#  Defect versus defect: phases of single file marching in periodic landscapes
with road blocks

Atri Goswami<EMAIL_ADDRESS>Gurudas College, 1/1, Suren Sarkar Road,
Jewish Graveyard, Phool Bagan, Narkeldanga, Kolkata 700054, West Bengal, India
Rohn Chatterjee<EMAIL_ADDRESS>Sudip Mukherjee<EMAIL_ADDRESS><EMAIL_ADDRESS>Barasat Government College, 10, KNC Road, Gupta Colony,
Barasat, Kolkata 700124, West Bengal, India

###### Abstract

We consider an inhomogeneous asymmetric exclusion processes in a closed
geometry with a point and an extended defects across which the hopping rates
are less than elsewhere. We study the stationary density profiles that result
from the competition between the point and extended defects. By using a
minimum current principle, we identify the conditions of the existence of a
localised shock or a localised domain wall. We further elucidate
delocalisation of these localised domain walls, when the strengths of the
point and extended defects follow a certain mutual relationship, which
originates again from the minimum current principle. We show that in that case
the system admits a pair of delocalised domain walls, none of which can
penetrate the extended defect. We calculate the fluctuation properties of the
domain walls and obtain the long time averaged shapes of the delocalised
domain walls.

##  I Introduction

Totally asymmetric simple exclusion process (TASEP) with open boundaries and
its related models in one dimension (1D) serve as simple models of restricted
1D transport. TASEP was originally introduced as a simple model describe
ribosome translocation along messenger RNA in eukaryotic cells macdonald1 .
TASEP has been used to describe a varieties of physical systems, e.g., ﬂuid
motion along artiﬁcial crystalline zeolitical structures fluid  and movement
of unidirectional vehicular traﬃc along roads traffic . It was subsequently
discovered to have novel boundary induced phase transitions tasep1 ; tasep2 .
See Refs. rev1 ; rev2  for reviews on TASEP. Natural systems are never pure.
Impurities or defects are ever present in condensed matter or biological
systems. In a TASEP, this can be in the form of a point or a finite fraction
of the channel (called an “extended defect”) having reduced hopping rates that
tend to hamper flow of particles. Open TASEPs with both point and extended
defects have been studied previously; see, e.g., Refs. def1 ; def2 ; def3 ;
def4 ; def5 . TASEP with periodic boundary conditions that necessarily
conserved total particle number has also been studied. A uniform TASEP on a
ring necessarily has a spatially uniform mean density in the steady states,
due to the underlying translational invariance. In contrast, an inhomogeneous
TASEP in a ring geometry explicitly breaks the translational invariance, and
hence is a candidate to produce macroscopically nonuniform steady state
densities. For instance, studies on TASEP in a ring geometry with either one
lebo  or two niladri-tasep  point defects reveal spatially nonuniform
stationary densities when the mean particle is intermediate, i.e., neither too
low (close to zero), nor too high (close to unity), the TASEP admits localised
(LDW) or delocalised (DDW) domain walls. Similarly, a periodic TASEP with one
or more extended defects can display one LDW mustansir  or more than one DDWs
tirtha-niladri  for intermediate mean densities. For sufficiently low or high
densities, the bulk density is macroscopically uniform, independent of whether
the defects in question are point or extended. Thus defects tend to become
controlling factors of the ensuing steady states in the systems when there are
sufficient number of particles in the system.

An important issue concerning the role of defects in TASEP is the interplay
and competition between multiple defects of different types that result into
nonuniform steady states in the TASEP. Being motivated by this generic issue,
we here study a periodic TASEP having one point defect and one extended
defect. Apart from its theoretical motivations, this study would be useful to
understand the phenomenologies of ribosome translocations along closed mRNA
loops (circular translation of polysomal mRNA) with clusters of slow codons
along which ribosome translocations are inhibited mrna1 ; mrna11 ; mrna2 ;
mrna3 ; mrna4 ; mrna5 . The principal result from our work is that for
intermediate values of the mean density, both there can be an LDW due to both
the point and extended defects. This leads to competition between them. We
invoke a minimum current principle to decide the dominant defect. Furthermore,
when the strengths (defined below) of the point and extended defects maintain
a particular relation, for intermediate densities one find a pair of DDWs
instead of a single LDW. For very low and high densities, LD and HD phases
follow.

