# A many-body Szilárd engine with giant number fluctuations

Omer Chor  Amir Sohachi  Raymond & Beverly Sackler School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel  Eran Rosen  Raymond
& Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 6997801,
Israel  Saar Rahav<EMAIL_ADDRESS>Schulich Faculty of Chemistry,
Technion–Israel Institute of Technology, Haifa 3200008, Israel  Yael Roichman
<EMAIL_ADDRESS>Raymond & Beverly Sackler School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel  Raymond & Beverly
Sackler School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

Szilárd’s engine, a model machine that converts information to work, is one of
the cornerstones of the thermodynamics of information. We report on an
experimental realization of a macroscopic many-particle Szilárd engine.
Crucially, the working substance of this engine consists of several active,
self propelling, particles that exhibit giant number fluctuations. We find
that the work per cycle that can be extracted from this engine is proportional
to number fluctuations in a system where pressure depends monotonically on
density. The differences between active and nonactive particles reveal
themselves through the dependence of the mean work per cycle on the number of
particles.

In recent years there has been a renewed interest in the deep connections
between information and thermodynamics [1]. This fundamental line of research
originated with Maxwell’s celebrated thought experiment [2] and was further
developed by Szilárd, Landauer, and Bennett [3, 4, 5]. Advances in the theory
of out-of-equilibrium systems led to the development of fluctuation theorems
and other new results for processes with measurement and feedback [6, 7, 8, 9,
10, 11, 12]. This revival led to several experimental realizations of
information engines [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. What
was a thought experiment for Maxwell is now a reality.

While many-body information engines were previously analyzed theoretically for
both classical [26, 27, 28] and quantum setups [29, 30, 31, 32], most existing
experimental realizations of information engines deal with single-particle
systems. Information engines are inherently out-of-equilibrium due to the
application of measurement outcome-dependent feedback. In most of the
literature on information engines, this is the sole mechanism that drives the
engine away from equilibrium. Only very recently, information engines
connected to a nonequilibrium environment [33], or an active working substance
[34], were considered. Here we report an experimental realization of an active
many-particle Szilárd engine. Our setup allows us to study how the presence of
active matter changes the behavior of an information engine, and show that
giant number fluctuations modify the information gained from measurements due
to the tendency of active matter to aggregate.

Collections of active particles serve as examples for far-from-equilibrium
systems, and a considerable amount of research was devoted to finding out
which thermodynamic concepts can be used to describe them [35, 36, 37, 38].
Among the remarkable properties of active particles are new emergent phenomena
[39, 40, 41, 42, 43], motility induced phase separation [44, 45, 43], and
flocking at large numbers [46, 47]. As a result, giant number fluctuations
were observed in many active systems [48, 49, 50, 51].

Figure 1: Experimental setup. a Shows the operation of Szilárd’s engine, b is
its generalization for a many-body system, and c is our experimental setup
with many active particles. Columns show the different steps in the operation
of the information machines: first, particles are allowed to move freely
within the full system. Then, a partition is placed in the center of the
arena. Finally, the partition is allowed to move until the system reaches
mechanical equilibrium. Finally, the partition is removed, and the system
returns to its original configuration.

Our experimental setup is a generalization of Szilárd’s engine (Fig. 1a) to
operate on many active particles instead of one passive particle. In a many-
body Szilárd engine [26] (Fig. 1b), $N$ ideal gas particles are placed in the
system, and a measurement is made to determine which side contains more
particles (for simplicity, an odd $N$ is assumed). Then, a partition is placed
in the center of the box and is allowed to move to its equilibrium position.
Mechanical equilibrium is reached when the mean densities are equal, namely

$V_{L,R}=\frac{N_{L,R}}{N}V_{tot}.$ (1)

Here $N_{L,R}$ and $V_{L,R}$ are the number of particles and volumes of the
left and right-hand side subsystems, respectively, with the constraints
$N_{L}+N_{R}=N$ and $V_{L}+V_{R}=V_{tot}$. As in Szilárd’s original single-
particle engine, work can be extracted during the motion of the wall from its
initial position at the center of the system to its equilibrium position (Eq.
1), but here this position is not at the edge of the system. We note that a
more general version of this setup was previously analyzed and optimized in
[28], where it was shown that, in some cases, more work could be extracted
when the wall is not placed in the center at the beginning of each cycle.

For an engine operating on an ideal gas, the maximum amount of work is
extracted by performing a quasistatic process connecting the initial state to
the equilibrium position of the partition. Larger imbalance in the initial
division of particles results in more extracted work. It can be argued that
this is the case for any system where the pressure is a monotonic function of
the density (see derivation in the Supplemental Material). This suggests that
operating information engines on active matter systems might result in
unusually large amounts of extracted work per cycle. Here, we test this
hypothesis by realizing a many-body Szilard engine consisting of self-
propelled particles which exhibit giant number fluctuations and compare its
operation with the ideal gas many-body Szilárd engine discussed above.

