium element, namely an isotropic
and often periodic driving mechanism that, by itself, does not introduce any
directionality [4]. Typical examples comprise the rocking (or “flashing”) of
the potential or the overall temperature [6, 3, 7, 4]. More complex
temperature fields have also been investigated [8, 9, 10, 11, 12].

The self-propulsion of an active Brownian particle (ABP) represents yet
another non-equilibrium mechanism that one ought to be able to exploit for
ratcheting. While it does transiently break the spatial and temporal symmetry
of equilibrium Brownian motion [13], it does not, by itself, give rise to a
net macroscopic current. One would however expect that one of the simplest
realizations of an active Brownian ratchet should consist of an ABP exposed to
a spatially asymmetric (periodic) activity landscape. Yet, even though a
number of ratchet designs with active particles have been discussed in the
literature [14, 17, 18, 19, 20, 21, 22, 23, 24, 15, 16], none of them was
based solely on a stationary activity landscape. Instead, some relied on ABPs
placed in a soft potential in one spatial dimension [19, 21], or in asymmetric
hard potentials in two-dimensions [14, 17, 18, 20, 16]. The asymmetric
potentials, so typical of conventional ratchets, can be relinquished entirely,
though, if one exploits the tendency of ABPs to polarize towards low-activity
regions and accumulate there [25, 15, 26, 27, 28]. The standard flashing
potential can then be replaced by a dynamic activity landscape. Examples
include propagating optical activation pulses that induce aligned or anti-
aligned drifts, depending on the persistence length of the ABP motion relative
to the pulse width and propagation speed [22, 23, 24]. In general, traveling
activity waves induce traveling density and orientation waves of the ABPs, and
can thus plainly be employed to sort ABPs, e.g., by size [15].

To sum up, ratcheting has been demonstrated for active particles in spatially
asymmetric potential landscapes or in space-and-time dependent activity
landscapes. However, no fundamental symmetry prevents ABPs from ratcheting
also in _stationary_ spatially asymmetric activity landscapes. In the
following, we show that such ratchets are indeed realizable and explore the
maximum current and rectification efficiency of a class of simple shapes,
numerically.

##  2 Model

We consider the motion of an ABP in a unit-square arena (thus taking its size
as the natural length unit) with periodic boundary conditions in two
dimensions (see Fig. 1). The state at time $t$ is fully characterized by the
position ${\bf{r}}(t)=[{x}(t),{y}(t)]$ and polarization
$\mathbf{n}(t)=[\cos\theta(t),\sin\theta(t)]$ of the ABP. The translational
and rotational Brownian motion are represented by mutually independent and
unbiased ($\left<\eta_{i}\right>=0$) Gaussian white noises $\eta_{i}(t)$ of
unit strength,
$\left<\eta_{i}(t)\eta_{j}(t^{\prime})\right>=\delta_{ij}\delta(t-t^{\prime})$,
and diffusion constants $D_{\text{t}}$ and $D_{\text{r}}$, respectively. The
stationary activity landscape enters via a superimposed deterministic speed
field $v[x,y]$. The dynamical equations for the ABP read

$\displaystyle\dot{x}$
$\displaystyle=v(x,y)\cos(\theta)+\sqrt{2D_{\text{t}}}\eta_{x},$ (1a)
$\displaystyle\dot{y}$
$\displaystyle=v(x,y)\sin(\theta)+\sqrt{2D_{\text{t}}}\eta_{y},$ (1b)
$\displaystyle\dot{\theta}$ $\displaystyle=\sqrt{2D_{\text{r}}}\eta_{\theta},$
(1c)

We only consider activity fields symmetric in the $y$-direction,
$v(x,1/2+y)=v(x,1/2-y)$, so that $\left<\dot{y}(t)\right>=0$ and the steady-
state current is a scalar $I=\left<\dot{x}(t)\right>$. Whenever $I$ is
consistently positive or negative, the device exhibits ratcheting.

A few general observations about the dynamics are gleaned directly from the
above equations. First, the essential stochastic ingredient of the model is
the rotational diffusion. If $D_{\text{r}}$ is taken to infinity, the ABP
motion looses its persistence. The model then reduces to a passive gas locally
equilibrated at a spatially modulated (effective) temperature
$T=D_{\text{t}}+v^{2}/2D_{\text{r}}$, with Boltzmann’s constant and the
friction coefficient set to unity. While such a gas can move
thermophoretically in the presence of a temperature gradient, it cannot
maintain a steady current in a periodic temperature profile. (We comment on
the more subtle limit of a Knudsen gas [29, 30], at the end of the paper.) The
ratcheting effect must thus entirely result from a clever combination of the
more or less persistent motion in the high- and low-activity regions,
respectively.

