# First results of Low-latitude Ionospheric Irregularities measured by NavIC
and GPS near the Anomaly Crest and the Magnetic Equator

###### Abstract

Ionospheric irregularity studies are important aspects for understanding
ionospheric physics and related processes, especially near the low-latitude
regions. However, simultaneous measurements (utilizing the L-band signals of
NavIC and GPS) of irregularity scale sizes over the Indian longitude sector,
has not been addressed extensively. To address this problem, the paper
presents simultaneous characterization of low-latitude ionospheric
irregularities over a location near the northern EIA crest (Indore: 22.52∘N,
75.92∘E geographic and magnetic dip of 32.23∘N) and a location (Hyderabad:
17.41∘N, 78.55∘E geographic and magnetic dip of 21.69∘N) between the crest and
the magnetic equator, utilizing the Indian navigation system, NavIC and GPS L5
signal C/No variations to determine the range of the ionospheric irregularity
scale sizes using Power Spectral Density (PSD) analysis. The study period
spans from September 2017- September 2019, covering both disturbed and quiet-
time conditions in the declining phase of solar cycle 24. Observations show
that the irregularity scale size ranges from about 500 m to 6 km. Furthermore,
no correlation was observed between the data derived scintillation
characteristics and the geomagnetic activity. This study for the first time,
shows the nature of the temporal PSD for ionospheric scintillation during
varying solar and geophysical conditions, by measuring the irregularity scale
sizes utilizing simultaneous observations from NavIC and GPS from locations
near the northern crest of the EIA and in between crest and the magnetic
equator, ensuring proper characterization of ionosphere over the geosensitive
Indian subcontinent.

be submitted soon

Department of Astronomy, Astrophysics and Space Engineering, Indian Institute
of Technology Indore, Simrol, Indore 453552, Madhya Pradesh, India

Space and Atmospheric Sciences Division, Physical Research Laboratory,
Ahmedabad, 380009 Gujarat, India

Sumanjit<EMAIL_ADDRESS>
Low-latitude ionospheric irregularity studies are performed using NavIC and
GPS signals for the first time.

Measurements of disturbed and quiet-time low-latitude ionospheric irregularity
scale sizes using actual transionospheric signals are presented.

No correlation found between the scintillation characteristics derived from
the data and the geomagnetic activity.

##  1 Introduction

The phenomenon of scintillation can be thought analogue with the stars that
twinkle in the night sky due as a result of variations in density of the
atmosphere due to turbulence. The phenomenon of plasma instability (post-
sunset) in the equatorial ionosphere generates irregularities and large scale
depletions in the electron density referred to as the Equatorial Plasma
Bubbles (EPBs) (sc:0 and references therein). Radio waves propagating through
these irregularities experience diffraction and scattering, which cause random
fluctuations in the VHF and L-band signal amplitude, phase, direction of
propagation, and polarization referred to as scintillations [Carrano . (2011),
Carrano, Groves Caton (2012), Priyadarshi (2015)]. It is well known that the
irregularities which produce scintillations are mainly found in the F layer of
the ionosphere ($\sim$ 300-350 km), where the plasma density has the maximum
value. Rayleigh–Taylor (R-T) instabilities are considered as the primary
mechanism generating EPBs with scale sizes from a few hundreds of meters to
several kilometers [Priyadarshi (2015)]. Activities of scintillations are
generally observed during the period of maximum solar activity, occurring near
the magnetic equator in the post-sunset-to-midnight sector [Basu, MacKenzie
Basu (1988), Basu, Basu . (1988)]. It is well known that ionospheric
scintillations are demonstrations of space weather effects, affecting the
performance of space-based navigation and communication systems which rely on
transionospheric radio-wave propagation [Carrano, Groves Caton (2012), de
Oliveira Moraes . (2017)].

For characterizing radio wave scintillation that travels in the ionosphere,
numerous phase screen theories have been developed, as early as the 1950s
following works done by [Booker . (1950), Hewish Bragg (1951)]. Generally,
scintillation of radio waves is calculated using the theory of wave
propagation in random media to obtain the parameters of the exiting wave from
the ionosphere. It is performed by solving the Fresnel diffraction theory
problem involving the propagation of these waves traveling between the
ionosphere and the Earth. Thus, these calculations replace the ionosphere by
an equivalent phase screen that is random, where the irregularities in the
ionosphere are considered as phase objects [Taylor Infosino (1976)].
Simultaneously, the ground pattern produced by radio waves propagating through
this screen is derived from the diffraction theory [Bhattacharyya (1999)].

In general, the one-and two-dimensional phase screens are analytic
representations of the intensity of the spectral density function in terms of
the various phase structure function combinations [Carrano Rino (2016)]. Phase
screen models have been extensively used to simulate the distortion of
wideband waveforms for communications [Knepp Nickisch (2009)], fading of GNSS
satellite signals, scintillation of satellite signals used in radio
occultation experiments, to name a few. These phase screen models’ inputs are
the in-situ electron density measurements, a radio receiver’s time series
phase measurements, and a turbulent ionospheric medium-based stochastic model.
Several researchers (sc:3,sc:4,sc:5,sc:29 and references therein) have
addressed in-situ electron density fluctuations measured by the satellites to
predict the propagation effects. However, the disadvantage lies in the fact
that measured values sample the electron density irregularities and
fluctuations at the satellite’s orbital altitude. The transionospheric
propagation effects can be attributed to the integrated development of the
density variation along the signal ray path at all altitudes. If any, the
relation between density fluctuations sampled at one altitude to another is
still not evident in the literature. Therefore, if the density irregularities
associated with an EPB have not risen to the satellite’s altitude that
measures the in-situ density values, detection of fluctuations will be absent
despite the EPB causing radio scintillations as a result of the irregularities
present at lower altitudes. There is an advantage of probing the ionosphere at
all altitudes, from the ground-based observations of signal phase transmitted
by a satellite at the topside ionosphere or beyond, over in-situ electron
density observations. Thus, it becomes essential to understand and quantify
the contribution from phase scintillations caused by diffraction, in addition
to interpreting the signal phase time series and the corresponding limitations
of judiciously representing the structure of irregularity at these altitudes
[Secan . (1995), Carrano, Groves Caton (2012), Bhattacharyya . (2000), Knepp
(2004), Caton . (2009), Costa . (2011)].

The presence of the northern crest of the Equatorial Ionization Anomaly (EIA)
and the geomagnetic equator that touches the southern tip of the Indian
peninsular region, accompanied by sharp latitudinal gradients of ionization,
makes the Indian longitude sector a highly geosensitive region of
investigation for ionospheric research during geomagnetically disturbed
periods when the low-latitude ionization is significantly affected as a result
of solar eruptions like the Coronal Mass Ejections (CMEs), the Coronal
Hole/High-Speed Solar Wind streams (CH/HSSW) and associated Corotating
Interaction Regions (CIRs) [Chakraborty . (2020a), Chakraborty . (2020b),
Chakraborty . (2020c)]. Previous studies show the development of power law
phase screen theories and models for scintillation based on weak and strong
conditions. The nature of phase screen, associated with both weak and strong
scattering, in the geosensitive region of India, during geomagnetically
disturbed and quiet conditions, has not been extensively studied in the
literature. This study for the first time, to the best of our knowledge, shows
the nature of the temporal Power Spectral Density (PSD) for ionospheric
scintillation associated with weak as well as strong scatter conditions,
during varying solar and geophysical conditions, by measuring the irregularity
scale sizes utilizing the Navigation with Indian Constellation (NavIC) as well
as the Global Positioning System (GPS) observations from locations near the
northern crest of the EIA and in between crest and the magnetic equator that
ensure proper simultaneous ionospheric irregularity characterization over the
geosensitive Indian longitude sector.