Our experimental setup is presented in Fig. 1c. Particles in our system are
self-propelled, elongated bristle robots (Hexbug, Nano). Each robot has $12$
flexible legs on which it jumps due to the rotation of an internal, battery-
powered motor [52]. The motion of a single bristle-robot can be described as
directed random motion with a persistence length of more than $1$ m [53]
(larger than the radius of our system, which is less than $0.2$ m).
Collections of bristle-robots exhibit complex behavior, including the
formation of giant clusters [52, 54, 55]. We place the bristle robots in a
half-circular arena with rounded edges to prevent particles from getting stuck
in corners. The arena is equipped with a detachable partition which can be
inserted at will and rotates freely on an axle. The experiments are recorded
by a camera (Logitech, Brio). Half of the videos were mirrored to account for
a slight chirality of particles making the system asymmetric around its
center. However, it was verified that this asymmetry does not affect forces
applied by particles from one side or the other beyond the range of
statistical error, and specifically that the mean force they apply on walls is
monotone in the density of particles, regardless of the side of the system in
which they are at (see Supplemental Material). Instead of mirroring, operating
the machine in a protocol where the wall is not placed in the center can also
be considered. However, to clarify, simplify calculations, and make the
comparison to an ideal gas more straightforward, we chose to mirror half of
the videos and operate the information engine according to the previously
described protocol.

In our experiments, we monitor the position of the partition and the number of
particles using image analysis [56]. We extract the number of particles in
each half of the system (left and right) by evaluating the relative area
fraction they occupy. Pressure is not a state function in a system of bristle
bots [57], so generally, it can depend on external properties like the shape
of the system. Nevertheless, these particles apply a force on the boundaries,
specifically on the rotating partition that we place in the system’s center.
Its motion depends on the momentary force balance between both sides, which is
random. In the following part, we refer to the system’s (two-dimensional)
pressure as the average force per unit length applied on the walls. It was
measured using a piezoelectric force sensor (PCB, 208C01) connected to an
oscilloscope (Agilent, DSO1024A).

In Fig. 2, we plot the time-averaged pressure as a function of density for
various system parameters ($N,V$). We find that pressure is linear in the
density for lower values and saturates for higher densities. We attribute this
saturation to a measurement error, which is discussed in depth in the SM. For
the lower density regime, we fit the pressure to an ideal-gas-like equation of
state $PV=\alpha N$, with a fitting parameter $\alpha=6.48\times 10^{-5}$ J
(Fig. 2, solid line), which is extrapolated to the higher density regime and
used to calculate the extractable work. A more detailed characterization of
the pressure is beyond the scope of this work since we are more interested in
the effect of number fluctuations on the distribution of work that can be
extracted from the system. To confirm that the exact dependence of the
pressure on the density does not significantly modify the extracted work, we
also fit pressure to a logarithmic dependence on density,
$P=a\log\left(b(N/V)+c\right)$ (Fig. 2, dashed line), which fits measured
pressure also in the erroneous high-density regime. Hereafter we show that the
dependence of work on the number of particles is similar for both pressure
functions.

Figure 2: Measured pressure (average force divided by wall length) as a
function of particle density (per unit area) in the system. Results show a
linear dependence of the pressure on the density for low densities until
reaching a plateau due to consequent collisions being too close apart so that
they cannot be distinguished. The linear fit (solid black line) was
extrapolated to the higher density regime. For comparison, we also fitted
pressure to a logarithmic dependence on density (dashed black line). Later on,
we show that both fits result in a similar dependence of work on the number of
particles, and thus conclude that the exact form of $P(N/V)$ does not
substantially modify our main results. A more detailed explanation regarding
how forces were measured and why there is a systematic error in measurements
for higher densities is specified in the supplementary material.

Next, we measured the distribution of $N_{R,L}$ for various values of $N$ and
extracted the standard deviations of these distributions $\Delta N_{R,L}$
(Fig. 3b). Note that while for a few particles (e.g., $N=6$, Fig. 3a), the
distribution is almost identical to the binomial distribution expected for an
ideal gas, as the number of particles increases, the distribution becomes
significantly wider than calculated for a system in equilibrium (e.g., $N=19$,
Fig. 3a). Quantitatively, for $N\geq 6$, where number fluctuations
significantly deviate from the expected behaviour for a system in equilibrium,
a power-law $\Delta N_{R}\sim\langle N_{R}\rangle^{1.34}$ was fitted, compared
to $\Delta N_{R}\sim\langle N_{R}\rangle^{0.5}$ for an ideal gas. Since our
experiments are limited to $N\leq 25$ particles, we cannot predict whether
this power-law persists for larger $N$ in our system. The growth in $\Delta
N_{L,R}$ indicates that giant number fluctuations are indeed present in the
system. The source for these number fluctuations is the tendency of the
bristle-bots to aggregate in corners and boundaries. This results in larger
fluctuations in the force applied to the partition, which strongly depends on
the location of the aggregates. Pictures of typical clusters in the system are
presented in Fig. 4.

Figure 3: a Histograms of the number of particles in the right half of the
system $N_{R}$ for $N=6,19$ particles (green line), compared to the expected
distribution for an ideal gas (blue line). b The standard deviation of the
distribution of the number of particles in the right half of the system
$\Delta N_{R}$ as a function of the mean amount of particles in it $\langle
N_{R}\rangle$, showing that the standard deviation in the experimental system
grows faster with the number of particles than it does for an ideal gas. For
an ideal gas, the standard deviation relates to the mean through a power-law
$\Delta N_{R}\sim\langle N_{R}\rangle^{0.5}$, whereas for the higher values of
$\langle N_{R}\rangle$ in the active gas, a higher power-law is observed.
Figure 4: Snapshots of clusters from typical experiments. a Shows a cluster
created at the corner of the system before the partition is inserted. Such
clusters contribute to the giant number fluctuations in the system. b Clusters
form near the partition, affecting the pressure that is applied to it.