For conceptual purposes, it is sufficient to consider discrete landscapes with
$v$ being represented by a step function, since the dynamics is anyway low-
pass filtered by the translational diffusion process. Any small scale details
and discontinuities in $v(x,y)$ will thereby effectively be washed out. Also
notice that setting the maximum value of $v$ to a very large (formally
infinite) value amounts to the idealization of strictly ballistic dynamics in
the high-activity (or simply “active”) regions. Similarly, retaining a non-
vanishing $D_{\text{t}}>0$ to avoid an absorbing state, the minimum value of
$v$ can safely be set to zero in the low-activity (or simply “passive”)
regions, without much loss of generality. This choice, which shall be adopted
for the remainder, simply amounts to purely diffusive dynamics, inside the
passive region. Notice that a periodic two-step function of a single scalar
variable is necessarily symmetric. In one dimension, one thus clearly cannot
achieve autonomous ratcheting with a corresponding stationary activity
field—nor (as shown below) with any other.

In summary, translational diffusion acts as a regularization for step-wise
constant activity profiles, so that the archetypal activity landscape
discretely jumps between $v=0$ and some finite or possibly even infinite value
$v$. In the latter case, the active region is traversed in no time, so that,
the total dwell time $\tau$ of the particle in the unit cell is equal to the
time spent in the passive region. The latter is independent of $v$ and, at
first sight, of $D_{\text{r}}$. However, $D_{\text{r}}$ limits the “take-off”
of ABPs emerging from the passive region, and in fact also the whole particle
distribution at the active-passive boundary. For example, the ABP cannot take
off if it emerges with a swim direction pointing back into the passive region.
Also it can “tunnel” through narrow edges of the passive region. One therefore
generally still expects the current $I\simeq\tau^{-1}$ (in our unit length
setup) and the dwell time $\tau$ to depend on $D_{\text{t}}$ and
$D_{\text{r}}$, even if one takes $v\to\infty$, in the active region. It is
however plausible, and indeed corroborated by our Brownian dynamics
simulations of the model presented below that for a given geometric shape of
the passive region, one can often find an optimum choice of
$D_{\text{t}}\simeq D_{\text{r}}$ (Fig. 2). Then
$\tau(D_{\text{t}},D_{\text{r}})\to\tau(D_{\text{r}})$ depends solely on
$D_{\text{r}}$, implying $I\simeq D_{\text{r}}$, with a purely geometric
prefactor. The latter can only depend on dimensionless features of the shape
(such as the parameters $\delta$ and $\varepsilon$ in Fig. 1). In other words,
under such idealized conditions, the task of an optimum ratchet design is
entirely reduced to a geometric optimization problem.

These general considerations based on an infinite step function $v(x,y)$ may
not always be practically useful, from an active-matter perspective. For
instance, an experimental realization of our idealized ABP might possibly only
allow for a maximum speed $v$, below the asymptotic regime alluded to above
(in which the dwell time in the arena equals the trapping time in the passive
region). This will clearly reduce the ratchet current from its maximum value,
and the dwell time will depend both on $D_{\text{t}}\simeq D_{\text{r}}$ and
the maximum attainable value of $v$. This “attenuated” transport regime, with
$D_{\text{t}}\simeq D_{\text{r}}\simeq v$ may be of particular practical
interest, if the active speed of the ABP is regarded as a costly input. The
most desirable _modus operandi_ of the ratchet will then not anymore be that
of maximum current $I\simeq D_{\text{r}}$, obtained in the limit $v\to\infty$,
because the ratio $I/v$ vanishes in this limit. Instead, one will then
typically be interested in conditions that optimize this ratio, which can be
interpreted as the rectification efficiency of the active ratchet, very much
in the spirit of ABP engines and bacterial motors [17, 18, 16]. The interested
practitioner will then generally have to find the corresponding optimum
parameter values $D_{\text{t}}$, $D_{\text{r}}$, and $v$ for a given ratchet
geometry, numerically.