##  II Model

We consider an inhomogeneous asymmetric exclusion process in a ring geometry
with $L$ sites; see Fig. 1 for a schematic diagram of the model. The model has
a single slow site at $i=1$ and an extended slow section from
$i=\epsilon_{1}\,L$ to
$(\epsilon_{1}+\epsilon_{2})L,\,\epsilon_{1},\epsilon_{2}<1$. The hopping rate
across the single slow site at $i=1$ is assumed to be $p<1$, and the extended
slow segment has a hopping rate $q<1$. In general $p\neq q$. Elsewhere in the
system, the hopping rate is unity. The particles are assumed move in the
anticlockwise direction. Periodic boundary condition is imposed on the system.
The following microscopic processes fully describe the dynamics of the model:

(a) At any site $i=1$ to $\epsilon_{1}\,L,\,\epsilon_{1}<1$, particles hop in
an anticlockwise manner to the neighboring site at rate unity, subject to
exclusion.

(b) At any site $i=\epsilon_{1}\,L$ to
$(\epsilon_{1}+\epsilon_{2})L,\,\epsilon_{1}+\epsilon_{2}<1$, particles hop in
an anticlockwise manner to the neighboring site at rate $q<1$, subject to
exclusion.

(c) At any site $i=(\epsilon_{1}+\epsilon_{2})\,L$ to $i=L$, particles hop in
an anticlockwise manner to the neighboring site at rate unity, subject to
exclusion.

(d) From site $i=L$ to 1, particles hop in an anticlockwise manner to the
neighboring site at rate $p<1$, subject to exclusion.

Figure 1: Schematic model diagram. The thick bluish green circle marks the
point defect across which the hopping rate is $p<1$. The arc in red is the
extended defect along which the hopping rate is $q<1$. In the remaining
segments, the hopping rate is unity (see text).

##  III Steady state density profiles

###  III.1 Mean-field theory

In this Section, we set up and analyze the MFT equations. Let $n_{i}$ be the
occupation at site $i$. In the MFT approximation, correlations are neglected
and averages of products are replaced by products of averages blythe . While
MFT is an adhoc approximation, it provides a good analytically tractable
guideline to the steady state densities and phase diagrams. It is convenient
to consider the system to be composed of three segments - $T_{1}$ between
$i=1$ to $\epsilon_{1}L$,$T_{2}$ between $\epsilon_{1}L$ to
$(\epsilon_{1}+\epsilon_{2})L$ and $T_{3}$ between
$(\epsilon_{1}+\epsilon_{2})L$ to $L$, connected serially. The MFT equation
for the density $n_{i}$ in the different segments reads

$\displaystyle\frac{dn_{i}}{dt}$ $\displaystyle=$ $\displaystyle
n_{i+1}[1-n_{i}]-n_{i}[1-n_{i-1}],\;1<i<\epsilon_{1}L,$ (3) $\displaystyle=$
$\displaystyle qn_{i+1}[1-n_{i}]-qn_{i}[1-n_{i-1}],$
$\displaystyle\;\;\;\;\;\;\epsilon_{1}L\leq
i\leq(\epsilon_{1}+\epsilon_{2})L,$ $\displaystyle=$ $\displaystyle
n_{i+1}[1-n_{i}]-n_{i}[1-n_{i-1}],$
$\displaystyle\;\;\;\;\;\;(\epsilon_{1}+\epsilon_{2})L<i\leq L.$