##  2 Data Collection and Analysis Procedure

In this work, we started by investigating the ionospheric activity above
Indore within the period September 01, 2017 through September 30, 2019, which
coincides with the declining phase of solar cycle 24. Data has been collected
using a multi-constellation and multi-frequency GNSS receiver (Septentrio
PolaRxS Pro) capable of receiving GPS L1 (1575.42 MHz), L2 (1227.60 MHz) and
L5 (1176.45 MHz) signals and specialized for ionospheric monitoring and space
weather research. Here, the amplitude and phase samples get generated at 50 Hz
in order to calculate the scintillation indices, C/No (dB-Hz), satellite lock
time, azimuth (deg), elevation (deg), and the Total Electron Content (TEC). We
followed the similar procedure as done by [Ayyagari . (2020)] We have also
taken data using a NavIC (ACCORD) receiver, capable of receiving L5 and S1
(2492.028 MHz) signals along with GPS L1 signal. The output of the receiver
includes the C/No, azimuth, elevation, pseudorange (in m), carrier cycles
(cycles). The receiver is provided by the Space Applications Centre, ISRO.
Both these receivers are operational in the Department of Astronomy,
Astrophysics and Space Engineering of Indian Institute of Technology Indore
(22.52∘N, 75.92∘E geographic; magnetic dip 32.23∘N). Additionally, data from
another NavIC receiver operational at the department of Electronics and
Communication Engineering, Osmania University, Hyderabad (17.41∘N, 78.55∘E
geographic; magnetic dip 21.69∘N), has been used for the analysis. From the
entire analysis of the period of 25 months, we have only found scintillation
events on 27 days as observed from NavIC Indore and validated by GPS Indore
and NavIC Hyderabad. In the current work, we have used the 27 scintillation
events as our parent sample. Among those 27 events, we have chosen three
representative events (with most coverage from all the GPS and NavIC
satellites). In the rest of the paper, we will show results and further
analysis of these three events which are suitably chosen across different
solar and geophysical conditions.

##  3 Results and Discussions

In this section, the strong storm day of September 8, 2017, the weak storm day
of August 17, 2018 and the quiet day of September 16, 2017 have been presented
as sample cases to study the C/No variation as observed by the NavIC
satellites; thereafter analysing the corresponding PSD and the respective
irregularity scale sizes with the spectral slopes, under varying geophysical
conditions over Indore: a location near the northern crest of the EIA of the
Indian longitude sector. It is to be noted that as the NavIC satellites are
geostationary/geosynchronous, the C/No data can be continuously observed
throughout the day. Similar approaches are taken to show the respective
variations as observed by the NavIC data over Hyderabad: a location away from
the northern crest, for the September 8, 2017 storm day. Additionally for
comparison of NavIC variation to that of the GPS, same methodology is adopted
for the GPS observations on these days over Indore.

###  3.1 September 8, 2017: a case of strong storm

Due to a CME that arrived on September 6, 2017, a G4 (Kp= 8, severe) level
geomagnetic storm was observed at 23:50 UT on September 7, 2017, at 01:51 UT
and 13:04 UT on September 8, 2017, according to the National Oceanic and
Atmospheric Administration (NOAA) (https://www.swpc.noaa.gov/). The
Disturbance storm time (Dst) index (available at
http://wdc.kugi.kyoto-u.ac.jp/dstdir/), which measures the strength of the
disturbance caused to the horizontal component of the geomagnetic field due to
incoming solar eruptions, showed dips during September 8, 2017, with values
going below -100 nT (-124 nT at 02:00 UT and -109 nT at 18:00 UT). This
signified that the nature of the geomagnetic disturbance had been strong in
severity and double-peaked.

Figure 1 shows the C/No variation (dB-Hz) over Indore located near the
northern crest of the EIA that passes over the highly geosensitive Indian
longitude sector, for the entire day of September 8, 2017, as observed by a
set of geostationary (PRNs 3,6 and 7) and geosynchronous (PRNs 2,4 and 5)
satellites of NavIC. The observations are taken by utilizing the L5 (1176.45
MHz) signal of NavIC. Upon closely observing panels showing PRNs 5 and 6 of
Figure 1, it is clearly visible that there had been strong fluctuations in the
C/No with values dropping from $\sim$52 dB-Hz to $\sim$35 dB-Hz around 17-18
UT (22:30-23:30 LT), between local post-sunset and pre-midnight sector.

Figure 1: The C/No (dB-Hz) variation during the disturbed day of September 8,
2017, as observed by L5 signal of NavIC satellite PRNs 2-7 over Indore. It is
to be noted here that LT=UT+05:30h.

To verify whether the observed C/No drops in Figure 1 are due to the
scintillations that had occurred during this period, top panel of Figure 2
shows the hourly binned variance plots of all the PRNs of NavIC during the
entire day of September 8, 2017, while the bottom panel of the same figure
shows the corresponding standard deviation of the normalized ratio
$\frac{(C/N_{o})}{<C/N_{o}>}$ ($<>$ indicating average values). This parameter
is similar to the scintillation index $S_{4}$. In the hourly bin of 18-19 UT
for PRN 5 and 17-18 UT for PRN 6, it is clearly visible that the variance over
the average values were significant. It is to be noted that the high value
observed in the 14-15 UT bin for PRN5 has not been considered as it is getting
manifested as a result of the C/No drop to zero observed by this PRN in Figure
1.

Figure 2: The hourly binned variance plots (top panel) of C/No and the
corresponding standard deviation plots (bottom panel) of
$\frac{(C/N_{o})}{<C/N_{o}>}$ for all PRNs of NavIC on September 8, 2017 as
observed from Indore.

The temporal power spectrum has been introduced by several researchers
[Strangeways (2009), Rino (19791), Rino (19792), Fougere (1985)] as follows:

$S=\frac{T}{(f_{0}^{2}+f^{2})^{\frac{p}{2}}}$ (1)

where f0 is the outer scale frequency, T is the spectral strength and p is the
spectral slope. Equation 1 can be simplified to the following (when
$f>>f_{0}$):

$S=Tf^{-p}$ (2)

Furthermore, the spectrum of electron density fluctuations ($\delta N$) can be
modeled as a power law, with an outer scale, given by [Rino (19791), Rino
(19792), Carrano, Valladares Groves (2012)]:

$S_{\delta N}(q)=\frac{C_{s}}{(q_{0}^{2}+q^{2})^{(m+\frac{1}{2})}}$ (3)

where, $C_{s}$ is the strength of the turbulence proportional to T [Rino
(19791), Rino (19792), Carrano, Valladares Groves (2012)], m is the
irregularity spectral index (p=2m-1) and $q_{0}$ is the wave number of the
outer scale and is related to the outer scale turbulence (L) as:

$L=\frac{2\pi}{q_{0}}$ (4)

Therefore, for $q>>q_{0}$, the spectrum in equation 3 modifies to:

$S_{\delta N}(q)=C_{s}q^{-(2m+1)}=C_{s}q^{-(p+2)}$ (5)

Figure 3 shows the PSD variation from PRN5 and PRN6. The absolute values of p
for the PRNs 5 and 6 are 3.690$\pm$0.009 and 3.596$\pm$0.008 respectively.
These values are obtained from the co-variance matrix of the least squares fit
performed to the PSD in the logarithmic domain by using equation 2. These
values are then verified with the approximate estimation [Rino (19791), Rino
(19792)] of $p=\frac{[P(0.05)-P(0.5)]}{10}$ from figure 3. The values of the
corresponding irregularity scale sizes (using equations 3 through 5) are found
out to be ranging from about 518 m to 6.026 km.

Figure 3: The PSD variations with least square fit (black solid line)
corresponding to the bins of intense C/No variation of Figure 2.

The GPS observations of C/No variance (top panel), standard deviation of
$\frac{(C/N_{o})}{<C/N_{o}>}$ (bottom panel) and the corresponding PSD are
shown in Figures 4 and 5 respectively. In Figure 4, it can be clearly observed
that the C/No variance was the most significant during the time bin of 16-17
UT.

Figure 4: The hourly binned variance plots (top panel) of C/No and the
corresponding standard deviation plots (bottom panel) of
$\frac{(C/N_{o})}{<C/N_{o}>}$ for the GPS satellites on September 8, 2017 over
Indore.