The remainder of the paper is dedicated to a more comprehensive analysis of
the above general considerations. In particular, we want to clarify why
stationary active Brownian ratchets can only be realized in at least two space
dimensions. We also estimate realistic values of the maximum dimensionless
current $I(v\to\infty)/D_{\text{r}}$ and rectification efficiency $I/v$, for a
simple wedge geometry.

##  3 One-dimensional activity patterns

Already in one spatial dimension, spatially varying activity profiles
accommodate non-intuitive effects. For example, the mean first passage time
may depend non-monotonically on the distance from a target and the target
finding probability can increase if the activity increases towards the target
[31]. This seemingly contradicts the known fact that active particles spend
less time in regions of higher activity. However, while the latter is a
steady-state property, the former relates to transient behavior. In fact, when
an ABP is oriented along an activity gradient, it accelerates and thus
increases its chance to reach a target before it looses its orientation.
Similarly, an ABP placed in the middle of a one-dimensional domain with a
linear activity gradient reaches the high-activity end faster and more often
than the low-activity end [32]. Although these effects look promising with
regard to designing autonomous active Brownian ratchets, e.g., with a
saw–tooth-shaped stationary activity landscape, there is a catch. In the cited
experiments [32, 31], the state is repeatedly reset externally, by placing the
particle back in its initial position upon reaching the target or the boundary
of the arena. For a genuine ratchet, such _“deus-ex-machina”_ type outside
interventions are clearly not a permissible option.

More formally, one can demonstrate the absence of ratcheting in one-
dimensional activity landscapes, as follows. Activity landscape can sort and
locally accumulate ABPs according to their orientation, but they do not
reorient them. Crucially, and quite in contrast to potential landscapes,
activity landscapes do not exert any forces or torques on the ABPs, which are
a crucial mechanism underlying the ratcheting of ABPs in one-dimensional
potential landscapes [19]. As all orientations are thus equally probable in an
unbiased ensemble, the spatially integrated total polarization must vanish.
Together with the continuity equation for particle number conservation [33],
this entails that the net current vanishes, too. More concretely, one may
evoke the continuity of the local polarization profile as a function of
position, for piece-wise continuous activity profiles [28, 27, 26]. From this
one concludes that, for a vanishing total polarization, there must be at least
be one position $x_{0}$ in the polarization profile at which the time-averaged
orientation vanishes. The time averaged current $I$ at this point is given by
the time-integral over $v[x(t)=x_{0}]\cos\theta(t)$. Up to a constant factor,
this is just the vanishing time-averaged orientation. And since, in one
spatial dimension, the continuity condition implies that the steady state
current is spatially constant, $I$ vanishes everywhere if it vanishes locally,
at $x_{0}$. We have corroborated this conclusion by extensive Brownian
dynamics simulations and by numerical solution of the Fokker–Planck equation,
associated with Eq. (1), using the method of Ref. [34].

##  4 Two-dimensional activity patterns

Compared to one-dimensional activity landscapes, the situation is much
different in two and higher-dimensional activity landscapes. The main reason
is that the inevitable zeroes of the polarization do now no longer constrain
the overall current to vanish, unless they cover a whole vertical line
$(x_{0},\\{y\\})$. The latter is by no means required by the condition on an
overall vanishing polarization. Around an isolated point of vanishing current,
the resulting systematic flow field (or, equivalently, polarization field)
takes the form of a vortex, as seen in Fig. 1. The sorting and accumulation of
ABPs according to their orientation along the $x$-direction, which is already
possible in one-dimensional activity landscapes [27, 26, 28], and exploited in
non-stationary active Brownian ratchets [22, 23, 24, 15], is now modulated
along the second spatial direction $y$. A particle moving along the
$y$-direction therefore experiences an effectively time-modulated activity
pattern along the transport direction $x$, which has a similar rectifying
effect as a dynamical one-dimensional activity profile.

The stationary but spatially periodically modulated activity-landscape
$v(x,y)$ shown in Fig. 1 provides a proof-of-principle example and serves as
an instructive illustration of a working ratchet. It features a piece-wise
constant activity field with a wedge-shaped passive region, where $v(x,y)=0$,
in an otherwise moderately active unit square with constant
$v(x,y)=D_{\text{r}}$. The landscape is asymmetric along the $x$-direction and
mirror-symmetric along the $y$-direction. The dimensionless numbers
$\delta_{x}$, $\delta_{y}$, and $w=\varepsilon(1-2\delta_{x})$, with
$\varepsilon\in[0,1]$, denote the distances of the edges from the periodic
boundaries and the width of the wedge along its mirror-symmetry axis,
respectively. The extreme geometries corresponding to an infinitely thin
passive region ($\varepsilon=0$) and a convex, triangular passive region
($\varepsilon=1$) both yield sub-optimal ratchets.