We note that the above mean-field equations are invariant under the
transformations $n_{i}\rightarrow 1-n_{L-i-1}$, which defines the particle
hole symmetry in this model. Before proceeding further it is convenient
introduce a quasi-continuous coordinate $x\equiv i/L$ in the thermodynamic
limit $L\rightarrow\infty$. Thus, we have $0\leq x\leq 1$. Now defining
$\rho_{a}(x)\equiv\langle n^{a}_{i}\rangle$, where $\langle...\rangle$ implies
temporal averages in the steady states and $a=1,\,2$ or $3$ for the three
segments, the MFT equations in the steady states yield the stationary currents
$J_{1},\,J_{2},\,J_{3}$ in the three segments as

$\displaystyle J_{1}$ $\displaystyle=$ $\displaystyle\rho_{1}(1-\rho_{1}),$
(4) $\displaystyle J_{2}$ $\displaystyle=$ $\displaystyle
q\rho_{2}(1-\rho_{2}),$ (5) $\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle\rho_{3}(1-\rho_{3}),$ (6)

where $\rho_{1},\,\rho_{2}$ and $\rho_{3}$ are the stationary densities in
$T_{1},\,T_{2}$ and $T_{3}$ respectively. Current conservation means

$J_{1}=J_{2}=J_{3}$ (7)

in the steady states. Each of $T_{1},T_{2},T_{3}$ can be in its LD, HD, MC
phase individually, similar to an open TASEP. In addition, they can also be in
their domain wall (DW) phase. Furthermore, particle number conservation (PNC)
gives

$\int_{0}^{\epsilon_{1}}\rho_{1}dx+\int_{\epsilon_{1}}^{\epsilon_{1}+\epsilon_{2}}\rho_{2}dx+\int_{\epsilon_{2}+\epsilon_{1}}^{1}\rho_{3}dx=n,$
(8)

where $n=\sum_{i}\,n_{i}/L$ is the mean particle density in the system.

Steady state currents (4)-(6) together with the particle number conservation
(8) can be used to calculate the steady state densities
$\rho_{1},\,\rho_{2},\,\rho_{3}$. A complete characterization of the steady
states of this model requires specifying the phases in all of
$T_{1},\,T_{2},\,T_{3}$, i.e., solving for all of
$\rho_{1},\,\rho_{2},\,\rho_{3}$. While an open TASEP can be in LD, HD and MC
phases, a TASEP in a closed system can be in domain wall (DW) phase in an
extended region of the space of the control parameters lebo ; niladri-tasep ;
mustansir ; hinsch ; tirtha-niladri ; parna-anjan ; astik-parna . While each
of $T_{1},T_{2},T_{3}$ can be in any of these four phases under various
conditions, we will see below that the conditions of current conservation and
PNC ensure that not all of the phases are actually admissible in the present
model. This forms a principal result from this work. In general, all of
$T_{1},\,T_{2},\,T_{3}$ may be in the same phases, or different phases. We
name the former as identical phase solutions and the latter as distinct phase
solutions. Before we make detailed quantitative analysis, we can make the
following general comments. We note that $J_{1}=J_{3}$ gives
$\rho_{1}=\rho_{3}$ or $\rho_{1}=1-\rho_{3}$ in the steady state. We shall see
below that the relation between $\rho_{2}$ and $\rho_{1}$ (or $\rho_{3}$) is
more complex due to $q<1$.

In order to validate our mean-field theory, we have also performed stochastic
simulations (using a random sequential update algorithm) to determine the
phase diagrams and calculate the stationary densities, and we compare with our
mean-field theory predictions; see Section III.6 for more details on the
numerical simulations.

####  III.1.1 Identical phase solutions in MFT

We first explore the possibilities of all the three channels are in the same
phase, which form the identical phase states of the system. In this case, the
possible phases of the system are LD-LD-LD, HD-HD-HD and MC-MC-MC phases. We
use (7) and (8) to determine the phases.