Subsequently, the corresponding PSD of that time bin has been plotted in
Figure 5. The value of p is 3.318$\pm$0.011 and the irregularity scale sizes
are ranging from about 503 m to 6.007 km as observed by the GPS satellites on
that day. It is to be noted that these observations are comparable to that
observed by NavIC and hence the suitability of NavIC to detect and measure the
ionospheric irregularities is quite evident.

Figure 5: The PSD variations with least square fit (black solid line)
corresponding to the time bins of intense C/No variation of Figure 4.

Taking similar approach for the storm day of September 8, 2017, the following
Figure 6 shows the NavIC measured C/No variation (dB-Hz) over Hyderabad
located away from the northern crest of the EIA (or closer to the magnetic
equator) of the Indian region. Upon closely observing panels showing PRNs 5
and 6, it is clearly visible that there had been strong fluctuations in the
C/No with values dropping from $\sim$51 dB-Hz to 36 dB-Hz around 17-18 UT
(22:30-23:30 LT) for PRN 5 and from $\sim$52 dB-Hz to $\sim$37 dB-Hz around
16-17 UT (21:30-22:30 LT), between local post-sunset and pre-midnight sector.

Figure 6: The C/No (dB-Hz) variation during the disturbed day of September 08,
2017, as observed by L5 signal of NavIC satellite PRNs 2-7 over Hyderabad.

Similarly, for verification of the observed C/No drops due to scintillation,
Figure 7 shows the hourly binned variance plot (top panel) and the
corresponding standard deviation of $\frac{(C/N_{o})}{<C/N_{o}>}$ plot (bottom
panel) for all the PRNs of NavIC during the entire day of September 8, 2017.
In the hourly bin of 17-18 UT for PRN 5 and 16-17 UT of PRN 6, it is clearly
visible that the variance over the average values are significant.

Figure 7: The hourly binned variance plots (top panel) of C/No and the
corresponding standard deviation plots (bottom panel) of
$\frac{(C/N_{o})}{<C/N_{o}>}$ for all PRNs of NavIC on September 8, 2017 over
Hyderabad.

Figure 8 shows shows the PSD variation from the PRNs 5 and 6. The values of p
for PRNs 5 and 6 are 3.072$\pm$0.011 and 3.008$\pm$0.010 and the irregularity
scale sizes are ranging from about 506 m to 6.039 km.

Figure 8: The PSD variations with least square fit (black solid line)
corresponding to the time bins of intense C/No variation of Figure 7.

###  3.2 September 16, 2017: a case of quiet condition

Following similar approach as presented for the previous case, the following
Figure 9 shows the C/No variation (dB-Hz) for the entire day of September 16,
2017, as observed by the L5 signal of NavIC. Drops in the C/No has been
observed at multiple time stamps throughout the day by all the PRNs.

Figure 9: The C/No (dB-Hz) variation during the entire quiet day of September
16, 2017 over Indore, as observed by L5 signal of NavIC satellite PRNs 2-7.

However, to confirm whether the C/No drops had been significant, Figure 10 has
been plotted that shows the hourly binned variance plots (top panel) and the
corresponding standard deviation plots (bottom panel) of
$\frac{(C/N_{o})}{<C/N_{o}>}$ of all the PRNs of NavIC during the entire day
of September 16, 2017. The hourly bin of 6-7 UT for all the PRNs show the most
significant rise and hence maximum variation among all the bins of the day.

Figure 10: The hourly binned variance plots (top panel) of C/No and the
corresponding standard deviation plots (bottom panel) of
$\frac{(C/N_{o})}{<C/N_{o}>}$ for all PRNs of NavIC on September 16, 2017 over
Indore.

Figure 11 shows the PSD variation from all the satellites of NavIC. It is to
be noted that only PRNs 3, 4, and 5 show a power law variation in the PSD and
hence are taken in consideration. The values of p for PRNs 3, 4 and 5 are
2.873$\pm$0.012, 2.949$\pm$0.009 and 2.829$\pm$0.008 respectively. The values
of the corresponding irregularity scale sizes are ranging from about 500 m to
6.057 km.

Figure 11: The PSD variations with least square fit (black solid line)
corresponding to the time bins of C/No variation of Figure 10.

###  3.3 August 17, 2018: a case of weak storm

Due to the passage of a HSSW stream around the Earth, from a CH on the Sun, a
G1 (Kp= 5, minor) level geomagnetic storm was observed on August 17, 2018. The
Dst index showed dip on this day with a value of -37 nT at 05:00 UT,
thereafter signifying this storm to be of a weak nature.

Very similar to the previous two cases, Figure 12 shows the C/No variation
(dB-Hz) for the entire day of August 17, 2018, as observed by the L5 signal of
NavIC. Drops in the C/No has been observed at multiple time stamps throughout
the day by all the PRNs.

Figure 12: The C/No (dB-Hz) variation during the entire day of August 17, 2018
over Indore, as observed by L5 signal of NavIC satellite PRNs 2-7.

However, for the verification of the significance of these C/No drops, Figure
13 shows the hourly binned variance plots (top panel) and the corresponding
standard deviation plots (bottom panel) of $\frac{(C/N_{o})}{<C/N_{o}>}$ of
all the PRNs of NavIC during the entire day of August 17, 2018. The time bin
of 21-22 UT for the all the PRNs show the most significant rise and hence
maximum variation among all the bins of this day.

Figure 13: The hourly binned variance plots (top panel) of C/No and the
corresponding standard deviation plots (bottom panel) of
$\frac{(C/N_{o})}{<C/N_{o}>}$ for all PRNs of NavIC on August 17, 2018 over
Indore.

Figure 14, in a way similar to the previous cases, shows the PSD variation
from all the NavIC satellites. It is to be noted that only PRNs 3, 4 and 6
show a power law variation in the PSD and hence are taken in consideration.
The values of p for PRNs 3, 4 and 6 are 2.306$\pm$0.009, 2.518$\pm$0.011 and
2.322$\pm$0.008 respectively. The values of the corresponding irregularity
scale sizes are ranging from about 497 m to 6.001 km, which is comparable to
the previous cases.

Figure 14: The PSD variations with least square fit (black solid line)
corresponding to the time bins of C/No variation of Figure 13.

Finally to understand the nature of the ionospheric variability and the
corresponding irregularity scale sizes from simultaneous observations over
locations near the northern crest of EIA and the magnetic equator, Tables 1
and 2 summarize the values of p and the range of the outer scale sizes (in km)
corresponding to the NavIC satellite PRNs and the GPS satellites that showed
power law variation from the temporal analysis, for the cases where there had
been C/No drops observed by the GPS L5 signals over Indore and NavIC L5
signals over both Indore and Hyderabad, under variable geophysical conditions
and during the period: September 2017- September 2019 falling in the declining
phase of solar cycle 24. The difference in the irregularity scale sizes
observed by these these PRNs can be attributed to the different ionospheric
pierce points corresponding these satellites.

Table 1: Values of p and the corresponding range of outer scale size of irregularities for the cases when GPS and NavIC L5 C/No drops were observed during varying geophysical conditions over Indore (a location near the northern crest of EIA). Day | Geophysical conditions | PRN | p | range of L (km)
---|---|---|---|---
Sep 08, 2017 | Strong storm condition | 5 | 3.690$\pm$0.009 | 0.518-6.026
|  | 6 | 3.596$\pm$0.008 | 0.518-6.014
|  | GPS | 3.318$\pm$0.011 | 0.503-6.007
Sep 16, 2017 | Quiet condition | 3 | 2.873$\pm$0.012 | 0.500-5.974
|  | 4 | 2.949$\pm$0.009 | 0.515-6.057
|  | 5 | 2.829$\pm$0.008 | 0.507-5.991
|  | GPS | 2.705$\pm$0.009 | 0.507-5.983
Aug 17, 2018 | Weak storm condition | 3 | 2.306$\pm$0.009 | 0.499-5.993
|  | 4 | 2.518$\pm$0.011 | 0.497-5.834
|  | 6 | 2.322$\pm$0.008 | 0.503-6.001
|  | GPS | 2.242$\pm$0.013 | 0.510-6.054
Table 2: Values of p and the corresponding range of irregularity scale size for the cases NavIC L5 C/No drops were observed along with power law during the varying geophysical conditions over Hyderabad (a location in between crest and the magnetic equator). Day | Geophysical conditions | PRN | p | range of L (km)
---|---|---|---|---
Sep 08, 2017 | Strong storm condition | 5 | 3.072$\pm$0.011 | 0.506-6.007
|  | 6 | 3.008$\pm$0.010 | 0.519-6.039
Sep 16, 2017 | Quiet condition | 3 | 2.910$\pm$0.018 | 0.496-5.948
|  | 4 | 2.816$\pm$0.007 | 0.511-6.025
Aug 17, 2018 | Weak storm condition | 5 | 2.509$\pm$0.009 | 0.509-5.992