Work can be extracted from the system by considering a quasistatic process in
which the partition is moved to the mechanical equilibrium position. This work
is given by

$W(k)=\int_{\frac{1}{2}V_{tot}}^{\frac{N/2+k}{N}V_{tot}}P_{R}dV_{R}+\int_{\frac{1}{2}V_{tot}}^{\frac{N/2-k}{N}V_{tot}}P_{L}dV_{L},$
(2)

where $k=N_{R}-N/2$ and $P_{L,R}$ are the pressure of the left/right-hand side
of the system. For an ideal gas, this gives

$W(k)=Nk_{B}T\left(\frac{1}{2}\ln\left(1-4\left(\frac{k}{N}\right)^{2}\right)+\frac{k}{N}\ln\frac{1+2\frac{k}{N}}{1-2\frac{k}{N}}\right),$
(3)

where $T$ is the temperature and $k_{B}$ is Boltzmann’s constant. The ensemble
average of the extracted work is defined as $\langle
W\rangle=\sum_{k=-N/2}^{N/2}W(k)\Pr(k)$, with $k$ distributed binomially
(shifted to $\langle k\rangle=0$),
$\Pr(k)=\left(\frac{1}{2}\right)^{N}\binom{N}{\frac{N}{2}+k}$, since particles
are non-interacting and their spatial distributions are uniform. We note that
for the engine operating on an ideal gas and $N\gg 1$, a useful approximation
for the mean work can be obtained by noting that $P(k)$ is appreciably large
when $|k/N|\ll 1$. As a result one can expand $W(k)$ in powers of $k/N$ [26],

$\displaystyle
W(k)=Nk_{B}T\left(2\left(\frac{k}{N}\right)^{2}+O\left(\left(\frac{k}{N}\right)^{4}\right)\right),$
(4) $\displaystyle\lim_{N\to\infty}\langle
W_{\mathrm{ideal}}\rangle=\frac{2k_{B}T}{N}\langle
k^{2}\rangle=\frac{1}{2}k_{B}T,$ (5)

and higher order terms vanish since they are of order $\frac{\langle
k^{4}\rangle}{N^{3}}=\frac{O(N^{2})}{N^{3}}\xrightarrow{N\to\infty}0$. We note
that Eq. 5 is consistent with a previous analysis of the same system [26], but
we repeat calculations to highlight how number fluctuations affect the result,
and because we use the same analysis for active particles. In Fig. 5 (blue
curve), we plot the transition of $\langle W_{\mathrm{ideal}}\rangle$ from
$k_{B}T\ln 2$ for $N=1$ to $k_{B}T/2$ for $N\gg 1$. The maximal efficiency of
the system is $\eta=\langle W_{\mathrm{ideal}}\rangle/k_{B}TI=1$, achieved for
one particle. Here $I=\ln 2$ is the Shannon information gained from the
measurement.

Figure 5: Mean work per cycle as a function of the number of particles for the
passive and active many-body Szilárd’s engines. The work is normalized by the
respective energy scales (ideal gas in a solid blue line, the experimental
active system with pressure linear in density in solid green line, the
experimental active system with pressure logarithmic in density in yellow
dashed line).

We calculate $\langle W_{\mathrm{linear}}\rangle$ for the active particle
system (where linear stands for linear dependence of pressure on density)
following the same analysis, using $\alpha$ instead of $k_{B}T$ for the
pressure and the experimentally measured distribution $P(k)$ (Fig. 3) instead
of the binomial distribution. Finally, we compute $\langle
W_{\mathrm{log}}\rangle$ for active particles with a logarithmic dependence of
pressure on density (see discussion above, Fig. 2 and the SM). Results for
different values of $N$ are shown in Fig. 5 (linear $P(N/V)$ in green points,
logarithmic $P(N/V)$ in yellow points).

In order to compare the active system to the ideal gas system, we normalize
each $\langle W\rangle$ by the relevant energy scale ($\alpha$, $aV_{tot}$ and
$k_{B}T$, for the linear active, logarithmic active, and ideal gas,
respectively). As the number of particles in the system grows in an ideal gas
system, the average work rapidly decreases to its value in the $N\gg 1$ limit.
The decrease in average work is due to the low probability of having large
number fluctuations in the system. A small imbalance between the number of
particles on each side of the system means that the wall can only be moved a
short distance from its initial position at the center to its equilibrium
position, so less work is extracted in this case. This trend is in contrast to
the trend in the active system, where the output work grows with the number of
particles in the system due to the increased number fluctuations with $N$
(Fig. 3). This result holds for both dependencies of the pressure on the
density that were considered, indicating that our result is robust and does
not depend on the exact details of pressure in the system. We expect that
similar results should hold for any system with giant number fluctuations (as
long as pressure is monotonically increasing in the density of particles). It
is noted that work in the active system with linear pressure is larger than
$\alpha\ln 2$, which does not contradict the second law of thermodynamics
since this is a nonequilibrium system where energy is constantly injected into
the system (via batteries) and dissipated to the environment.

