#  Current reversal in polar flock at order-disorder interface

Jay Prakash Singh1,2<EMAIL_ADDRESS>Partha Sarathi Mondal1
<EMAIL_ADDRESS>Vivek Semwal1
<EMAIL_ADDRESS>Shradha Mishra1<EMAIL_ADDRESS>1
Indian Institute of Technology (BHU) Varanasi, India 221005
2 Israel Institute of Technology,Technion Hafia Israel

###### Abstract

We studied a system of polar self-propelled particles (SPPs) on a thin
rectangular channel designed into three regions of order-disorder-order. The
division of the three regions is made on the basis of the noise SPPs
experience in the respective regions. The noise in the two wide region is
chosen lower than the critical noise of order-disorder transition and noise in
the middle region or interface is higher than the critical noise. This make
the geometry of the system analogous to the Josephson Junction (JJ) in solid
state physics. Keeping all other parameters fixed, we study the properties of
the moving SPPs in the bulk as well as along the interface for different
widths of the junction. On increasing interface width, system shows a order-
to-disorder transition from coherent moving SPPs in the whole system to the
interrupted current for large interface width. Surprisingly, inside the
interface we observed the current reversal for intermediate widths of the
interface. Such current reversal is due to the strong randomness present
inside the interface, that makes the wall of the interface reflecting. Hence
Our study give a new interesting collective properties of SPPs at the
interface which can be useful to design devices like switch using active
agents.

##  I Introduction

Emergence of collective motion Ndlec  _et al._ (1997); Harada  _et al._
(1987); Turkina and Sokolov (2001); Kron and Spudich (1986); Laub and Loomis
(1998) and global ordering Pattanayak  _et al._ (2020a); Vicsek  _et al._
(1995); Singh  _et al._ (2021a); Mishra and Mishra (2022); Toner and Tu
(1995); Singh and Mishra (2020) among the various living or non-living
systems, are well known phenomena. Each particle show systematic motion at the
cost of its internal energy. All individuals in a group synchronize themselves
to show different behavioral state, exhibiting host of interesting properties
like pattern formation Marchetti  _et al._ (2013), non equilibrium phase
transition Vicsek  _et al._ (1995), large density fluctuation Marchetti  _et
al._ (2013); Grégoire and Chaté (2004); Chaté  _et al._ (2008a); Bhattacherjee
_et al._ (2015); Narayan  _et al._ (2007), enhance dynamics Bechinger (2016);
Angelani  _et al._ (2009); Harder  _et al._ (2014); Pattanayak  _et al._
(2020b); De Castro and Sollich (2017); Semwal  _et al._ (2021a); Pattanayak
_et al._ (2019); Semwal  _et al._ (2021b); Baskaran and Marchetti (2008);
Singh  _et al._ (2022), motility induced phase separation Buttinoni  _et al._
(2013); Cates and Tailleur (2015); Sese-Sansa  _et al._ (2018); Dikshit and
Mishra (2022a, b) etc,. Interestingly, different real biological systems are
encountered with different kinds of confined geometry Chaffey (2003); Hawkins
_et al._ (2009); Das  _et al._ (2015); Uspal  _et al._ (2015); Volpe  _et al._
(2011); Mozaffari  _et al._ (2016). Confinement and boundary play significant
role in variety of biological systems Hawkins  _et al._ (2009), sheared
systems Saracco  _et al._ (2011), other places like in fluid dynamics.
Boundary can induce many interesting phenomena like, spontaneous flow inside
the channel Mishra and Pattanayak (2017), and another classic example include
Rayleigh-Benard convection in the fluid Bodenschatz  _et al._ (2000). There
are variety of practical applications based on confined geometry like; mass
transport in nanofluids to enhance the microfluidic devices Zumdieck  _et al._
(2008a); Voituriez  _et al._ (2005), geophysical applications etc.,
There are few studies where researchers have seen the behaviour of SPPs at the
interface of two different substrate media. Most of studies involves the media
of two dissimilar fluids Zumdieck  _et al._ (2008b); Wang  _et al._ (2015);
Dominguez  _et al._ (2016); Malgaretti  _et al._ (2016). For example,
Dirichlet et al. Dietrich  _et al._ (2017) observed that catalytic active
Brownian microswimmer at different solid-liquid interface shows inhomogeneity
in the particles speeds with respect to the orientation of catalytic substrate
at different interface. Another well known classical example of interface
between superconductor and insulator with boundary is Josephson Junction (JJ)
in solid state system Kontos  _et al._ (2002); Josephson (1962); Buzdin  _et
al._ (1982); Kontos  _et al._ (2001); Kulik (1966).