##  4 Conclusions

The ionosphere over the Indian longitude sector is highly dynamic and
geosensitive due to the presence of the northern crest of EIA and the magnetic
equator. An extensive study of the ionospheric irregularities, that have
drastic impacts on the performance of the global navigational satellite
systems in and around these locations under varied solar and geophysical
conditions, is essential. The detailed study of these ionospheric
irregularities brings forward important aspects to understand the ionospheric
physics and related processes. In order to address this problem, this paper
presents the simultaneous characterization of the low-latitude ionospheric
irregularities over locations chosen such as to cover the zones of both the
near the northern crest of EIA (Indore: 22.52∘N, 75.92∘E geographic and
magnetic dip of 32.23∘N, with observations from NavIC and GPS L5 signals) and
in between crest and the magnetic equator (Hyderabad: 17.41∘N, 78.55∘E
geographic and magnetic dip of 21.69∘N, with observations from NavIC L5
signals) of the Indian subcontinent. This study utilizes the L5 signal C/No
variations to analyse the power spectrum and determine the spectral slope and
range of the ionospheric irregularity scale sizes.The study has been performed
during a span of September 2017- September 2019 (declining phase of solar
cycle 24) that consisted of weak, strong geomagnetic storms and quiet-time
condition. Observations show that the irregularity scale size ranges from
about 500 m to 6 km. These results are consistent with the previously reported
[Bhattacharyya (2019)] irregularity scale sizes associated to L-band
scintillations over regions surrounding the EIA. This study uses actual
transionospheric signals rather than signals based on in situ measurements
and/or phase screen model derived analysis. The observations also show no
correlation between the scintillation observed and the geomagnetic activity.
The observations are indeed consistent with the fact that scintillations are
associated with the formation and evolution of EPBs in and around the EIA
region. While the formation of plasma depletions can be either suppressed or
stimulated by geomagnetic storms, once formed the turbulence and associated
scintillation is determined by the local plasma properties. This study for the
first time, shows the nature of the temporal PSD for ionospheric scintillation
during varying solar and geophysical conditions, by measuring the intermediate
irregularity scale sizes that generally occur in the low-latitude regions,
utilizing simultaneous observations from NavIC and GPS over locations near the
northern crest of the EIA and in between crest and the magnetic equator which
ensures proper ionospheric irregularity characterization over the geosensitive
Indian longitude sector.

###### Acknowledgements.

SC acknowledges Space Applications Centre (SAC), Indian Space Research
Organization for providing research fellowship under the project NGP-17. DA
acknowledges the INSPIRE fellowship from the Department of Science and
Technology, which she used to pursue her research. The authors acknowledge
SAC, ISRO for providing the NavIC receivers to Department of Astronomy,
Astrophysics and Space Engineering, IIT Indore under this project. Authors
also acknowledge the Department of ECE, Osmania University for providing us
with the NavIC data. Further acknowledgements go to World Data Center for
Geomagnetism, Kyoto for the Dst index freely available at
http://wdc.kugi.kyoto-u.ac.jp/kp/index.html and for the sunspot numbers
available at http://www.sidc.be/silso/ (SILSO data/image, Royal Observatory of
Belgium, Brussels). The authors acknowledge the reviewers for their
suggestions in improving the quality of the manuscript.

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Decomposing Linear Representations of Finite Groups

Kaashif Hymabaccus Wadham College University of Oxford

Supervisor: Dr Dmitrii Pasechnik

Final Honour School of Mathematics and Computer Science Part C Trinity 2019

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††thanks: Equal contribution††thanks: Equal contribution

# Shape-dependent motility of polar inclusions in active baths

Pritha Dolai<EMAIL_ADDRESS>Friedrich-Alexander-Universität, Erlangen-
Nürnberg, Germany 91054  International Centre for Theoretical Sciences, Tata
Institute of Fundamental Research, Bengaluru, India 560089  Aditya Singh
Rajput<EMAIL_ADDRESS>K. Vijay Kumar<EMAIL_ADDRESS>International Centre for Theoretical Sciences, Tata Institute of Fundamental
Research, Bengaluru, India 560089

###### Abstract

Collections of persistently moving active particles are an example of a
nonequilibrium heat bath. One way to study the nature of nonequilibrium
fluctuations in such systems is to follow the dynamics of an embedded probe
particle. With this aim, we study the dynamics of an anisotropic inclusion
embedded in a bath of active particles. By studying various statistical
correlation functions of the dynamics, we show that the emergent motility of
this inclusion depends on its shape as well as the properties of the active
bath. We demonstrate that both the decorrelation time of the net force on the
inclusion and the dwell time of bath particles in a geometrical trap on the
inclusion have a non-monotonic dependence on its shape. We also find that the
motility of the inclusion is optimal when the volume fraction of the active
bath is close to the value for the onset of motility induced phase separation.

##  I Introduction

The study of systems driven out of equilibrium by a throughput of energy at
the level of the individual units has revealed a rich set of nonequilibrium
collective states [1, 2, 3, 4]. The physical realization of such active
materials range from molecular motors inside cells driven by ATP to
mechanically agitated granular materials. Active particles which form the
basic constituents of these materials are typically anisotropic in shape with
associated orientational degrees of freedom. The hydrodynamic theory of such
active liquid crystals has been a topic of great interest in the past decades
[2, 5].

A complementary direction of research considers isotropic particles wherein
the relationship between dissipation and noise in their dynamics is not in
accordance with the fluctuation-dissipation theorem [6, 7]. Examples of such
systems include Janus particles with asymmetric surface reactions, spherical
droplets with asymmetric internal flows, and driven isotropic granular
particles. Such active particles form the basis of scalar active matter [8].
Three commonly studied models of active particles are the Run-and-Tumble
Particles (RTPs), Active Brownian Particles (ABPs), and Active Ornstein-
Uhlenbeck Particles (AOUPs). An emergent property seen in systems of such
particles with purely repulsive interactions is a nonequilibrium “condensed”
phase arising due to the interplay between activity and density. This
motility-induced-phase-separation (MIPS) is, in fact, seen in homogeneous
systems of RTPs, ABPs and AOUPs [9, 10, 11, 12, 13].

Interacting systems of active particles provide examples of nonequilibrium
heath baths and studying the dynamics of probes embedded in such reservoirs
provide insight into the statistical properties of active baths. For instance,
the dynamics of tagged active particles have recently been observed to display
universal behaviors across the three models mentioned above both in a single-
file geometry [14] and in harmonic chains [15]. Another approach would be to
embed tracers or inclusions in active heat baths, and follow their dynamics.
Systems comprised of active particles with embedded isotropic passive
inclusions can have depletion forces that can be either attractive or
repulsive [16, 17, 18]. Essentially, the accumulation of active particles near
boundaries changes the nature of the depletion forces between passive
colloidal particles [3]. It is well appreciated that these fluctuation-induced
depletion forces are sensitive to the shape of the inclusions in equilibrium
systems [19]. Such depletion forces also arise in active baths [20, 21, 22].
However, for passive inclusions embedded in active heat baths, the role of
geometry of embedded objects has not received much attention [23, 24, 25, 26,
27, 28, 29, 30, 31].