In summary, we experimentally demonstrated that operating Szilárd’s engine on
active matter allows extracting more work per cycle than its passive
counterparts. This is attributed to giant number fluctuations which ultimately
result from the tendency of active particles to spend more time near
boundaries and corners. These differences between active and passive Szilárd
engines reveal themselves in the dependence of the work per cycle on the
number of particles, which increases with particle number in our active
engine, while it decreases for an ideal gas. We further note that our system
is small ($N\leq 25$ in all experiments), and does not show ordered motion for
long periods of time, as observed in some larger active systems [58]. In a
system where all particles align and move in the same direction, there would
also be more efficient ways of harvesting work from the motion of particles.

To the best of our knowledge, this is the first experimental realization of an
information machine operating on an active gas and the first implementation of
an information machine at the macroscopic level (previous experiments were
done at the single electron or colloidal particle level, whereas our system
contains $4.5$ cm long particles). In this regard, we note a recent work in
which a macroscopic Maxwell’s demon was proposed through a mechanism of
amplifying thermal fluctuations [59]. For a macroscopic information machine
such as ours, however, amplification is not required since fluctuations are
athermal and can therefore occur at any scale.

Previous studies of the use of active matter to extract work from a system
were based on the interaction between active matter and boundaries, for
example in active ratchets [60, 61, 62, 63]. Here, we use a different
attribute of active matter: their tendency to aggregate. We note that the
energy necessary to maintain an active system in a steady state was not taken
into account here and in these previous studies.

Finally, the efficiency of the active information engine is very low because
particles constantly consume energy from batteries, and only a fraction of
this energy is eventually used to push the wall. In our system, directly
connecting the batteries to an electric engine would yield a significantly
higher power output than the output one can extract from the movement of the
partition. However, understanding the various mechanisms by which energy can
be extracted from active matter is fundamentally interesting. It will also
assist in the design of microscopic engines operating in naturally occurring
active systems, i.e., where the activity is given ”for free”.

###### Acknowledgements.

We thank Issac Yekutiel and the Chemistry Machine shop of Tel Aviv University
for the design and construction of the experimental system. OC and YR
acknowledge support from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 101002392). AS and YR acknowledge support from the Israel
Science Foundation (Grant No. 385/21). SR is grateful for support from the
Israel Science Foundation (Grant No. 1929/21).