LD-LD-LD phase:- For sufficiently low mean densities $n$, all the channels
should be sparsely populated, and hence we expect all the three segments to be
in their LD phases with uniform densities $\rho_{1}=\rho_{3}<\rho_{2}<1/2$
respectively. These may be obtained as follows. By using $J_{1}=J_{2}=J_{3}$

$\rho_{1}(1-\rho_{1})=q\rho_{2}(1-\rho_{2})=\rho_{3}(1-\rho_{3}).$ (9)

Since $q<1$, we expect $\rho_{1}=\rho_{3}<\rho_{2}$ in the LD phase. Form (9),
we get

$\rho_{1}=\rho_{3}=\frac{1}{2}[1\pm\sqrt{1-4q\rho_{2}(1-\rho_{2})}].$ (10)

Since we are looking for the LD-LD-LD phase, we choose

$\rho_{1}=\rho_{3}=\frac{1}{2}[1-\sqrt{1-4q\rho_{2}(1-\rho_{2})}]<\frac{1}{2}.$
(11)

Now using (8) we get

$[1-\epsilon_{2}]\rho_{1}+\rho_{2}\epsilon_{2}=n<\frac{1}{2}$ (12)

for the LD phase.

For explicit solutions of the densities in the LD-LD-LD phase, (9) together
with (12) can be used to give

$\rho_{1}=\left[-B\pm\sqrt{B^{2}-4AC}\right]\frac{1}{2A},$ (13)

as the two general solutions, where

$\displaystyle A$ $\displaystyle=$
$\displaystyle(1-\epsilon_{2})^{2}-\epsilon_{2}^{2}/q,$ (14) $\displaystyle B$
$\displaystyle=$
$\displaystyle\frac{\epsilon_{2}^{2}}{q}-(1-\epsilon_{2})(2n-\epsilon_{2}),$
(15) $\displaystyle C$ $\displaystyle=$ $\displaystyle n^{2}-\epsilon_{2}n.$
(16)

So far we have not imposed any conditions of the LD-LD-LD phase on the
solutions (13). The pertinent question then is: Which of the two solutions in
(11) is to be considered as the LD-LD-LD phase density solution? To settle
this, we use the fact that in the limiting case with vanishing particle
number, i.e., with $n\rightarrow 0$, the LD-LD-LD phase solution must smoothly
go to zero. This consideration allows us to pick the right solution in (13)
for the LD-LD-LD phase: we choose the solution that vanishes as $n\rightarrow
0$. Which one among the two in (13) does that depends upon the signs of $A$
and $B$, i.e., will be decided by $\epsilon_{2}$ and $n$. We thus note that
the point defect has no macroscopic effect on the steady state density
profiles. Instead at $x=0$, the location of the point defect, there is a local
peak of height $h$ and vanishing width in the thermodynamic limit, with
$h=n(1-p)/p$, such that the local density at $x=0$ is $\rho_{1}+h$ niladri-
tasep . This essentially acts as a boundary layer between $T_{1}$ and $T_{3}$.
Steady state density profiles in the LD-LD-LD phase with
$n=0.15,p=0.2,\,q=0.45$ and $n=0.15,p=0.15,\,q=0.6$ with a system size
$L=9000$ and $\epsilon_{1}=\epsilon_{2}=1/3$ are shown in Fig. 2. Good
agreement between MFT and MCS results are observed.

Figure 2: LD-LD-LD density plots with $n=0.15$: (top) $p=0.2,\,q=0.45$,
(bottom) $p=0.15,\,q=0.6$. MFT (blue lines) and MCS (red points) results are
shown. Good agreement between MFT and MCS results are observed.