In this study, we explore the emergent dynamics of a passive polar inclusion
embedded in a nonequilibrium active bath. In particular, we focus on how the
shape of the inclusion affects its macroscopic behavior. Generically, fore-aft
asymmetry and an energy flux can lead to net currents in many-body systems.
Does a polar inclusion in an active heat bath display persistent motion? How
does the persistent velocity of the inclusion depend on its shape? How does
the interaction between shape polarity and activity translate to emergent
motility of the inclusion? Active particles are known to accumulate near
static boundaries [32]. In fact, static wedge-shape objects are known to trap
active particles moving around them [33, 34]. How does the accumulation of the
active bath particles around a polar and _movable_ inclusion lead to its
emergent motility? How do the parameters of the active bath affect the
movement of the embedded inclusion? Our results are as follows: (i) we find
emergent persistent dynamics of the inclusion in active baths composed of
RTPs, ABPs and AOUPs, (ii) there is an optimum shape of the inclusion that
leads to an enhanced motility; the correlations of the net force on the
inclusion and the typical dwell time of the active particles in a trap around
the inclusion have a similar non-monotonic dependence on the inclusion
geometry, (iii) the motility of the inclusion is controlled by the volume
fraction of the active particles in the heat bath; in particular, the motility
is optimal for a volume fraction close to the onset of MIPS in the heat bath.

The paper is organised as follows. In section II, we discuss the construction
of the polar inclusion and the parameters that characterize its shape, and
present the governing dynamical equations for both the inclusion and the
active particles that make up the heat bath. Next in section III, we describe
our results on the emergent motility of the inclusion and its dependence on
geometry. We end with a summary of our study and an appropriate discussion in
section IV.

##  II Model

We study a passive polar inclusion embedded in an active nonequilibrium “heat
bath” with a distinct fore-aft asymmetry as shown in FIG. 1. The impacts of
the active particles on the inclusion are the driving force for its dynamics.
The inclusion is constructed by rigid bonds between $N_{I}$ particles placed
in an appropriate geometry. The nonequilibrium heat bath consists of $N_{B}$
scalar active particles with short-range repulsive interactions between
themselves and also with the particles that constitute the inclusion. Without
loss of generality, we choose the purely repulsive interaction to be the
Weeks-Chandler-Anderson (WCA) potential

$\displaystyle
U(r)=\epsilon\left\\{\begin{array}[]{lr}\frac{1}{4}+\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6},&r<a\\\
0,&r>a,\end{array}\right.$ (3)

where $\epsilon$ and $\sigma$ are the characteristic energy and length scales
of the potential, and $a=2^{1/6}\sigma$ is the interaction range which also
determines the effective size of the particles. The overdamped dynamics of
$\mathbf{r}_{i}$, the position $i^{\rm th}$ active particle is

$\displaystyle\frac{d{\mathbf{r}}_{i}}{dt}$
$\displaystyle={\mathbf{v}}_{i}-\mu\left[\sum_{j=1,\,j\neq
i}^{N_{B}}\nabla_{\mathbf{r}_{i}}U({\mathbf{r}}_{i}-{\mathbf{r}}_{j})+\sum_{p=1}^{N_{I}}\nabla_{\mathbf{r}_{i}}U(\mathbf{x}_{p}-\mathbf{r}_{i})\right]$
$\displaystyle\qquad+\sqrt{2\mu k_{B}T}\,{\bm{\eta}}_{i}(t)$ (4)

where $\mathbf{v}_{i}$ is the active self-propulsion velocity, $\mu$ is the
translational mobility, and ${\bm{\eta}}_{i}(t)$ is a Gaussian white noise
with zero mean and unit variance. In the above equation, $\mathbf{x}_{p}$ is
the position of the $p^{\rm th}$ particle making up the inclusion where
$p=1,2,\ldots N_{I}$. The rigid inclusion is characterised by its center of
mass position $\mathbf{R}$ and its orientation $\Phi$ [see FIG. 1]. As such,
$\mathbf{R}$ and $\Phi$ evolve in response to the net force and torque applied
on the inclusion from the impacts of the bath particles. The overdamped
evolution equations for these quantities are

$\displaystyle\frac{d\mathbf{R}}{dt}$
$\displaystyle=\mathcal{M}_{t}\;\mathbf{F}(t)+\sqrt{2\mathcal{M}_{t}k_{B}T}\;\bm{\mathcal{N}}(t)$
(5) $\displaystyle\frac{d\Phi}{dt}$
$\displaystyle=\mathcal{M}_{r}\,\mathsf{T}(t)+\sqrt{2\mathcal{M}_{r}k_{B}T}\;\mathcal{Z}(t)$
(6)

where $\mathcal{M}_{t}$ ($\mathcal{M}_{r}$) is the translational (rotational)
mobility of the inclusion, while $\bm{\mathcal{N}}(t)$ and $\mathcal{Z}(t)$
are Gaussian white noises with zero mean and unit variance, and the force
$\mathbf{F}$ and the torque $\mathsf{T}$ on the center-of-mass of the
inclusion due to the impacts of the bath particles are

$\displaystyle\mathbf{F}$
$\displaystyle=-\sum_{p=1}^{N_{I}}\sum_{i=1}^{N_{B}}\nabla_{\mathbf{x}_{p}}U(\mathbf{x}_{p}-\mathbf{r}_{i}),$
(7) $\displaystyle\mathsf{T}$
$\displaystyle=-\hat{\mathbf{z}}\cdot\sum_{p=1}^{N_{I}}\sum_{i=1}^{N_{B}}(\mathbf{x}_{p}-\mathbf{R})\times\nabla_{\mathbf{x}_{p}}U(\mathbf{x}_{p}-\mathbf{r}_{i}).$
(8)

Notice that if the active noise $\mathbf{v}_{i}=0$, then the equations (4),
(5) and (6) represent the dynamics of a polar inclusion in an equilibrium heat
bath. In this case, both the bath particles and the inclusion will have the
Boltzmann distribution as their steady-state probability. However, when the
bath-particles are active, the polar shape of the inclusion can lead to non-
trivial emergent dynamics. Before we discuss this dynamics, we first describe
the kind of active particles that we consider in this study.

The active nature of the nonequilibrium heat bath arises from the persistent
self-propelled motion of its constituent particles. We consider three kinds of
active baths: those comprised of (1) run-and-tumble particles (RTPs), (2)
active Brownian particles (ABPs), and (3) active Ornstein-Uhlenbeck particles
(AOUPs). These three scalar active particle models differ in the nature of the
active stochastic forces that propel them. Specifically, the active velocity
of the $i^{\rm th}$-RTP is given by $\mathbf{v}^{\rm RTP}_{i}=v_{\rm
R}\,(\cos\theta_{i},\sin\theta_{i})$, where the direction of self-propulsion
$0\leq\theta_{i}<2\pi$ tumbles at a Poisson rate $\gamma$ and $v_{\rm R}$ is
the speed during the active runs. In other words, the orientation $\theta_{i}$
of each RTP changes abruptly and stochastically after a mean run-time
$1/\gamma$. On the other hand, the internal orientation $\varphi_{i}$ of the
$i^{\rm th}$-ABP performs rotational diffusion and also governs the
instantaneous active velocity

$\displaystyle\frac{d\varphi_{i}}{dt}=\sqrt{2D_{r}}\;\xi_{i}(t),\qquad\mathbf{v}^{\rm
ABP}_{i}=v_{\rm A}\,(\cos\varphi_{i},\sin\varphi_{i})$ (9)

where $v_{\rm A}$ is the self-propulsion speed, $D_{r}$ is the rotational
diffusion constant and $\xi_{i}(t)$ is a Gaussian white noise with zero mean
and unit variance. Thus the orientation of the ABPs evolves continuously in
time. Finally, the active velocity of AOUPs is governed by the following
equation

$\displaystyle\tau\frac{d{\mathbf{v}}^{\rm AOUP}_{i}}{dt}=-{\mathbf{v}}^{\rm
AOUP}_{i}+\sqrt{2\Delta}\;\zeta_{i}(t)$ (10)

where $\tau$ is the persistence time of self-propelled motion, $\Delta$ is the
noise strength and $\zeta_{i}(t)$ is a Gaussian white noise. A characteristic
active speed for AOUPs can be defined as $v_{\rm O}=\sqrt{\Delta/\tau}$. Note
that, the mean-squared-displacement (MSD) for free active particles (of the
three kinds discussed above) have the same universal form [14]. All three
active models can be characterized by a generalized active speed $u$ and a
persistence rate $\omega$. One can easily show that
$u=v_{R}=v_{A}/\sqrt{2}=v_{O}$ and $\omega=\gamma=D_{r}/2=1/(2\tau)$.