## References

  * Parrondo  _et al._ [2015] J. M. Parrondo, J. M. Horowitz, and T. Sagawa, Nat. Phys. 11, 131 (2015).
  * Maxwell [1871] J. C. Maxwell, UK: Longmans (1871).
  * Szilard [1929] L. Szilard, Zeitschrift für Physik 53, 840 (1929).
  * Landauer [1961] R. Landauer, IBM J. Res. Dev. 5, 183 (1961).
  * Bennett [1973] C. H. Bennett, IBM J. Res. Dev. 17, 525 (1973).
  * Sagawa and Ueda [2008] T. Sagawa and M. Ueda, Phys. Rev. Lett. 100, 080403 (2008).
  * Cao and Feito [2009] F. J. Cao and M. Feito, Phys. Rev. E 79, 041118 (2009).
  * Horowitz and Vaikuntanathan [2010] J. M. Horowitz and S. Vaikuntanathan, Phys. Rev. E 82, 061120 (2010).
  * Sagawa and Ueda [2010] T. Sagawa and M. Ueda, Phys. Rev. Lett. 104, 090602 (2010).
  * Ponmurugan [2010] M. Ponmurugan, Phys. Rev. E 82, 031129 (2010).
  * Fujitani and Suzuki [2010] Y. Fujitani and H. Suzuki, J. Phys. Soc. Jpn. 79, 104003 (2010).
  * Kundu [2012] A. Kundu, Phys. Rev. E 86, 021107 (2012).
  * Admon  _et al._ [2018] T. Admon, S. Rahav, and Y. Roichman, Phys. Rev. Lett. 121, 180601 (2018).
  * Jun  _et al._ [2014] Y. Jun, M. Gavrilov, and J. Bechhoefer, Phys. Rev. Lett. 113, 190601 (2014).
  * Bérut  _et al._ [2012] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Nature 483, 187 (2012).
  * Toyabe  _et al._ [2010] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nat. Phys. 6, 988 (2010).
  * Paneru  _et al._ [2018] G. Paneru, D. Y. Lee, T. Tlusty, and H. K. Pak, Phys. Rev. Lett. 120, 020601 (2018).
  * Saha  _et al._ [2021] T. K. Saha, J. N. Lucero, J. Ehrich, D. A. Sivak, and J. Bechhoefer, Proc. Nat. Acad. Sci. 118, e2023356118 (2021).
  * Paneru and Kyu Pak [2020] G. Paneru and H. Kyu Pak, Adv. Phys.: X 5, 1823880 (2020).
  * Ribezzi-Crivellari and Ritort [2019] M. Ribezzi-Crivellari and F. Ritort, Nat. Phys. 15, 660 (2019).
  * Koski  _et al._ [2014a] J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola, Phys. Rev. Lett. 113, 030601 (2014a).
  * Koski  _et al._ [2014b] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin, Proc. Nat. Acad. Sci. 111, 13786 (2014b).
  * Masuyama  _et al._ [2018] Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi, S. Kono, Y. Tabuchi, R. Yamazaki, M. Ueda, and Y. Nakamura, Nat. Commun. 9, 1 (2018).
  * Cottet  _et al._ [2017] N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq, Q. Ficheux, J. Anders, A. Auffèves, R. Azouit, P. Rouchon, and B. Huard, Proc. Nat. Acad. Sci. 114, 7561 (2017).
  * Vidrighin  _et al._ [2016] M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. Kim, V. Vedral, and I. A. Walmsley, Phys. Rev. Lett. 116, 050401 (2016).
  * Kim and Kim [2011] K. H. Kim and S. W. Kim, Phys. Rev. E 84, 012101 (2011).
  * Horowitz and Parrondo [2011] J. M. Horowitz and J. M. Parrondo, New J. Phys. 13, 123019 (2011).
  * Song  _et al._ [2021] J. Song, S. Still, R. D. H. Rojas, I. P. Castillo, and M. Marsili, Phys. Rev. E 103, 052121 (2021).
  * Kim  _et al._ [2011] S. W. Kim, T. Sagawa, S. De Liberato, and M. Ueda, Phys. Rev. Lett. 106, 070401 (2011).
  * Bengtsson  _et al._ [2018] J. Bengtsson, M. N. Tengstrand, A. Wacker, P. Samuelsson, M. Ueda, H. Linke, and S. Reimann, Phys. Rev. Lett. 120, 100601 (2018).
  * Cai  _et al._ [2012] C. Cai, H. Dong, C. Sun, _et al._ , Phys. Rev. E 85, 031114 (2012).
  * Jeon and Kim [2016] H. J. Jeon and S. W. Kim, New J. Phys. 18, 043002 (2016).
  * Saha  _et al._ [2022] T. K. Saha, J. Ehrich, M. Gavrilov, S. Still, D. A. Sivak, and J. Bechhoefer, arXiv preprint arXiv:2208.00288 (2022).
  * Malgaretti and Stark [2022] P. Malgaretti and H. Stark, arXiv preprint arXiv:2203.13075 (2022).
  * Ramaswamy [2010] S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  * Takatori and Brady [2015] S. C. Takatori and J. F. Brady, Phys. Rev. E 91, 032117 (2015).
  * Fodor  _et al._ [2016] É. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. Van Wijland, Phys. Rev. Lett. 117, 038103 (2016).
  * Fodor and Cates [2021] É. Fodor and M. E. Cates, Europhys. Lett. 134, 10003 (2021).
  * Wang  _et al._ [2015] W. Wang, W. Duan, S. Ahmed, A. Sen, and T. E. Mallouk, Acc. Chem. Res. 48, 1938 (2015).
  * Zöttl and Stark [2016] A. Zöttl and H. Stark, J. Phys.: Condens. Matter 28, 253001 (2016).
  * Bär  _et al._ [2020] M. Bär, R. Großmann, S. Heidenreich, and F. Peruani, Annu. Rev. Condens. Matter Phys. 11, 441 (2020).
  * Crosato  _et al._ [2019] E. Crosato, M. Prokopenko, and R. E. Spinney, Phys. Rev. E 100, 042613 (2019).
  * Gompper  _et al._ [2020] G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. Löwen, R. Golestanian, U. B. Kaupp, L. Alvarez, _et al._ , J. Phys.: Condens. Matter 32, 193001 (2020).
  * Cates and Tailleur [2015] M. E. Cates and J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  * Digregorio  _et al._ [2018] P. Digregorio, D. Levis, A. Suma, L. F. Cugliandolo, G. Gonnella, and I. Pagonabarraga, Phys. Rev. Lett. 121, 098003 (2018).
  * Marchetti  _et al._ [2013] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013).
  * Liebchen and Levis [2017] B. Liebchen and D. Levis, Phys. Rev. Lett. 119, 058002 (2017).
  * Deseigne  _et al._ [2010] J. Deseigne, O. Dauchot, and H. Chaté, Phys. Rev. Lett. 105, 098001 (2010).
  * Ginelli  _et al._ [2010] F. Ginelli, F. Peruani, M. Bär, and H. Chaté, Phys. Rev. Lett. 104, 184502 (2010).
  * Toner  _et al._ [2005] J. Toner, Y. Tu, and S. Ramaswamy, Ann. Phys. 318, 170 (2005).
  * Zhang  _et al._ [2010] H.-P. Zhang, A. Be’er, E.-L. Florin, and H. L. Swinney, Proc. Nat. Acad. Sci. 107, 13626 (2010).
  * Giomi  _et al._ [2013] L. Giomi, N. Hawley-Weld, and L. Mahadevan, Proc. R. Soc. London, Ser. A 469, 20120637 (2013).
  * Dauchot and Démery [2019] O. Dauchot and V. Démery, Phys. Rev. Lett. 122, 068002 (2019).
  * Deblais  _et al._ [2018] A. Deblais, T. Barois, T. Guerin, P.-H. Delville, R. Vaudaine, J. S. Lintuvuori, J.-F. Boudet, J.-C. Baret, and H. Kellay, Phys. Rev. Lett. 120, 188002 (2018).
  * Sánchez and Díaz-Leyva [2018] R. Sánchez and P. Díaz-Leyva, Phys. A 499, 11 (2018).
  * Bradski [2000] G. Bradski, Dr. Dobb’s Journal: Software Tools for the Professional Programmer 25, 120 (2000).
  * Solon  _et al._ [2015] A. P. Solon, Y. Fily, A. Baskaran, M. E. Cates, Y. Kafri, M. Kardar, and J. Tailleur, Nat. Phys. 11, 673 (2015).
  * Bricard  _et al._ [2013] A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Nature 503, 95 (2013).
  * Freitas and Esposito [2022] N. Freitas and M. Esposito, arXiv preprint arXiv:2204.09466 (2022).
  * Hiratsuka  _et al._ [2006] Y. Hiratsuka, M. Miyata, T. Tada, and T. Q. Uyeda, Proc. Nat. Acad. Sci. 103, 13618 (2006).
  * Angelani  _et al._ [2009] L. Angelani, R. Di Leonardo, and G. Ruocco, Phys. Rev. Lett. 102, 048104 (2009).
  * Di Leonardo  _et al._ [2010] R. Di Leonardo, L. Angelani, D. Dell’Arciprete, G. Ruocco, V. Iebba, S. Schippa, M. P. Conte, F. Mecarini, F. De Angelis, and E. Di Fabrizio, Proc. Nat. Acad. Sci. 107, 9541 (2010).
  * Angelani  _et al._ [2011] L. Angelani, A. Costanzo, and R. Di Leonardo, Europhys. Lett. 96, 68002 (2011).