HD-HD-HD phase:- The HD-HD-HD phase can be analysed by applying the particle-
hole symmetry on the LD-LD-LD phase. The specific solution for the HD-HD-HD
phase for a given set of $\epsilon_{2},\,n$ can be obtained by first obtaining
the corresponding LD-LD-LD phase solution following the above arguments and
then applying the particle-hole symmetry. Physically, for very high $n$, all
of $T_{1},T_{2}$ and $T_{3}$ should be nearly filled with particles, and hence
HD-HD-HD phase is expected. In this case, $\rho_{1},\rho_{2},\rho_{3}>1/2$ and
$\rho_{1}=\rho_{3}>\rho_{2}$. Using current conservation, we find

$\rho_{1}=\rho_{3}=\frac{1}{2}[1+\sqrt{1-4q\rho_{2}(1-\rho_{2})}]>\frac{1}{2}.$
(17)

For an explicit solution for the HD-HD-HD phase, we again consider (13) and
note that if $n=1$, unsurprisingly $\rho_{1}=1$ is a solution, which in turn
means $\rho_{2}=1$ and $\rho_{3}=1$, i.e., a completely filled up system.
Thus, when the system is nearly filled and all of $T_{1},T_{2},T_{3}$ are in
their HD phases, we should accept that particular solution in (13) which
smoothly reaches unity when $n\rightarrow 1$. Which of the two solutions in
(13) will satisfy this property depends on the signs of $A,\,B$.

Similar to the LD-LD-LD phase, the point defect has no macroscopic effect on
the stationary densities. Instead, one has a local valley of vanishing width
in the thermodynamic limit and a depth $h^{\prime}=(1-n)(1-p)/p$, such that at
$x=0$, the density is $\rho_{1}-h^{\prime}$. Steady state density profiles in
the HD-HD-HD phase with $n=0.8,p=0.2,\,q=0.45$ and $n=0.8,p=0.15,\,q=0.6$ with
a system size $L=9000$ and $\epsilon_{1}=\epsilon_{2}=1/3$ are shown in Fig.
3. Good agreement between MFT and MCS results are observed.

Figure 3: HD-HD-HD density plots: (top) $n=0.8,\,p=0.2,\,q=0.45$, (bottom)
$n=0.8,\,p=0.15,\,q=0.6$. MFT (blue lines) and MCS (red points) results are
shown. Good agreement between MFT and MCS results are observed.

MC-MC-MC phase:- In the MC phase, the density should be 1/2. This means in
these putative MC phases, $J_{1}=J_{3}=1/4,\,J_{2}=q/4$. This immediately
shows that there is no MC-MC-MC phase in the model, i.e., all the three
segments cannot be simultaneously in the MC phases, as that would violate the
current conservation condition (9).

####  III.1.2 Distinct phase solutions in MFT

We now consider the possibilities when each of $T_{1},T_{2}$ and $T_{3}$ can
be in different, spatially uniform or even nonuniform stationary states
separately. These are different phase states of the system. Let us first
consider what happens when we start adding particles when the system is in its
LD-LD-LD phase with a sufficiently low $n$. With addition of particles,
$\rho_{1}=\rho_{3}<1/2$ and $\rho_{2}<1/2$ will rise, maintaining
$J_{1}=J_{2}=J_{3}$ and $\rho_{1}=\rho_{3}<\rho_{2}<1/2$ (i.e., the LD-LD-LD
phase). As $n$ keeps rising, the LD-LD-LD phase ultimately gives way to
nonuniform phases. We argue below that this can happen in two distinct ways.

Point defect dominated steady states:- As $n$ rises, the LD-LD-LD phase is no
longer the desired stationary state. This can be seen from the fact that the
height $h$ of the local peak at $x=0$ in the LD-LD-LD phase continues to rise
and the steady state current decreases as $p$ decreases. This cannot continue
indefinitely. For instance, as argued in Refs. lebo ; niladri-tasep , in a
periodic TASEP with just one point defect, as $n$ reaches a lower threshold