Figure 1:  A polar inclusion in an active heat bath. The anisotropic inclusion
is constructed from $N_{I}$ particles rigidly connected to each other in the
geometry shown schematically. The inclusion, with opening angle $\psi$ and the
radius $\mathcal{R}$ defining its overall shape, has its center-of-mass at
$\mathbf{R}$ and its orientation is given by the unit normal
$\hat{\mathbf{n}}=-(\cos\Phi\,\hat{\mathbf{x}}+\sin\Phi\,\hat{\mathbf{y}})$
where $\Phi$ is the angular orientation. The nonequilibrium heat bath is
composed of active particles either of the RTP, ABP or AOUP kind. These active
particles (whose instantaneous direction of self-propulsion is indicated by
the blue/grey semicircles) exert forces on the embedded inclusion due to their
persistent motion. Note that the active particles interact amongst themselves
and also with the particles that make up the inclusion via the same repulsive
WCA potential (3).

##  III Results

We study the system of active particles and anisotropic passive inclusion with
an opening angle $\psi$ and a radius $\mathcal{R}$ in a two-dimensional
periodic box of side $L$. The schematic of our model system is shown in FIG.
1. The inclusion has different shapes depending upon its opening angle $\psi$.
The active bath is characterized by a Pèclet number $\textrm{Pe}=u/(a\omega)$
and an area fraction $\phi=(N_{B}\pi a^{2}+A_{\rm inc})/L^{2}$ where we
approximate the area enclosed by the inclusion $A_{\rm
inc}\approx(\pi-\psi/2)\mathcal{R}^{2}$. We choose $\sigma$,
$\sigma^{2}/\mu\epsilon$ and $\epsilon/\sigma$ respectively as the units of
length, time and force. The equations (4), (5) and (6) together with the
dynamical equations for the bath particles are numerically integrated using
the Euler-Maruyama scheme with a non-dimensional time step $\Delta t=10^{-3}$.
We fixed the thermal energy at $k_{B}T/\epsilon=0.01$, the radius of the
inclusion at $\mathcal{R}/\sigma=5$ (unless specified), and the linear size of
the simulation domain at $L/\mathcal{R}=10$. All simulations are run for long
times ($\sim 150\,\omega^{-1}$) and various statistical averages and
correlations, denoted by $\langle\ldots\rangle$, are computed after the system
reaches a steady-state, and the errorbars are calculated over several
realizations.

###  III.1 Emergent active dynamics of the polar inclusions

Figure 2: Typical trajectories (all of duration $25\,\omega^{-1}$) of the
center-of-mass $\mathbf{R}$ of the inclusion at $\mathrm{Pe}=66$ and
$\phi=0.3$ for various $\psi$ in a bath of (a) RTPs, (b) ABPs and (c) AOUPs.
The colors of the trajectories correspond to various values of the opening
angle $\psi$, and thus the shape of the inclusion. Notice that the persistent
motion of the inclusion scales upto several $\mathcal{R}$. This persistent
motion is pronounced when $\psi\sim\pi/2$. See SI Movies.

We find that for a large range of parameter values, i.e., the opening angle of
the inclusion $\psi$, the area-fraction $\phi$ and the Péclet number
$\mathrm{Pe}$, the polar inclusions show persistent motion for short times. In
FIG. 2, we plot the typical trajectories of the center-of-mass of the
inclusion $\mathbf{R}$ at three different opening angles $\psi$ for the three
models that we study. We notice that for small $\psi$ the trajectories do not
display persistent motion. This is also true for large $\psi$. When
$\psi\sim\pi/2$, we see long stretches of persistent motion of the inclusion.
However, at long times, the movement of the inclusion is diffusive in nature
for all angles $\psi$. In other words, a polar inclusion in an active bath
shows persistent motion at short times and crosses over to diffusive dynamics
at long times – the inclusion itself behaves as an ‘emergent active particle’.

Figure 3: Dynamical correlations of the polar inclusion at $\mathrm{Pe}=66$
and $\phi=0.3$. Mean-squared displacement of the inclusion in a bath of (A)
RTPs, (B) ABPs, and (C) AOUPs. All models display a transition from an initial
ballistic behavior $\sim t^{2}$ to an asymptotic diffusive behavior $\sim t$
confirming the persistent motion of the inclusion at short times. (D-F) The
orientatation correlations as a function of time for three different models.
The orientation correlation $C_{\mathrm{\mathbf{n}\mathbf{n}}}$ decays
monotonically for the AOUP bath (F) while for (D) RTP and (E) ABP baths, it
has a negative minima around $\omega t\sim 2$ for inclusion shapes that show
enhanced persistent motion. The two-point correlation function of the force on
the inclusion (11) shows approximately an exponential decay in time for all
three models (G-I). However, the decay time $\tau_{F}$ is a non-monotonic
function of $\psi$ (see FIG. 4).

To further study this persistent motion of the inclusion, we plot in FIG. 3
(A-C) the mean-squared displacement (MSD)
$\langle\left[\Delta\mathbf{R}(t)\right]^{2}\rangle=\langle\left[\mathbf{R}(t)-\mathbf{R}(0)\right]^{2}\rangle$
for the three models and various opening angles $\psi$ at $\phi=0.3$ and
$\textrm{Pe}=66$. The MSD shows ballistic behavior
$\langle\left[\Delta\mathbf{R}(t)\right]^{2}\rangle\sim t^{2}$ for short times
which eventually crosses over to diffusive dynamics
$\langle\left[\Delta\mathbf{R}(t)\right]^{2}\rangle\sim t$ at times larger
than $\omega^{-1}$.

An alternate measure of the persistent motion of the inclusion is to calculate
the auto-correlation
$C_{\mathbf{n}\mathbf{n}}(t)=\langle\hat{\mathbf{n}}(0)\cdot\hat{\mathbf{n}}(t)\rangle$
of the unit-vector
$\hat{\mathbf{n}}=-\cos\Phi\,\hat{\mathbf{x}}-\sin\Phi\,\hat{\mathbf{y}}$ (see
FIG. 1). The dependence of $C_{\mathbf{n}\mathbf{n}}(t)$ is shown in FIG. 3
(D-F) for the three active bath models. We observe that
$C_{\mathbf{n}\mathbf{n}}$ decays to zero at long times $t\gg\omega^{-1}$.
However, while $C_{\mathbf{n}\mathbf{n}}(t)$ decays monotonically to zero for
the AOUP bath, it shows oscillations for the RTP and ABP baths. This is
attributed to the different stochastic nature of the active velocity: the
magnitude of the active velocity is fixed for RTPs and ABPs whereas for AOUPs
it is drawn from a Gaussian distribution with a characteristic value $u$. As
such the orientation of the inclusion decorrelates quickly for the AOUP bath.
This is also observed in the MSD of the inclusion angle
$\langle[\Delta\Phi(t)]^{2}\rangle$ where the crossover to the diffusive
regime occurs at timescales $t\lesssim 1/\omega$ for AOUP bath whereas it
occurs at timescales $t\gtrsim 1/\omega$ for RTP and ABP baths.