4848.25 |  | 3.864 |  $-$1.14 | - | - | - | - | 35.9(4.36) | - | - |
5305.853 |  | 3.827 |  $-$2.357 | - | - | - | - | - | - | 36.1(5.40) |
5308.404 |  | 4.072 |  $-$1.81 | - | - | - | - | - | - | 37.8(5.16) |
5334.869 |  | 4.073 |  $-$1.562 | - | - | - | - | - | - | 43.5(5.04) |
4470.472 | Ni I | 3.399 |  $-$0.31 | 30.8(4.19) | - | - | - | - | - | - |
4686.207 |  | 3.597 |  $-$0.64 | - | - | - | - | - | 44.7(4.59) | 75.0(5.77) |
4703.803 |  | 3.658 |  $-$0.735 | - | - | - | - | - | - | - |
4714.408 |  | 3.38 | 0.23 | - | - | - | - | 30.5(3.59) | - | - |
4731.793 |  | 3.833 |  $-$0.85 | - | - | - | - | - | - | 61.0(5.92) |
4752.415 |  | 3.658 |  $-$0.7 | - | - | - | - | - | - | 76.4(5.91) |
4756.51 |  | 3.48 |  $-$0.34 | - | - | - | - | - | - | 90.8(5.69) |
4980.166 |  | 3.606 |  $-$0.11 | - | - | - | 28.10(4.00) | - | - | - |
5035.357 |  | 3.635 | 0.29 | 78.9(4.21) | - | - | - | - | - | - |
5081.107 |  | 3.847 | 0.3 | 52.8(4.02) | - | - | 38.7(4.23) | - | - | - |
5082.35 |  | 3.657 |  $-$0.54 | - | - | - | 32.2(4.06) | - | - | - |
5084.089 |  | 3.678 | 0.03 | 75.0(4.45) | - | 15.6(3.92) | - | 20.7(3.87) | - | - |
5099.927 |  | 3.678 |  $-$0.1 | - | - | - | - | - | - | 97.4(5.78) |
5102.96 |  | 1.676 |  $-$2.62 | - | 44.9(4.07) | - | - | - | 89.0(4.45) | - |
5115.389 |  | 3.834 |  $-$0.11 | - | - | 11.7(4.09) | - | - | - | 83.3(5.64) |
5146.48 |  | 3.706 | 0.12 | - | 55.4(4.15) | - | - | - | - | - |
6175.36 |  | 4.089 |  $-$0.53 | - | - | - | - | - | - | 52.0(5.64) |
6176.807 |  | 4.088 |  $-$0.26 | - | - | - | - | - | - | 70.5(6.01) |
6204.6 |  | 4.088 |  $-$1.13 | - | - | - | - | - | - | 31.3(5.82) |
6327.593 |  | 1.676 |  $-$3.15 | - | - | - | - | - | 66.8(4.34) | 75.7(5.84) |
6378.247 |  | 4.154 |  $-$0.89 | - | - | - | - | - | - | 37.8(5.79) |
6643.629 |  | 1.676 |  $-$2.3 | - | 102.2(4.14) | 14.6(3.89) | 61.3(4.31) | 34.8(3.98) | - | - |
4722.15 | Zn I | 4.029 |  $-$0.37 | - | 27.4(2.33) | - | - | 29.9(2.50) | - | - |
4810.53 |  | 4.08 |  $-$0.17 | 48.3(3.00) | - | - | - | - | 29.6(2.48) | 64.1(3.48) |
4883.684 | Y II | 1.084 | 0.07 | - | - | 73.5(0.67) | - | - | - | 169.0(2.97) |
5087.416 |  | 1.084 |  $-$0.17 | - | - | - | - | - | - | 164.9(3.12) |
5119.112 |  | 0.992 |  $-$1.36 | - | - | - | - | - | 89.2(1.55) | 104.3(3.11) |
5200.406 |  | 0.992 |  $-$1.36 | - | - | 35.5(0.55) | - | - | - | - |

The numbers in the parenthesis in columns 5 - 11 give the derived abundances
from the respective line.