While even this simple wedge model is not exactly solvable, its performance
can qualitatively be understood, using simple physical arguments. First, the
above-mentioned saturation of the ratchet current for infinite speed
$v\to\infty$ in the active region is simply due to the fact that the time
spent by the ABP in the active region becomes negligible compared to the time
$\tau$ spent diffusing in the passive region. This limit is thus amenable to
event-driven simulations. Below, we go one step further and exploit it to
construct a simplified geometric toy model that can provide semi-analytical
estimates for the ratcheting current. Unfortunately, as already pointed out
above, the conceptually convenient large-speed limit is somewhat academic. The
practitioner will be interested in more affordable, finite values of $v$.
Therefore, one should also consider the rectification efficiency $I/v$, which
is the current produced by the ratchet relative to that of a perfectly
polarized ABP.

To understand the pertinence of the limits of infinite or vanishing
diffusivities $D_{\text{r}}$, $D_{\text{t}}$, recall that ratcheting is all
about the geometric rectification of stochastic motion. In the limit
$D_{\text{r}}\to 0$ (perfect persistence), the initial polarization is however
entirely conserved, while the limit $D_{\text{r}}\to\infty$ (vanishing
persistence) corresponds to thermophoresis within an effective temperature
field. So both limits do not correspond to genuine active ratcheting.
Similarly, passive regions, with vanishing speed $v=0$, would all become
absorbing for $D_{\text{t}}\to 0$, while in active regions with a finite
$v<\infty$, $D_{\text{t}}\to\infty$ would wipe out the persistent active
motion. Again, both limits are irrelevant for the discussion of active
ratcheting. And even though one could set $D_{\text{t}}=0$ without creating an
absorbing state if a non-vanishing speed $v>0$ was maintained in the passive
(or less active) region, this choice would be unnatural, as it requires
passive regions with vanishing (or even “small”) $v$ to be administratively
forbidden. On the other hand, allowing for some finite $D_{\text{t}}\lesssim
D_{\text{r}}$ is not very consequential for the transport in the (more) active
regions, where it merely partially degrades the persistence induced by the
activity. This exposes $D_{\text{t}}$ as a parameter of minor physical
relevance except for its regularizing role in the passive regions. There are
however two more reasons for including a non-vanishing $D_{\text{t}}$, in the
discussion. Firstly, it will actually matter for the comparison to practical
physical realizations of an ABP ratchet. And secondly, it also serves to
regularize some fine-grained details of the ratchet geometry, thereby putting
a limit on an otherwise potentially limitless ornamentation of the ratchet
design that would in practice have to be cut off by a physical particle
radius. In contrast to the indispensable finite rotational diffusivity
$D_{\text{r}}$, the translational diffusivity $D_{\text{t}}$ thus plays a
rather technical role, as a model regularization parameter.

In conclusion, a pertinent discussion of a stationary ABP ratchet in two
dimensions is best conducted for finite diffusivities $D_{\text{r}}$ and
$D_{\text{t}}$. While $D_{\text{t}}^{-1}$ may at first suggest itself as the
natural time unit of the ratchet (its dwell time), it turns out that its
physical impact can, for a conceptual analysis, effectively be taken largely
out of the game. The trick is to set it to an optimum value that maximizes the
rectification efficiency $I/v$. Our numerical analysis (see Fig. 2) confirms
the expectation that this “best” value is unique and on the order of
$D_{\text{r}}$, for the simple geometry shown in Fig. 1. Its physical origin
may be understood from the role played by $D_{\text{t}}$ for controlling the
ABP’s escape time from the passive region. As already pointed out, above, if
$D_{\text{t}}\gg D_{\text{r}}$, the ABP will not have lost its polarization
when it leaves the passive region, and therefore typically swim right back
into it, unless that region is narrow enough to be traversed with a
substantial (“tunneling”) probability. Additionally, the dominance of
translational diffusion for $D_{\text{t}}\gg D_{\text{r}}$ will unduly degrade
the persistence in the active region beyond the inevitable minimum, set by
$D_{\text{r}}$. In contrast, if $D_{\text{t}}\ll D_{\text{r}}$, the
regularizing effect of the translational diffusion onto the absorbing state
may become less than optimal, as the initial particle polarization will then
have been lost long before the ABP reemerges from the passive region.
Altogether, this suggests an optimum value of $D_{\text{t}}$ on the order of
$D_{\text{r}}$, as indeed numerically confirmed in Fig. 2.