Motivated with the JJ in solid state, here in this article we will discuss the
collective properties of SPPs by designing a setup analogous to JJ. We have
modeled a system of polar SPPs with alignment interaction through a thin
rectangular narrow channel. Further, the thin channel is divided into three
regions wherein two opposite regions, SPPs move coherently. In the middle
region SPPs diffuse randomly and no net current. Although the comparison
between our setup and Josephson junction is not very common, since the
superconducting phenomena are macroscopically quantum in nature, but we still
designed an analogous model system for collection of SPPs and observe the
properties of it.
We also studied the case where a small external field is introduced, along the
long axis of the system which gives an easy direction for particle alignment.
System is studied for different widths of the intermediate disorder region
with and without easy direction (perturbation). On tuning the width of the
disorder region, SPPs with perturbation shows a non-equilibrium phase
transition whereas without perturbation, it shows a weak dependence on the
interface width. Further, at the junction, particles get reflected from it’s
walls and we observed the current reversal. This lead to the some of the SPPs
to move in opposite direction and hence contribute in negative current. We
also showed one application of the system geometry to use it for sorting two
different type of SPPs.

Rest of the manuscript is divided as follows; we have discussed the details of
the model in section II. The results are discussed in section III, one
possible application in section IV and conclusion of the paper with summary is
discussed in section V.

Figure 1: (color online) We show a model picture of the system obtained from
the simulation in which different color shows particle’s orientation $\theta$
($\in[0,2\pi]$). The region II is the interface (junction) with disorder
region where as the region to the left and right of the interface (I $\&$ III)
is the ordered region. Noise strength in region II is $\eta_{II}=0.7$ and in
regions I, III $\eta_{I}=\eta_{III}=0.3$. $d$ is the width of the
interface.The two vertical lines in the middle indicate the boundary of the
interface. The $W$ and $L$ are the short and long dimensions of the system.
The axis for the two directions of the system is drawn on the left. Periodic
boundary condition is used in both the directions.

##  II Model

We consider a collection of polar self-propelled particles (SPPs) moving on a
two-dimensional substrate on a rectangular narrow channel with periodic
boundary conditions (PBC) in both directions. The short and long axis of the
channel are denoted by $W$ and $L$, respectively as shown in fig.1. Particles
interact through a short-range alignment interaction within a small
interaction radius $R_{0}=1$Vicsek  _et al._ (1995). Moreover, the strength of
interaction of each SPP is the same. The system is partitioned into three
regions: the two regions on the left and right represent the ordered region
and third middle section shows the disordered region. The middle disordered
region is termed as junction or interface and the width $d$ of the interface
is our tuning parameter. The width of the junction is varied from $d$
$=(1-30)$. In three regions, each particle is defined by its position ${\bf
r}_{i}(t)$ and orientation $\theta_{i}(t)$ at time $t$ and they move along the
direction of their orientation with a fixed speed $v_{0}=0.5$. The position
and orientation updates of a particle are given by:

${\bf{r}}_{i}(t+\Delta{t})={\bf{r}}_{i}(t)+{v_{0}}{\bf n}_{i}{\Delta{t}}$ (1)
${\bf n}_{i}(t+\Delta{t})=\frac{{}\sum_{{j\in R_{0}}}{\bf
n}_{j}(t)+\eta_{i,k}N_{i}(t){\bf{\bf{{\xi}}}}_{i}(t)}{w_{i}{(t)}}$ (2)

where, $\Delta t=1.0$ is the unit-time step and ${\bf
n}_{i}=(\cos\theta_{i},\sin\theta_{i})$ is the unit direction vector of the
$i^{th}$ particle. In eq.2, the first term in the right hand side represents
the short-range alignment interaction inside the interaction radius ($R_{0}$)
of the $i^{th}$ particle. The second term
$\xi_{i}(t)=(\cos(\phi_{i}(t),\sin(\phi_{i}(t)))$ on the right hand side of
eq.2 denotes the vector noise which measures the error made by the particle,
following its neighbors. $\phi_{i}$ is uniform random angle $\in(-\pi,\pi)$,
$N_{i}(t)$ denote the number of neighbors within the interaction radius of the
$i^{th}$ particle at time $t$. Further, $\eta_{i,k}$ $(k=I$, $II$ and $III$)
($\eta_{I}=\eta_{III}=0.3$ and $\eta_{II}=0.7$) shows the strength of the
randomness present in the system for the three regions. We choose the
mentioned values of noise, because for the clean polar SPPs interacting
through Vicsek type alignment interaction with vector noise, the order-
disorder transition occurs at $\eta\sim 0.6$ (for the same set of parameters
used here)Singh  _et al._ (2021b); Chaté  _et al._ (2008b). $w_{i}(t)$ is the
normalisation factor which reduces the right hand side of the eq.2 to a unit
vector. The above model we called as system without perturbation (WOP).
Due to the rectangular geometry of the system, particle experiences an easy
axis for their motion (along the long axis of the system). We also introduced
an external perturbation along the long-axis of the channel. It gives an easy
direction for the SPPs motion; hence the orientation update equation will
become

${\bf n}_{i}(t+\Delta{t})=\frac{{}\sum_{{j\in R_{0}}}{\bf
n}_{j}(t)+h_{0}{\bf{n_{p}}}+\eta_{k}N_{i}(t){\bf{\bf{{\xi}}}}_{i}(t)}{w_{i}{(t)}}$
(3)

here, $h_{0}$ is the strength of the external field and kept fixed to a small
value with direction ${\bf{n_{p}}}=(1,0)$. Using the above equation eq.3, the
model is referred to as system with perturbation (WP). Further, number density
of SPPs is defined by $\rho=\frac{N}{L\times W}{=1.0}$, where $N$ is the total
number of particles in the system. All the particles are allowed to move
throughout the system and they experience the noise of different regions
accordingly. We let the system evolve from random homogeneous state of density
and orientation of particles. All the results discussed below are in the
steady state, and total time step of the simulation is taken ${10^{6}}$. One
simulation step is counted after the update of all the particles once.
Numerical details and parameters are as chosen as $R_{0}=1.0$, $L=200,400$,
$W=5$, and $h_{0}$ is varied from $2\%$ to $6\%$ of the strength of alignment;
which is fixed to $1$. A total 20 independent realisations are used for better
statistics.

##  III Results

###  III.1 Global ordering and junction width ($d$)

First, we study the effect of junction width $d$ on the global orientation in
the whole system of size $L\times{W}=200\times 5$ for different junction
widths $d$. Ordering in the system is characterized by the orientation order
parameter,

$\Psi(t)=\frac{1}{N}|{\sum^{N}_{i=1}}n_{i}(t)|$ (4)