Wavelength(Å) | El |  $E_{low}$(eV) | log gf |  BD$-$19 132 |  BD$-$19 290 |  HE 1157$-$0518 |  HE 1304$-$2111 |  HE 1354$-$2257 | BD+19 3109 | HD 202851 |
---|---|---|---|---|---|---|---|---|---|---|---
5205.724 |  | 1.033 |  $-$0.34 | - | - | 41.8(0.48) | - | - | - | - |
5289.815 |  | 1.033 |  $-$1.85 | - | - | - | - | - | 44.4(1.23) | 78.2(2.99) |
5402.774 |  | 1.839 |  $-$0.51 | - | 25.9(0.46) | - | - | - | - | - |
5544.611 |  | 1.738 |  $-$1.09 | 101.0(2.64) | - | - | 14.1(0.83) | - | - | - |
5546.009 |  | 1.748 |  $-$1.11 | - | - | - | - | - | - | - |
5662.925 |  | 1.944 | 0.16 | 156.8(2.64) | 52.7(0.31) | 22.7(0.61) | - | - | - | - |
6613.733 |  | 1.748 |  $-$1.1 | 123.5(2.85) | - | - | 12.1(0.67) | - | 54.5(1.47) | - |
4739.48 | Zr I | 0.651 | 0.23 | - | 8.3(0.95) | - | 30.9(1.80) | - | - | - |
4772.323 |  | 0.623 | 0.04 | - | 8.3(1.10) | - | - | - | - | - |
6134.585 |  | 0 |  $-$1.28 | - | - | - | 19.2(1.75) | - | - | - |
4317.321 | Zr II | 0.713 |  $-$1.38 | 98.3(2.24) | - | - | - | - | 53.7(1.28) | - |
5112.297 |  | 1.665 |  $-$0.59 | 88.0(2.34) | - | - | - | - | - | - |
4336.244 | Ce II | 0.704 |  $-$0.564 | - | 47.3(0.64) | - | - | - | - | - |
4364.653 |  | 0.495 |  $-$0.201 | - | - | - | - | - | 94.5(1.08) | - |
4407.273 |  | 0.701 |  $-$0.741 | - | - | - | 21.6(0.67) | - | - | - |
4483.893 |  | 0.864 | 0.01 | - | - | 38.5(0.99) | - | - | - | - |
4497.846 |  | 0.958 |  $-$0.349 | - | - | - | - | 48.2(1.11) | - | 80.5(2.90) |
4508.079 |  | 0.621 |  $-$1.238 | - | - | - | - | - | 36.1(1.04) | 57.4(2.58) |
4560.28 |  | 0.91 | 0 | - | - | 34.4(0.96) | - | - | - | - |
4562.359 |  | 0.477 | 0.081 | - | - | 62.9(0.87) | - | 97.8(1.20) | - | 103.5(2.30) |
4628.161 |  | 0.516 | 0.008 | 174.3(2.31) | - | - | - | - | 105.1(0.95) | 115.7(2.69) |
4725.069 |  | 0.521 |  $-$1.204 | - | 32.1(0.65) | - | - | - | 40.6(0.87) | - |
4747.167 |  | 0.32 |  $-$1.246 | - | - | - | - | - | 53.7(0.86) | - |
4773.941 |  | 0.924 |  $-$0.498 | - | - | - | - | - | 42.3(0.77) | - |
4873.999 |  | 1.107 |  $-$0.892 | - | - | - | 12.5(0.84) | - | - | 53.6(2.67) |
5187.458 |  | 1.211 |  $-$0.104 | - | 60.9(0.78) | - | - | 58.5(1.25) | - | 82.9(2.68) |
5274.229 |  | 1.044 |  $-$0.323 | - | 89.2(0.46) | 62.2(1.19) | - | - | 89.2(0.84) | - |
5330.556 |  | 0.869 |  $-$0.76 | 103.2(2.06) | - | - | - | 39.8(1.12) | - | 70.9(2.60) |
6034.205 |  | 1.458 |  $-$1.019 | 53.6(2.23) | - | - | - | - | - | 35.2(2.74) |
5188.217 | Pr II | 0.922 |  $-$1.145 | 53.4(1.52) | - | - | - | - | - | 36.0(2.17) |
5219.045 |  | 0.795 |  $-$0.24 | 112.1(1.47) | - | 11.9(0.31) | 21.3(-0.06) | 24.0(0.06) | 60.3(0.37) | - |
5259.728 |  | 0.633 |  $-$0.682 | 93.4(1.30) | - | - | - | - | - | 72.7(2.15) |
5292.619 |  | 0.648 |  $-$0.3 | - | 53.1(-0.06) | 10.8(0.15) | - | 33.3(0.13) | 54.5(0.10) | 77.3(1.90) |
5322.772 |  | 0.482 |  $-$0.315 | - | 55.9(-0.23) | 12.4(0.04) | - | 49.5(0.23) | 79.3(0.28) | 86.9(1.94) |
6165.891 |  | 0.923 |  $-$0.205 | - | - | 8.70(0.21) | - | - | 50.3(0.18) | - |
4446.384 | Nd II | 0.204 |  $-$0.59 | - | - | 49.4(0.84) | - | - | 93.6(0.75) | - |
4451.563 |  | 0.38 |  $-$0.04 | - | - | - | - | - | 112.6(0.84) | 119.8(2.52) |
4556.133 |  | 0.064 |  $-$1.61 | - | - | - | - | - | 58.1(0.76) | - |