To summarize, the natural length unit of the stationary active ratchet is set
by the domain size, its natural time unit by the inverse rotational diffusion
coefficient $D_{\text{r}}^{-1}$. And it is conceptually convenient (if not
generally highly advisable) to work with an optimized translational
diffusivity $D_{\text{t}}\simeq D_{\text{r}}$ of comparable magnitude. The
natural scale for the maximum ratchet current $I\simeq\tau^{-1}\simeq
D_{\text{r}}$ is then $D_{\text{r}}$ itself, while that of the natural
efficiency $I/v$ is $(\tau v)^{-1}\simeq D_{\text{r}}/v$. In practice, both
quantities may be expected to be somewhat reduced by a dimensionless
geometrical shape factor. The crucial message is then that determining the
optimum current $I/D_{\text{r}}$ and efficiency $I/v$ boils down to an
infinite dimensional geometric optimization problem intertwined with the
“thermodynamic” optimizations of the parameters $D_{\text{t}}$ and
$D_{\text{t}}$, $v/D_{\text{r}}$, respectively.

Figure 1: Unit cell of a (unit width) two-dimensional square ratchet with
$\delta_{x}=\delta_{y}=0.1$, $\varepsilon=0.75$, $v=D_{\text{r}}$, and
$D_{\text{t}}=10^{-4}D_{\text{r}}$. The background color encodes the
probability density for the position of the ABP that predominantly dwells in
the wedge-shaped passive region. Arrows show the mean orientation
$\left<\mathbf{n}\right>$ of the ABP obtained from Brownian dynamics
simulations, colors coding for the angular variance
$1-\Bigl{(}\left<n_{x}\right>^{2}+\left<n_{y}\right>^{2}\Bigr{)}^{1/2}$; small
values indicate strong alignment and O(1)-values a random orientation.

##  5 Numerical study

To provide a specific but instructive example, Fig. 1 illustrates the working
principle of the active Brownian ratchet and its polarization field
$\mathbf{n}(t)$ for a wedge-shaped passive region in the unit square, with
periodic boundary conditions. As already alluded to above, the orientation
field is indeed seen to form vortices around the points with vanishing average
orientation, which help to defy the no-go theorem for one-dimensional active
ratchets. To create the figure, we solved Eq. (1) by a Brownian dynamics
simulation with time-step $dt=10^{-4}/v$. The central observable is the
ratchet current $I=x(T)/T$, evaluated as the final traversed $x$-distance of
the ABP divided by the total simulation time $T=10^{7}/v$. We checked that the
vertical current $y(T)/T$ in the $y$-direction vanishes, as expected. As
demonstrated in Refs. [28, 26, 27], along the active–passive boundary, the ABP
points on average towards the passive region. This may seem surprising, since
it seems to imply a net particle influx into the passive region. It is an
illusion, however, since the swim pressure acting onto an active-passive
boundary is not exerted across it [26]. Actually, the particle can therefore
“escape” from the passive region, against this swim pressure. If it escapes
along the tip-side (right in Fig. 1), it likely ends up in the indented
concave part of the passive region (left in Fig. 1). On the other hand, if the
ABP escapes in the vertical direction towards the horizontal active channels
of width $2\delta_{y}$ (top and bottom in Fig. 1), it can generate a net
current from right to left. As a result, the passive region blocks particle
paths to the right more than those to the left. Remarkably, active Brownian
ratchets relying on potential forces acting like hard walls [35, 36, 20, 16,
14, 18] are based on the very same principle. The important difference here is
that our setup does not involve any potential forces, and the ABP can thus
freely pass back and forth between the passive and active region. With hard
walls, the ABP would slide along the wedge until it gets trapped in the pocket
or escapes into the channel, thereby generating a net ratchet current. In our
force-free active ratchet, the sliding motion is replaced by the diffusive
spreading inside the passive region.