In the ordered state, i.e., when majority of particles are moving in the same
direction, then $\Psi$ will be closer to $1$, and of the order of
$\frac{1}{\sqrt{N}}$ for a random disordered state. First, we show the
variation of $\Psi=<\Psi(t)>$, where, $<..>$ means average over time in the
steady state and over $20$ independent realisations. We first study the system
WOP. In fig.2(a), we plot $\Psi$ vs. junction width $d$ and found that with
increase in $d$, $\Psi$ shows small decay, which have been further confirmed
by orientation probability distribution function (PDF) $P(\Psi)$ in fig2(b).
To understand the small decay of $\Psi$ with width $d$, we have shown the
snapshots for two different junction widths $d=2$ and $18$ in fig.2(b) and
(c), respectively. The circles represent the particle and color of the circle
shows their orientations. In general the SPPs form the ordered band inside the
ordered region as shown by the dense moving SPPs along the channel in fig.2(b)
(from right direction). The size of the ordered band or cluster depends on the
chosen set of system parameters Mishra  _et al._ (2010); Chaté  _et al._
(2008a); our . Hence for small width as shown in fig.2(b), when the width of
the interface is smaller or of the order of the size of the band, the SPP
passes the interface before it experiences the disorder present inside the
interface. Hence the global order parameter retains high value $\sim 0.8$ and
direction of moving SPPs band remains unaffected after interaction with the
interface (shown by the almost a clear common orientation of all particles in
the system in fig.2(b)). As we increase the width of the interface, and the
width of the disordered region is larger than the size of the band. Then
before the band can pass the interface it experiences the disturbance and
front of the band randomise. Due to that the particles moving at the back also
feel random orientation before they enter the interface. Hence, a part of the
band reorient before it can pass the interface as shown in fig.3(a-b). The
whole process looks like a reflection due to the interface as in fig.3(c-d).
Some part of the band able to come out of the interface from the other side
and they contribute the forward moving current, but a finite fraction of
particles from the band observe reflection from the walls of the interface as
shown in fig.3(e-f). A clear animation of the interaction of a band with the
interface is shown in SM1. The interface acts like a partially reflecting
wall, it leads the SPPs to avoid the junction. Also to avoid the interface
their orientation develops some contribution in $Y-$ direction as well. As
shown in the snapshots fig.3(e-f), the orientation of particles are not
strictly along $\theta=(0,\pi$ or $2\pi)$. This reduces their frequency of
going inside the interface. This results that mostly the SPPs are moving in
the ordered region only, and we find weak dependence of global order parameter
on the width $d$.

Figure 2: (color online) All the plots (a)-(c) shown here for without
perturbation. (a) We plot the global orientation order parameter $\Psi$ vs.
width of the disorder region $d$. Inset;(a) plot shows the global order
parameter distribution $P(\Psi)$ for different width of the junction $d$.
Different colored break lines are for $d$ = 4 (black), $d$ = 8 (red), $d$ = 12
(green) and $d$ = 18 (blue). Plot (b) and (c) show the space snapshots of the
system for width $d$ = 5 and $d$ = 18 respectively. color of each particle
represents it’s orientation $\theta$ ($\in[0,2\pi]$) according to the color
bar. $L\times{W}=200\times 5$ and $\rho=1.0$ Figure 3:  (color online)
Snapshots of particles (for system WOP) with their orientation angle $\theta$
shown in color bar at times $t$ = 2122, 2163, 2204, 2223, 2253 and 2271 (a-f)
respectively. The time is chosen such that a dense band of particles moving
towards the interface (blue color) in (a), enter inside the junction (b) and a
part of the particles inside the band start getting reflected (having
orientation towards $-x$ direction) (c-d) and macroscopic fraction of
particles are reflected from the interface as shown in (e-f). Some part of
particles are transmitted through the junction (e-f). The width of the
interface $d=14$, $L\times{W}=200\times 5$ and $\rho=1.0$

Further, we have studied the system WP. The perturbation introduced in such a
way that flock is biased to move along the +ve direction of long axis
(x-axis). Interestingly, we have found that global orientation order parameter
$\Psi$ decay sharply with an increase in the junction width $d$ as shown in
fig.4(a). Which has been confirmed by plotting the probability distribution
function (PDF) $P(\Psi)$ for different junction widths $d$ in inset of
fig.4(a).