$n=\frac{p}{1+p}$ (18)

the point defect starts to have macroscopic effects in the form of a domain
wall; see also Ref. erwin-defect . Physically, for a very low $n$, the system
should be in its LD-LD-LD phase. The upon addition of particles, $n$ rises. As
it rises, it eventually reaches a lower threshold, a macroscopically
nonuniform steady state in the shape of a DW is formed. This threshold should
obviously depend on the defect strengths. For a single point defect in an
otherwise homogeneous ring (i.e., no extended defect, $q=1$) is as given in
(18). This should however be different in the present model (marked
$n_{cL}^{p}$ below; see below), when $q<1$. A DW, essentially a pile up of
particles due to a bottleneck or a defect, should form behind the point
defect, extending an HD phase up to a point $x_{w}$ and an LD phase from
$x_{w}$ to 1. Hence it is first formed in $T_{1}$, as $n$ just exceeds
$n_{cL}^{p}$ with the DW position $x_{w}$ being at $x=0$. As $n$ rises further
above this lower threshold, the size of this pile or the DW increases by
shifting its position $x_{w}$ (see below for a formal definition) along the
ring, moving from $T_{1}$ first to $T_{2}$ and then to $T_{3}$, finally
reaching $x=1$, at which point the DW ends at an threshold $n_{cU}^{p}$ of $n$
(see below), bringing the model to its HD-HD-HD phase. If $n$ is still
increased, the system continues to remain in its HD-HD-HD phase till $n=1$ is
reached, when the system gets fully filled. See sm1  for a related movie
(Movie 1) that visually presents this picture. Below we provide a quantitative
analysis that corroborates this intuitive picture.

First let us assume that the DW is located outside $T_{2}$, and is either in
$T_{1}$ or in $T_{3}$, i.e., $0<x_{w}<\epsilon_{1}$, or
$\epsilon_{1}+\epsilon_{2}<x_{w}<1$. In this case, $\rho_{2}$, the density of
$T_{2}$, is spatially uniform. The DW being located either in $T_{1}$ or
$T_{3}$, connects a low density segment with density $\rho_{\text{LD}}$ and a
high density segment with density $\rho_{\text{HD}}$. We have

$\rho_{\text{LD}}=\frac{p}{1+p},\,\rho_{\text{HD}}=\frac{1}{1+p}.$ (19)

Using current conservation (7),

$q\rho_{2}(1-\rho_{2})=\rho_{\text{LD}}(1-\rho_{\text{LD}})=\frac{p}{(1+p)^{2}}.$
(20)

This is quadratic equation in $\rho_{2}$ that has two solutions

$\displaystyle\rho_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1\pm\\{1-\frac{4p}{q(1+p)^{2}}\\}^{1/2}\right]$
(21) $\displaystyle=$ $\displaystyle\rho_{2+}(>1/2),\,\rho_{2-}(<1/2).$

Clearly, these solutions are physically acceptable, so long as
$4p/[q(1+p)^{2}]<1$, or equivalently,

$\frac{p}{(1+p)^{2}}<\frac{q}{4}.$ (22)

We will come back to this condition below and re-interpret this in terms of
the stationary currents.

Since the particles are hopping in the anticlockwise direction, subject to
exclusion, any pile up of particles due to the point defect at $x=0$ will
start just behind it, i.e., in $T_{1}$, and a DW is created. As more and more
particles are added, the size of this “pile” of particles or the DW grows, and
the DW position $x_{w}$ starts to move away from $x=0$. As a specific example,
consider the case when the DW is inside $T_{1}$: $0\leq
x_{w}\leq\epsilon_{1}$. Since the particles flow in the anticlockwise
direction, and a DW should form behind a bottleneck (which in the present case
is the point defect at $x=0$), we write

$\displaystyle\rho_{1}(x)$ $\displaystyle=$
$\displaystyle\rho_{\text{HD}},\;\;0\leq x\leq x_{w},$ (23) $\displaystyle=$
$\displaystyle\rho_{\text{LD}},\;\;x_{w}\leq x\leq\epsilon_{1},$