For the setup illustrated in Fig. 1, we also investigated the rectification
efficiency $I/v$ for finite activity, $v<\infty$, as a function of the
diffusivities $D_{\text{r}}$ and $D_{\text{t}}$. In accord with our foregoing
qualitative considerations, the numerical results shown in Fig. 2 feature a
maximum around $I/v\sim 0.014$ for $D_{\text{r}}\sim 0.3v$ and
$D_{\text{t}}\sim 0.001v$. These optimum values are specific for the chosen
geometry and cannot be found without performing the numerical simulation.

Figure 2: Rectification efficiency $I/v$ as function of the inverse Péclet
numbers $D_{\text{r}}/v$ and $D_{\text{t}}/v$, for the active Brownian ratchet
depicted in Fig. 1.

A more challenging task is to find the most efficient ratchet geometry. Here,
we restrict this infinite dimensional optimization problem to the class of
wedge or arrowhead shapes illustrated in Fig. 1. We ask for the optimum depth
of the concave indentation, which is parametrized by $\varepsilon$. For
shallow indentations, the ABP spends more time in the passive region as needed
to loose its polarization. This reduces the current and the rectification
efficiency compared to a design with a stronger indentation. However, for very
deep indentations, the passive region becomes too narrow to allow for a
substantial reorientation of the traversing ABP, and the corresponding
“tunneling” of the polarization eventually nullifies the ratcheting effect
($I\propto\varepsilon\to 0$). In other words, there is necessarily a non-
monotonic dependence of the rectification efficiency on $\varepsilon$. As
illustrated in Fig. 3, this implies that the intermediate optimum value of
$\varepsilon$, once again, needs to be found numerically. This result also
nicely demonstrates the difference between our force-free active ratchet and
its siblings operating with potential forces. In particular, for ratchets with
hard walls around an exclusion zone of the same shape as our passive region,
the ratcheting would always be maintained, regardless of the wall thickness.
The figure demonstrates that the non-monotonic dependence of the rectification
strength on the indentation depth is robust against the fine tuning of the
diffusivities, and that the optimization depends on the interplay between the
geometry and the inverse Péclet numbers $D_{\text{t}}/v$ and $D_{\text{r}}/v$.

Beyond the indentation depth, one can also consider the effect of the
parameter $\delta_{y}$ for the lateral width of the horizontal active
channels. The current decreases both as $\delta_{y}\to 0$, when the channel
width vanishes, and for $\delta_{y}\gtrsim 1/2$, when the passive volume
becomes marginal relative to the overall domain size. Similarly, as for
$\varepsilon$, $D_{\text{r}}/v$ and $D_{\text{t}}/v$, the rectification
efficiency $I/v$ thus also exhibits a maximum as a function of $\delta_{y}$.
Finally, the remaining parameter $\delta_{x}$ measures the overall width of
the passive region in the $x$-direction. When $\delta_{x}\to 1/2$, the width
of the passive region vanishes, and therefore also the current $I$, similarly
as for $\varepsilon\to 0$. On the other hand, the current monotonically
increases with decreasing $\delta_{x}\to 0$, until the passive region spans
across the whole domain. Together, the shape parameters $\delta_{x}$,
$\delta_{y}$, and $\varepsilon$ control how pointed and asymmetric the passive
region may become. Generally speaking, $I/v$ grows with increasing asymmetry.

Figure 3: Rectification efficiency $I/v$ as a function of the indentation
depth, parametrized by $1-\varepsilon$. Various combinations of $D_{\text{r}}$
and $D_{\text{t}}$ are shown, with colors coding for the value of
$D_{\text{r}}/v$: $0.1$ (red), $0.3$ (blue) and $1$ (green), and line style
for $D_{\text{t}}/v\mathrel{\mathop{\mathchar 58\relax}}$ $10^{-4}$ (solid)
and $10^{-3}$ (dotted).

##  6 Geometric toy model

A more mechanistic insight into the effects of the ratchet geometry on the
current can be obtained from a schematic, purely geometrical toy model. It is
defined by the idealized rules that the particle moves with infinite speed
$v\to\infty$ in the active region and rotates and spreads sufficiently fast
throughout the passive region to emerge from its surface with uniform spatial
and orientational distributions, after a dwell time $\tau$. The ensuing
simplifications enable us to bypass the computationally expensive Brownian
dynamics simulations for qualitative estimates. The path of the ABP in the
active region is then uniquely determined by the ratchet geometry alone. Once
the ABP leaves the passive region with randomized orientation and position, it
immediately hits either another part of the same passive region or one of its
periodoc images, as sketched in Fig. 4.