Figure 4: (color online) All the plots (a)-(c) shown here for with
perturbation. (a) We plot the global orientation order parameter $\Psi$ vs.
width of the disorder region $d$ with perturbation. Inset;(a) plot shows the
global order parameter distribution $P(\Psi)$ for different width of the
junction $d$. Different colored break lines are for $d$ = 4 (black), $d$ = 8
(red), $d$ = 12 (green) and $d$ = 18 (blue). color of each particle represents
it’s orientation $\theta$ ($\in[0,2\pi]$) according to the color bar. Plot (b)
and (c) show the space snapshots of the system for width $d$ = 5 and $d$ = 18
respectively. The two vertical lines indicate the boundary of the interface.
$L\times{W}=200\times 5$ and $\rho=1.0$

-

Figure 5: (color online) Snapshots of particles with their orientation
$\theta$ (shown in colorbar) inside the junction at times $t$ = 2101, 2125,
2150, 2175, 2195 and 2220 from (a-f) respectively for system WP ($h_{0}=6\%$).
There is no clear reflection of the particles from the interface. Other
details of the system is the same as in fig.3.

Fig.4(b)-(c) show the plot the snapshots for different junction widths $d$
$=2$ and $18$. For lower width of junction flock does not experience any
hurdle and passes coherently with different bands, leads to higher values of
$\Psi$. Moreover, for higher values of junction width, most of the SPPs
trapped into the junction with random directions hence decreases in the value
of $\Psi$. This is very different from what we observed for system WOP as
shown in fig.2(c) and fig.3, where the boundary of the interface acts like
reflecting walls. But for the system WP, due to an easy direction for the
moving band of SPPs, they are forced to enter inside the interface. Inside the
interface the strength of the external perturbation is not strong enough to
help them to pass. But it oppose their orientation in other direction. Hence
particles feel a kind of frustration inside and spend more time in the
junction as shown in the fig.4(c) and leads to small order parameter for large
junction width $d$. In fig.5(a-f) we show the interaction of a band moving
towards the interface at different times. We found that a moving band
experiences some fluctuation from the interface, but density inside the
junction is large as shown in fig.5(c-d), and finally the particles get out of
the junction from the other side without getting a clear reflection as found
for the case WOP (as shown in fig.5(e-f)). The details of interaction of a
band with interface for system WP is shown in SM2. Further we study the
properties of the flock along the interface in system without perturbation.

Figure 6: (color online) Plot (a)-(c) show the time variation of event current
$\Psi_{xd}$ along the long axis with increasing width $d$. (d) event current
distribution $P(\Psi_{xd})$. (e) Shows the x-orientation current auto
correlation for three junction width $d$. Black, red and green colors show the
results for junction width $d$ = 4, 8 and 12 respectively.
$L\times{W}=200\times 5$ and $\rho=1.0$. Figure 7: (color online) Plot (a)-(c)
show the time variation of orientation event current $\Psi_{yd}$ along the
short axis with increasing width d. (d) Orientation event current distribution
$P(\Psi_{yd})$. Black, red and green colors show the junction width $d$ =4, 8
and 12 respectively. $L\times{W}=200\times 5$ and $\rho=1.0$.