The short-time ballistic dynamics of the MSD seen in FIG 3 (A-C) and the long-
lived two-point correlations of the unit normal vector FIG 3 (D-F) clearly
indicate persistent motion of the inclusion. However, this behavior is in the
configurational degrees of freedom, i.e., $\mathbf{R}$ and $\hat{\mathbf{n}}$,
of the inclusion. How does this persistence emerge from the coordinated
impacts on the inclusion due to active particles constituting the bath? To
answer this question, we look at the force $\mathbf{F}$ on the center-of-mass
of the inclusion defined in (7). In the steady-state, the average force
vanishes $\langle\mathbf{F}(t)\rangle=\mathbf{0}$. To quantify the temporal
dynamics of $\mathbf{F}(t)$, we measured the normalized two-point force
correlation function:

$\displaystyle
C_{\mathbf{F}\mathbf{F}}(t)=\frac{\langle\mathbf{F}(t)\cdot\mathbf{F}(0)\rangle}{\langle\mathbf{F}(0)\cdot\mathbf{F}(0)\rangle}.$
(11)

Figure 3 (G-I) shows that $C_{\mathbf{F}\mathbf{F}}(t)$ decays exponentially
for all opening angles $\psi$ and the decay time $\tau_{F}\sim\omega^{-1}$. In
other words, the dynamical force acting on the inclusion has the
characteristics of an exponentially correlated noise [29]. The torque
$\mathsf{T}$, defined in (8), on the center-of-mass of the inclusion vanishes
on the average. And its two-point correlation function decays exponentially as
well. However, we found that the decay time of this torque-torque correlation
function is largely independent of $\psi$. This suggests that the dynamics of
the orientational degree of the inclusion $\Phi$ does not depend on its shape.

We note from FIG. 3(G-I) that the decay time $\tau_{F}$ of
$C_{\mathbf{F}\mathbf{F}}(t)$ has a non-monotonic dependence on $\psi$. In
fact, $\tau_{F}$ has a maximum around $\psi\sim\pi/2$ as seen in FIG. 4(A). In
other words, the persistent active forces on the inclusion from the heat bath
depend in a non-trivial way on the shape of the inclusion. It is important to
realize that the inclusion is a polar object for all angles $\psi$. However,
for $\psi<\pi$ the region enclosed by the inclusion is a non-convex region,
while for $\psi\geq\pi$ the enclosed region is convex. Specifically, the non-
convex shapes represent a wedge-like trap that can lead to enhanced
accumulation of the active particles in this region thus leading to stronger
persistent motion of the inclusion.

Figure 4: (A) The decay time $\tau_{F}$ of the two-point correlation function
$C_{\mathbf{F}\mathbf{F}}(t)$ of the force shows a peak as the shape of the
inclusion is changed by varying the opening angle $\psi$. Although the
location of the peak coincides for all the three models, the maximal value of
$\tau_{F}$ for the AOUP bath is lower than that for the RTP and ABP baths. (B)
The probability distribution of the normal component of the force
$F_{n}=\mathbf{F}\cdot\hat{\mathbf{n}}$ on the inclusion at various values of
$\psi$ for the ABP bath. Notice that the location of the peak of this
distribution changes with $\psi$ while its width is largely unchanged. We
observe a similar result for the RTP and AOUP baths.

###  III.2 Shape dependent motility of the polar inclusion

Though $\langle\mathbf{F}\rangle=\mathbf{0}$ in a fixed frame, the component
of $\mathbf{F}$ along the normal to the inclusion defined as
$F_{n}\equiv\mathbf{F}\cdot\hat{\mathbf{n}}$ has a non-zero average. In FIG.
4(B), we plot the distribution of $F_{n}$ for the ABP model at various opening
angles $\psi$. We note that while $\langle F_{n}\rangle$ depends on $\psi$,
the variance of the distribution is largely independent of the opening angle
of the inclusion. These observations hold for the RTP and AOUP models as well.

Clearly, $F_{n}$ controls the emergent motility of the inclusion. We define
the motility of the inclusion as

$\displaystyle
V_{\textrm{inc}}=\left\langle\frac{d\mathbf{R}}{dt}\cdot\hat{\mathbf{n}}\right\rangle=\mathcal{M}_{t}\;\left\langle\mathbf{F}\cdot\hat{\mathbf{n}}\right\rangle.$
(12)

Note that, in principle $V_{\textrm{inc}}$ can change sign when the shape of
the inclusion changes, unlike the active speed $u>0$. In FIG. 5(A), we show
the variation of $V_{\textrm{inc}}$ as a function of the opening angle $\psi$
for an area fraction $\phi=0.3$ and $\mathrm{Pe}=66$. It is immediately
obvious from the figure that $V_{\textrm{inc}}$ has a maximal value around
$\psi_{\textrm{opt}}\sim\pi/2$. Several points are to be noted from FIG. 5(A).
First, the variation of $V_{\textrm{inc}}$ as a function of $\psi$ has a
similar behavior across all three models. In fact, even the numerical values
are also very similar. This trend is consistent with the variation of
$\tau_{F}$ seen in FIG. 4(A). Second, $V_{\textrm{inc}}>0$ for all angles
$\psi$. This is rather surprising since it is possible that the emergent
velocity of the inclusion could change sign with its shape. In particular, for
$\psi\leq\pi$, the inclusion has a non-convex shape and changes over to a
convex shape for $\psi>\pi$ (see FIG. 2). As such, one might have expected a
sign reversal of $V_{\textrm{inc}}$ around $\psi\sim\pi$. Our numerical
simulations, however, show that this is not the case. Third, as might be
expected, $V_{\textrm{inc}}$ vanishes both at $\psi\sim 0$ and $\psi\sim
2\pi$. For small $\psi$, the opening wedge like region is not wide enough to
trap sufficient active particles, while for $\psi\sim 2\pi$ the inclusion
approaches the shape of a line. In both cases, the inclusion does not have any
net polarity in its shape. Fourth, to check how $V_{\textrm{inc}}$ depends on
the overall size of the inclusion, we considered the ABP model and varied the
radius $\mathcal{R}$. We found that the optimum speed $V_{\rm
opt}=V_{\textrm{inc}}(\psi_{\rm opt})$ is largely independent of
$\mathcal{R}$. However, the optimal angle $\psi_{\rm opt}$ decreases with
$\mathcal{R}$ as shown in FIG. 5(B). Finally, we note that $V_{\textrm{inc}}$
is comparable in magnitude to the coefficient of the ballistic term in the MSD
of the inclusion shown in FIG. 3(A-C).

Figure 5: (A) The variation of the mean persistent motility of the inclusion
defined in (12) as a function of its shape. $V_{\mathrm{inc}}$ peaks around
$\psi\sim\pi/2$ for all models. Interestingly, the numerical values of
$V_{\mathrm{inc}}$ are also similar for all $\psi$ across the three models.
Remarkably, we observe that $V_{\mathrm{inc}}$ _does not_ change sign as the
shape of the inclusion changes from a non-convex region (for $\psi<\pi$) to a
convex region (for $\psi>\pi$). In other words, the emergent mean velocity of
the anisotropic inclusion is always along $\hat{\mathbf{n}}$. (B) The optimum
angle $\psi_{\mathrm{opt}}$ at which $V_{\mathrm{inc}}$ peaks decreases with
the radius $\mathcal{R}$ of the inclusion. (C) The time spent by the active
particles in the wedge-like trapping region follows an exponential
distribution with a characteristic dwell time $\tau_{\mathrm{dwell}}$. This
dwell time depends on the shape of the inclusion and shows a peak around
$\psi\sim\pi/4$. The value of $\tau_{\mathrm{dwell}}$ is higher for RTP and
ABP baths compared to the AOUP bath. In this plot, $\mathrm{Pe}=66$ and
$\phi=0.3$.