4556.133 |  | 0.064 |  $-$1.61 | 123.1(2.14) | - | 17.0(0.99) | - | - | - | - |
4703.572 |  | 0.38 |  $-$1.07 | - | 58.8(0.50) | - | - | - | - | - |
4706.543 |  | 0 |  $-$0.88 | - | - | - | - | - | - | - |
4797.153 |  | 0.559 |  $-$0.95 | - | - | - | 29.1(0.67) | 41.7(0.82) | - | - |
4811.342 |  | 0.064 |  $-$1.14 | 148.9(2.05) | 90.7(0.50) | - | - | 79.9(1.15) | 85.3(0.68) | 98.4(2.59) |
4825.478 |  | 0.182 |  $-$0.86 | - | - | 43.0(0.90) | - | - | - | - |
4859.039 |  | 0.32 |  $-$0.83 | - | - | 41.1(0.99) | - | 63.6(0.80) | - | 87.5(2.29) |
4947.02 |  | 0.559 |  $-$1.25 | 109.0(2.08) | 28.9(0.40) | - | - | 29.5(0.85) | 56.0(0.96) | 62.3(2.35) |
4961.387 |  | 0.631 |  $-$0.71 | - | 58.7(0.40) | 39.0(1.18) | - | 63.0(1.04) | - | - |
5130.59 |  | 1.3 | 0.1 | - | - | 37.0(1.09) | - | - | - | - |
5212.361 |  | 0.204 |  $-$0.87 | - | 123.9(0.70) | 60.1(1.17) | - | 87.5(1.13) | 103.3(0.80) | 99.6(2.41) |
5255.506 |  | 0.204 |  $-$0.82 | - | - | 41.9(0.82) | - | 83.5(0.98) | 105.9(0.78) | - |
5276.869 |  | 0.859 |  $-$0.44 | - | - | 22.4(0.82 | - | - | 65.8(0.67) | - |
5287.133 |  | 0.744 |  $-$1.3 | 101.5(2.17) | - | - | - | - | 42.4(0.96) | 49.5(2.31) |
5293.163 |  | 0.822 |  $-$0.06 | - | - | 46.9(0.86) | - | - | - | - |
5311.453 |  | 0.986 |  $-$0.42 | - | - | - | 28.2(0.71) | 49.8(1.04) | 58.1(0.84) | 66.8(2.23) |
5319.815 |  | 0.55 |  $-$0.21 | - | - | 63.0(0.94) | - | - | - | - |
5356.967 |  | 1.264 |  $-$0.25 | 126.0(2.34) | - | - | - | 30.90(2.51) | - | 72.2(2.38) |
5361.51 |  | 0.68 |  $-$0.4 | - | 106.9(0.65) | 51.3(1.10) | - | - | 94.2(0.85) | - |
5371.927 |  | 1.412 | 0.003 | - | - | - | - | - | - | - |
5442.264 |  | 0.68 |  $-$0.91 | 143.8(2.39) | - | 28.1(1.19) | - | - | 71.0(0.92) | - |
5718.118 |  | 1.41 |  $-$0.34 | - | - | - | - | - | 37.4(0.78) | - |
5825.857 |  | 1.08 |  $-$0.76 | - | - | - | - | - | 49.4(0.93) | 59.3(2.33) |
4424.337 | Sm II | 0.485 |  $-$0.26 | - | - | - | 34.7(0.08) | - | - | - |
4434.318 |  | 0.378 |  $-$0.576 | 108.3(1.14) | 52.8(-0.21) | - | - | - | 80.1(0.50) | - |
4458.509 |  | 0.104 |  $-$1.11 | - | 44.3(-0.19) | - | 28.0(0.05) | 56.1(0.51) | 67.7(0.34) | 71.8(1.74) |
4499.475 |  | 0.248 |  $-$1.413 | 81.6(1.18) | - | 16.6(0.73) | - | 28.3(0.42) | 51.2(0.52) | 59.4(1.89) |
4519.63 |  | 0.543 |  $-$0.751 | - | - | - | - | - | 59.6(0.46) | - |
4615.444 |  | 0.544 |  $-$1.262 | - | - | - | - | - | 25.0(0.21) | 53.9(1.94) |
4674.593 |  | 0.184 |  $-$1.055 | - | - | - | - | - | 66.3(0.28) | - |
4676.902 |  | 0.04 |  $-$1.407 | 94.2(1.04) | - | 16.3(0.46) | - | - | - | - |
4704.4 |  | 0 |  $-$1.562 | 99.7(1.23) | - | - | - | - | - | - |
4726.026 |  | 0.333 |  $-$1.849 | - | - | - | - | - | 22.6(0.40) | - |
4791.58 |  | 0.104 |  $-$1.846 | 84.7(1.35) | - | - | - | 25.6(0.57) | - | 42.5(1.73) |
4815.805 |  | 0.185 |  $-$1.501 | - | 23.2(-0.17) | - | - | - | 45.9(0.30) | - |
4844.209 |  | 0.277 |  $-$1.558 | - | - | - | - | - | - | 53.2(1.87) |
4854.368 |  | 0.379 |  $-$1.873 | 44.4(1.07) | - | - | - | - | - | 29.1(1.79) |

The numbers in the parenthesis in columns 5 - 11 give the derived abundances
from the respective line. The line information are taken from Kurucz database
of atomic line lists
(https://lweb.cfa.harvard.edu/amp/ampdata/kurucz23/sekur.html).