###  III.2 Current inside the junction

Junction current:- In this section, we discuss the junction current within the
junction along the long $x$-axis as well as $y$-axis with the variation of the
junction width $d$. The junction current is calculated when at least 25%
particles of the whole system are within the junction, and we named this
current as event current. The event current in the junction along $x$ and
$y$-directions is defined by $\Psi_{xd}(t)=\frac{1}{n}\sum^{n}_{i}v_{xi}$,
$\Psi_{yd}=\frac{1}{n}\sum^{n}_{i}v_{yi}$ and finally total event current
$\Psi_{d}(t)=\frac{1}{n}|{\sum^{n}_{i}}n_{i}(t)|$. where $v_{xi},v_{yi}$, and
$n$ represent the components of velocity vector along the long and short axis,
and the total number of particles within the junction respectively. In the
fig.6(a)-(c) We show the time series of $\Psi_{xd}$ for different values of
junction width $d$. We observe, with increased $d$, the amplitude of
$\Psi_{xd}$ decreases and also positive and negative current changes in a
periodic fashion with the decreased period. Here current is carried by the
particles along $+ve$ and $-ve$ $x$-direction, we call them positive and
negative currents respectively. Further, in fig.6(d); we show the current
probability distribution function $P(\Psi_{xd})$ of $\Psi_{xd}$. It clearly
suggests that with the increase in the width of the junction, there is a clear
signature of current reversal. Also, in fig.6(e) we plot the current-current
auto-correlation function $C(\Psi_{xd})=<\Psi_{xd}\cdot\Psi_{xd}>$. Sharper
decay of auto-correlation with the increase in the junction width $d$. Hence
we find that in the channel the current along the long axis alternate from
$+ve$ to $-ve$ on tuning the width $d$. Here our claim regarding current
reversal phenomena is very interesting properties of the flock at the
junction. For small widths $d<8$, coherent flock enters into the junction and
crosses without significant deviation. For intermediate widths $8\leq d\leq
16$, we observed that once coherent moving SPPs enters into the junction, it
faces a randomness inside the junction. Since inside the junction all the
directions are equally probable, but flock prefers to move in $+ve$ or $-ve$;
$x$-direction, that lead to the quicker escape from the disorder region.
Further SPPs try to come out from the junction and stabilizes back and forth
oscillations within the junction. This oscillation along $+ve$ and $-ve$
$x$-direction, we are calling alternating change in the orientation event
current. Interestingly this oscillation is more prominent for the intermediate
junction widths $8\leq d\leq 16$. For small width of the junction the extend
of the moving bands of SPPs is of the order or larger than the width of the
interface and SPPs can easily pass through it with small disturbance hence the
$x-$current $\Psi_{xd}$ shows small oscillations with time and no negative
current. But as we increase the width of the interface, when the size of the
interface is larger than the size of the ordered band, then for some distance
the moving band of SPPs is able to penetrate (which is analogous to the
penetration depth in solid state) and then experience randomness. Which leads
to a fraction of particles from the moving band reverses its direction of
motion and we experience a negative current and hence negative $\Psi_{xd}$ as
shown in fig.6(b-c). This leads to the phenomena of current reversal inside
the junction. For widths $d>20$, moving SPPs experience more and more
reflection and unable to enter inside the junction and gets reflected from the
wall itself and hence we have weak junction current. Due to that the magnitude
of the junction current $\Psi_{xd}$ decreases with increasing $d$.
Furthermore, in fig.7(a)-(c), we show the junction current $\Psi_{yd}$ and
current PDF $P(\Psi_{yd})$ along the small axis with respect to junction width
$d$. We observe that there is no current reversal with the increase in $d$.
However, for higher $d$, there is a periodicity which is further confirmed by
the current distribution $P(\Psi_{yd})$ is shown shown in fig.7(d).

Figure 8: (color online) Probability distribution, $P(f)$, inside the
junction: Plot (a) is for WOP case and plots (b),(c),(d) are for WP case with
perturbation strength ($h_{0}$) 2$\%$, 4$\%$ and 6$\%$ respectively on $\log$
$\log$ scale. Colors ’black’,’red’,’green’,’blue’ and ’yellow’ corresponds to
$d$ = 4, 8, 12, 16 and 20 respectively. Insets shows the plot of $P(f)$ vs $f$
in $\log-y$ scale. In plot (a) the dashed line (cyan) is the power law with
exponent 1.25 and in plot (d) dashed line (magenta) is power law with exponent
1. $L\times W=200\times 5$ and $\rho=1.0$.

###  III.3 Fraction of particles inside the junction

Till now, we have discussed about the orientation of moving SPPs inside the
junction. We also find interesting results for the fraction of particles
inside the junction $f(t)=\frac{\mathcal{N}(t)}{N}$. Here $\mathcal{N}(t)$ is
the number of particles inside the junction at time $t$. We compared the
probability distribution function (PDF) of $f(t)$ for the systems WOP and WP.
The PDF, $P(f)$ is obtained by calculating the normalised distribution of
fraction of particles inside the junction and then PDF is averaged different
independent realisations.