As remarked above, the accumulation of the bath particles in the wedge-like
region has a non-trivial effect on the motility of the inclusion. This wedge-
like region exists only for opening angles $\psi<\pi$. The bath particles
dynamically enter and leave this region as the inclusion moves around. What is
the mean dwell time of the bath particles within this region? The dynamics of
a bath particle deep inside this wedge-like region, i.e.,
$\rho\ll\mathcal{R}$, is hindered by the subsequent accumulation of other
active particles at $\rho\lesssim\mathcal{R}$. As such, the dwell time would
be an increasing function of $\mathcal{R}$. At a given $\mathcal{R}$, the
opening angle $\psi$ also influences the dwell time with small values of
$\psi$ being more effective. However, the impacts due to the small number of
particles at small $\psi$ would not lead to a substantial net force on the
inclusion. On the other hand, for larger values of $\psi\sim\pi$, the trap
does not present a confining region. As such, we expect that there is an
optimum opening angle at which this trapping effect is maximal. To quantify
this effect, we measured the amount of time $t_{\rm dwell}$ that a particle
spends in the wedge-like region. Specifically, we considered a bath particle
to be “trapped” whenever $\rho/\mathcal{R}<\cos(\psi/2)$ where $\rho$ is its
distance from the geometrical center of the inclusion and $\omega\,t_{\rm
dwell}>1/20$. We find that the dwell times follow an exponential distribution
with a characteristic time scale $\tau_{\rm dwell}$. We plot $\tau_{\rm
dwell}$ as a function of $\psi$ in FIG. 5(C) and find that there is an optimum
opening angle when this characteristic dwell time is a maximum. Thus the
anisotropic shape and the wedge-like trapping region of our inclusion serve to
trap the bath particles and lead to its emergent shape-dependent motility
characteristics.

To decipher how the characteristics of the active bath controls the motility
of the inclusion, we plot the variation of $V_{\textrm{opt}}$ with the volume
fraction $\phi$ in FIG. 6(A). We observe that $V_{\textrm{opt}}$ is a non-
monotonic function of $\phi$ and has a peak around $\phi=0.3$. How do we
understand this optimum volume fraction? At small $\phi$, most of the active
particles accumulate in the wedge-like region of the inclusion. As such,
increasing $\phi$ leads to an initial increase in $V_{\textrm{opt}}$. This
trend continues until the wedge-like region is completely occupied by the
bath-particles. With a further increase in $\phi$, the large number of
particles present in the active bath now start clustering around the inclusion
in an isotropic manner. In other words, the anisotropic inclusion is
increasingly shielded by a cloud of active particles at large $\phi$. This
effectively nullifies the shape anisotropy of the inclusion and leads to a
decrease in $V_{\textrm{opt}}$. Note that this behavior is displayed by all
the three active bath models and, concomittant with the results in FIG. 5,
$V_{\textrm{opt}}$ is larger for AOUP baths compared to the RTP and ABP baths.
Thus the volume fraction of the bath has a non-trivial effect on the emergent
motility of the inclusion. On the other hand, an increase in the Péclet number
Pe leads to an expected monotonic increase in $V_{\textrm{opt}}$ as shown in
FIG. 6(B).

Figure 6: (A) The variation of the optimal velocity $V_{\mathrm{opt}}$
(defined as the value of $V_{\mathrm{inc}}$ at the optimal angle
$\psi_{\mathrm{opt}}$; see FIG. 5) with the volume fraction $\phi$ at
$\mathrm{Pe}=100$. The peak seen in $V_{\mathrm{opt}}$ results from a
shielding effect of the active particles on the inclusion (see text). The
dashed line denotes the value of $\phi$ at the MIPS phase boundary [35]. (B)
At $\phi=0.5$, $V_{\mathrm{opt}}$ is a monotonic function of the Péclet number
$\mathrm{Pe}$.

##  IV Discussion

We observe the following points from our study. First, the emergent motility
of the inclusion arises from a combined effect of the dynamics of the
inclusion as well as the dynamics of the active particles in the heat baths.
In other words, the polar inclusion in our study is _not_ a passive tracer
[30, 23], nor a stationary trap designed to capture active particles [33] or
traps with prescribed dynamics [34]. Rather, the degrees of freedom of our
inclusion evolve naturally due the impacts of the bath particles. It should
also be noted that the net motility of the inclusion is a consequence of the
nonequilibrium nature of the heat bath. In an equilibrium heat bath, the
inclusion would not display any net motility, even if it had a polar shape.
Second, the active particles in our heat baths are isotropic in shape. This
should be contrasted with baths consisting of rod-like active particles. In
the case of active baths with polar particles, the orientational degrees of
freedom of the bath particles has a significant effect on their trapping
dynamics in a static wedge [36]. Third, previous studies have considered
wedge-like inclusions (non-closed shape) in active baths and have measured
their emergent motility [37, 38] . While the emergent dynamics depends on the
opening angle of the wedges, the open shape does not allow constructing shapes
like that of our inclusion for $\psi>\pi$. As remarked earlier, our polar
inclusion always moves along $\hat{\mathbf{n}}$ and does not reverse its
direction of motion even when $\psi>\pi$. This is a novel feature of the
closed shape of our inclusion. Fourth, the motility of the inclusion arises
from the net force on it due to impacts of the bath particles. This impact
force has non-trivial correlations in time [29, 30]. This is not surprising
since the active forces on the bath particles themselves are correlated in
time, i.e., they have persistent driving. But what we have demonstrated in our
study is that the shape of the inclusion controls, in a non-trivial manner,
the temporal correlations of the net force due the bath particles, and thus
its emergent motility. Fifth, concomitant with our observation that increasing
$\phi$ beyond a critical value leads to a decrease in $V_{\mathrm{inc}}$ at a
fixed $\mathrm{Pe}$, we observe that the emergent motility of the inclusion is
optimum at the value of $\phi$ near the MIPS phase boundary [35, 10]. We
notice that below the MIPS transition, the suboptimal accumulation of
particles on the inclusion will lead to a decrease in motility. On the other
hand, deep in the MIPS regime, enhanced accumulation of the active particles
around the inclusion will “isotropize” the polar shape of the inclusion and
thus reduce its emergent motility. As such, the region around the MIPS phase
boundary in the $\mathrm{Pe}-\phi$ plane seems optimal to transduce the forces
from the active bath into enhanced motility of the inclusion. However, we do
not see a similar peak when $\mathrm{Pe}$ is varied at a fixed area fraction
$\phi=0.5$. Sixth, all three models of active particle leads to similar
emergent dynamics of the inclusion. This is in line with the observation that
these models also display universal scaling features in a one-dimensional
single file geometry [14].

Our study reveals optimum shapes for inclusions that leads to enhanced
motility. These results could potentially have implications for designing the
shapes of embedded objects in active baths [39]. Thus, an inclusion that
controls its shape in a dynamical manner can control its emergent active
motility when placed in an otherwise isotropic and unbiased active heat bath.
For instance, moving cells or droplets, wherein their shape is controlled by
other mechanisms, could display varied motility characteristics in medium
wherein the impacts of the bath particles are not governed by a Boltzmann
distribution.

In summary, we have studied the dynamics of a polar inclusion in three
different active baths, namely ABPs, RTPs and AOUPs. We see that the inclusion
behaves as an emergent active particle with short-time ballistic behaviour and
long-time diffusive behaviour. This emergent motility arises from the non-
trivial correlations of the impact force exerted by the bath particles on the
inclusion. These correlations, in turn, are controlled by the shape of the
inclusion. In particular, the inclusion (with $\mathcal{R}/a=5$) has a maximum
motility around an opening angle of $\psi\sim\pi/2$. Surprisingly, the
inclusion does not reverse the sense of its motion relative to its normal
vector even when it changes from a non-convex shape to a convex shape.
Remarkably, we find that the inclusion motility peaks close to the volume
fraction of the bath particles that corresponds to the onset of MIPS. An
emergent shielding effect of the active particles deep in the MIPS regime
seems to lead to a reduction in the inclusion motility. It would be
interesting to study the emergent interactions between such passive inclusions
embedded in active baths to explore the interplay between active fluctuations
and the geometry of shape [40]. Finally, our predictions can be tested in
various experimental systems, for instance, in vibrated granular systems with
embedded inclusions.

##  V Acknowledgements

We acknowledge support of the Department of Atomic Energy, Government of
India, under project number RTI4001. PD thanks Arghya Das for many useful
discussions. ASR thanks the SN Bhatt summer program of the ICTS-TIFR. KVK’s
research is supported by the Department of Biotechnology, India, through a
Ramalingaswami re-entry fellowship and by the Max Planck Society through a
Max-Planck-Partner-Group at ICTS-TIFR. KVK also acknowledges the financial
support of the John Templeton Foundation (#62220). The opinions expressed in
this paper are those of the authors and not those of the John Templeton
Foundation.

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