One can therefore evaluate the probabilities $P_{\leftarrow}$,
$P_{\rightarrow}$, and $P_{\updownarrows}$ that the ABP leaving the passive
region travels to the left, right, or merely vertically, respectively. If the
dwell time $\tau$ is approximated by the average reorientation time
$\tau=D_{\text{r}}^{-1}$ of the ABP, as would be the case for an optimum
choice of $D_{\text{t}}$, one estimates the current as

$I=D_{\text{r}}(P_{\leftarrow}-P_{\rightarrow}).$ (2)

The resulting probabilities are shown in Fig. 4 as functions of the
dimensionless horizontal width $\varepsilon$ of the symmetry axis of the
wedge-shaped passive region. One sees that $P_{\leftarrow}>P_{\rightarrow}$
for all values of $\varepsilon$, so that the model always predicts a leftward
current $I$ that is numerically roughly comparable to the optimum currents
obtained from the Brownian dynamics simulations. It naturally overestimates
the current for extreme values of $\varepsilon$, corresponding to concave and
vanishing passive volumes, respectively. The actual reorientation of the ABP
is then much less efficient than assumed by the stylized model, so that the
comparison further corroborates the primary role played by the optimized
destruction of the particle polarization in the passive region, for the
rectificaton efficiency of the ratchet.

Figure 4: Geometric toy model for the ratchet of Fig. 1; a: dependent on the
emission site and direction from the passive region, the ABP contributes a
current to the left (red), to the right (blue), or a “neutral” vertical
current; b: overall probabilities for the traversals depicted in a.

An iterative evaluation of the toy model provides further insight into the
role played by the active channels separating the passive image regions.
Starting from an initially uniform position distribution, one can find the
distribution of positions where an ABP ensemble leaving the active-passive
boundary will become trapped on the boundary again. The resulting position
distribution can be used as the initial condition for the next step, again
assuming uniformly distributed orientations, for simplicity. After many
iterations of this procedure, the position distribution no longer changes and
one can consider it as an approximate stationary position distribution of the
ABP. The resulting stationary distribution is similar to the one obtained from
the Brownian dynamics simulations, depicted in Fig. 1. It exhibits a maximum
in the indentation pocket of the passive region and, for
$\delta_{x}=\delta_{y}=0$, also at the reverse indentations connecting the
passive region with its periodic images. These particle accumulations hint at
the sensitivity of ratcheting to the precise geometry, as they would leak out
into the horizontal active channels to constitute the ratchet current, for
$\delta_{y}>0$.

Let us finally come back to the similarities and differences between our toy
model and gases in similar geometries. Dense gases or fluids, in which
frequent mutual particle collisions can be relied on for establishing local
equilibrium, should not exhibit ratcheting in spatially periodic setups, like
ours. But in so-called rarefied or Knudsen [29] gases, without an efficient
local equilibration mechanism, particles move ballistically in the space
between boundaries, similarly as ABPs in the active region of our geometric
toy model, so that the analysis of transport largely boils down to the problem
of boundary conditions. This is then a more subtle issue [30, 37] that would
deserve further study.

##  7 Conclusion

Spatially inhomogeneous activity profiles can be used to sort active Brownian
particles according to their orientations [28, 26, 27]. In one spatial
dimension, the requirements for the overall system’s polarization to vanish,
together with particle conservation, prevent ratcheting in time-constant
spatially-periodic activity landscapes. In two and more dimensions, such
active ratcheting is possible. We analyzed a proof-of-principle realization of
a wedge shaped two-dimensional autonomous force-free active Brownian ratchet.
It demonstrates that active ratcheting does not require a dynamic activity
profile, nor help from potential forces or walls.

Our study can be generalized in several ways. For example, it seems worthwhile
to find out whether the wedge-shaped ratchet design maximizes the current or
can be surpassed by more optimized geometries. Another potentially interesting
extension could be to ABPs with translational and/or orientational inertia
[38]. And, eventually, it would be nice if the ratcheting currents in rarefied
gases, hinted at by our toy model, could be experimentally demonstrated.

##  8 Acknowledgements

###### Acknowledgements.

We acknowledge financial support by the pre-doc award program at Leipzig
University, as well as by the Czech Science Foundation (project No.
20-02955J), and Charles University (project PRIMUS/22/SCI/009).

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