In fig.8(a-d) we show the plot of $P(f)$ vs. $f$ for different junction widths
$(d=4-20)$ for the WOP and WP for three different strengths of perturbations
$2\%$, $4\%$ and $6\%$ respectively. For all the cases the tail of the
distribution extends on increasing width of the interface $d$, due to the
increased area of the interface and hence more number of particles inside the
junction. For all junction widths the PDF is power law with slope $f^{-1.25}$
for system WOP (as shown by $\log-\log$ plot in the main plot of fig.8(a)). It
suggests probability of finding all possible fractions of particles for system
WOP. The inset fig.8(a) shows the same plot on $\log-y$ scale, to compare that
the distribution is clearly power law, and not an exponential. For the system
WP, the distribution clearly has deviation from power law and shows a clear
peak for $f$ close to $1$ as shown in fig.8(b-d). The height of peak increases
on increasing width $d$. For larger width and small perturbation $2\%$
(fig.8(b)) a macroscopic fraction of particles spend time inside the junction.
The appearance of peak close to $f\simeq 1$ is visible more clearly in the
inset plot of fig.8(b), which is shown on $\log-y$ scale. As we increase the
perturbation further the peak at larger $f$ starts to weaken and distribution
get flattens for intermediate $f^{\prime}s$. It starts to appear more
exponential in nature for larger perturbations $\geq 4\%$ as shown in the
insets of fig.8(c-d), which are drawn on $\log-y$ scale. The exponential
nature of the tail of the distribution represents a critical fraction of
particles inside the junction. For $6\%$ perturbation (fig.8(d) (main
figure)), the distribution shows a power law decay with power $f^{-1}$ for
moderate $f^{\prime}s$. It suggests the moderate fraction of particles inside
the junction.
Here we summarise the behaviour of density of particles inside the junction.
Adding a finite perturbation ease the SPPs to get in the interface. For weak
perturbation although perturbation is enough for SPPs to enter the junction,
but not sufficient for them to overcome the randomness present there. This
lead to accumulation of particles inside the junction for some time. Hence a
macroscopic fraction of particles spend time inside the junction and $P(f)$
shows a peak at $f\simeq 1$. As we increase the perturbation it lead to
quicker entry of SPPs to the junction but now perturbation is comparable to
the randomness present inside and they experience frustration inside the
junction. This is visible by flattened power law feature of $P(f)\simeq
f^{-1}$ for higher perturbations.

Till now, we have discussed the results for the same type of SPPs in the
system. In the next section IV we show a one good application of such
geometry, where junction can be used for sorting of particles.

##  IV Junction as a particle sorter

In this section we propose that such geometry of system for the case WOP: can
also be used for the sorting two type of particles. As described earlier, for
low and intermediate junction widths the particles travel in bands even after
passing through the junction. Because of the periodic boundary condition, with
time the band passes through the junction multiple times and after some time
the particles separate into multiple narrow bands which have some dynamics in
the transverse direction as well. This motivated us to think: what will happen
if we place a mixture of two different types of particles in the system ?
To investigate that we considered a mixture of two different type of particles
(1 $\&$ 2) distributed with random orientation and position in the system.

Two type of particles differ in their response to the noise: angle of the
random vector noise $\phi\in(-0.9\pi,+1.1\pi)$ for one type and
$\phi\in(-1.1\pi,+0.9\pi)$ for the other. Hence one type particles have noise
with mean $0.1\pi$ and the noise has mean mean $-0.1\pi$ for the second types.
Hence through noise, we have introduced a random clockwise and anticlockwise
chirality for the two type of particles. Also the strength of alignment
interaction is much stronger for particles of it’s own type compared to the
other type ($1.0$ and $0.5$ respectively). Results showed that after some time
two different types of particles form separate bands (data not shown). To
characterize this, we define the phase separation order parameter for both
types ($P_{1,2}$) as:

$P_{k}=\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\bigg{|}\dfrac{N_{s}(i)-N_{d}(i)}{N_{s}(i)+N_{d}(i)}\bigg{|}$
(5)

where,$k=1,2$ is the index used to denote two different types with
$N_{1},N_{2}$ being total number of particles of respective types. For
$k^{th}$ type, $N_{s}(i),N_{d}(i)$ respectively denote number of particles of
similar and dissimilar type inside the interaction radius of $i^{th}$
particle.

$P_{k=1,2}$ has value close to $1$ if particles of $k^{th}$ type are separated
and is close to $0$ if they are mixed. We observed that for $d\in(10,15)$ two
different types of particles form bands separated from each other on two sides
of the interface and mainly moving along the $y-$ direction. We checked the
system for two other choices of random noise: $\phi\in(-0.8\pi,+1.2\pi)$ and
$\phi\in(-1.2\pi,+0.8\pi)$ for two types and find the same results. In fig.9
we show the plot of $P(d)=<P_{1,2}>$ as a function of junction width $d$.
$<..>$, stands for average over two type of particles. $P(d)$ shows a peak for
$d=14$ and then decreases on increasing and decreasing $d$ from it. Hence for
intermediate values of interface width $d\simeq 14$, the two types of
particles are maximally separated from each other. Remember this range of $d$
where we find the maximum separation is the same where we find a transition to
switching current.

Figure 9: (color online) Plot of phase separation order parameter $P(d)$ vs.
interface width d. $L\times W=200\times 5,\rho=1.0$

##  V Summary and discussion

We have studied the properties of collection of polar self-propelled particles
moving on a two dimensional rectangular channel along an order-disorder
interface with periodic boundary condition in both directions. The interaction
among the particles is taken as Vicsek type viz; particles move with constant
speed and interact through short range alignment interaction. Inside the
junction or disorder region, particles experience a high noise disorder state,
and outside they are in the ordered state. The width of the junction is
adjusted by the junction width $d$. The model is motivated by the Josephson
junction, an analogous equilibrium system in solid state Josephson (1962). We
studied the system for the two cases: (i) system WOP, where we do not impose
any easy direction for moving SPPs and (ii) system WP where a small external
biased direction of motion along the long axis of the channel is introduced.
Interestingly, flock experience more disturbance at wider junction width in
the system WP in comparison to the system WOP. On increasing width of the
junction, the system WOP shows a very small change in the global orientation
of particles, whereas for the system WP, it shows transition from macroscopic
ordered to disordered state. At the junction, we have found the current
reversal for a range of intermediate widths of the junction. Such current
reversal is due to the reflection of particles from the walls of the interface
for intermediate junction widths.

Further, we also modeled a binary system of two-types of the particle in the
system and find that the two type of particles show macroscopic phase
separation for the same intermediate width of the interface. Hence such
geometry can also be used for the sorting of different types of particles.

To the best of our knowledge this kind of study for SPPs at interface has not
been explored yet. Although similar setups have been explored in experiments
and theory for magnetic devices showing interesting properties Gurzhi  _et
al._ (2003); Schmidt  _et al._ (2001). A detail comparison of our results
obtained here with these studies is our future work. We believe that the
results presented here can be tested in experiments by designing such system.
Our study also provide new scopes in active matter systems where particles
experiences different enviournment along their move
The results presented here can be useful to understand the manufacturing of
variety of practical devices using biological agents: mechanical circuits,
switching devices, geophysical sensors, etc.

##  VI Acknowledgement

J.P, P.S.M, V.S. and S. M., thanks PARAM Shivay for computational facility
under the National Supercomputing Mission, Government of India at the Indian
Institute of Technology, Varanasi and the computational facility at I.I.T.
(BHU) Varanasi. P.S.M. thanks UGC for research fellowship. V.S. thanks DST
INSPIRE (INDIA) for the research fellowship. S.M. thanks DST, SERB (INDIA),
Project No.: CRG/2021/006945, MTR/2021/000438 for financial support.

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