In this situation, we must have the entire $T_{2}$ to be its LD phase:
$\rho_{2}(x)=\rho_{2-}$ for $\epsilon_{1}\leq x\leq\epsilon_{1}+\epsilon_{2}$.
In addition, $T_{3}$ too should be in its LD phase, with
$\rho_{3}=\rho_{\text{LD}}$. We can now apply PNC to determine $x_{w}$. We get

$\rho_{\text{HD}}x_{w}+\rho_{\text{LD}}(\epsilon_{1}-x_{w})+\rho_{2-}\epsilon_{2}+\rho_{\text{LD}}(1-\epsilon_{1}-\epsilon_{2})=n.$
(24)

Equation (24) gives $x_{w}$ as a function of $p,q$ and $n$. Simplifying (24),
we obtain

$x_{w}=\left(\frac{1+p}{1-p}\right)[n-\rho_{2-}\epsilon_{2}-\rho_{\text{LD}}(1-\epsilon_{2})].$
(25)

Since a unique solution for $x_{w}$ can be obtained from (25), the DW has a
fixed position and hence is an LDW. For $x_{w}>0$ but $<1$, there is an LDW in
the bulk of $T_{1}$, making $\rho_{1}$ nonuniform, corresponding to the DW-LD-
LD phase. See Fig. 4 for a plot showing an LDW in $T_{1}$
($p=0.2000,n=0.35,q=0.65,L=9000,\epsilon_{1}=\epsilon_{2}=1/3$). Both MFT and
MCS results are shown. If we set $x_{w}=0$, all of $T_{1},\,T_{2}$ and $T_{3}$
are in their LD phases, and is in fact the boundary between the LD-LD-LD phase
and DW-LD-LD phase. We get

$\frac{p}{1+p}(1-\epsilon_{2})+\frac{\epsilon_{2}}{2}\left[1-\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]=n=n_{cL}^{p},$
(26)

a lower threshold on $n$, such that as $n$ exceeds $n_{cL}^{p}$ distinct phase
solutions appear with the disappearance of the LD-LD-LD phase. As an example,
if $T_{1},\,T_{2},\,T_{3}$ have the same size, i.e., if
$\epsilon_{1}=\epsilon_{2}=1/3$, we get

$\frac{2}{3}\frac{p}{1+p}+\frac{1}{6}\left[1-\sqrt{1-\frac{4p}{q(1+p)^{2}}}\right]=n_{cL}^{p}.$
(27)

Furthermore, if we set $\epsilon_{2}=0$, i.e., effectively no extended defect,
then $n_{cL}=p/(1+p)$, in agreement with Ref. niladri-tasep . On the other
hand, at $x_{w}=\epsilon_{1}$ entire $T_{1}$ is in its HD phase; as $x_{w}$
exceeds $\epsilon_{1}$, $T_{2}$ is now its DW phase. Thus,
$x_{w}=\epsilon_{1}$ gives HD-LD-LD phase and is the boundary between the DW-
LD-LD phase and HD-DW-LD phase. We get

$\displaystyle\frac{\epsilon_{1}}{1+p}+\frac{\epsilon_{2}}{2}\left[1-\sqrt{1-\frac{4p}{q(1+p)^{2}}}\,\right]$
$\displaystyle+\frac{p}{1+p}(1-\epsilon_{1}-\epsilon_{2})=n=n_{p1}.$ (28)

In the particular case with $\epsilon_{1}=\epsilon_{2}=1/3$, we get

$\frac{1}{3(1+p)}+\frac{1}{6}\left[1-\sqrt{1-\frac{4p}{q(1+p)^{2}}}\,\right]+\frac{p}{3(1+p)}=n=n_{p1}.$
(29)

When the LDW is inside $T_{2}$, $T_{1}$ is in its HD phase, hence
$\rho_{1}=\rho_{\text{HD}}$, and $T_{3}$ is in its LD phase, with
$\rho_{3}=\rho_{\text{LD}}$. Further, since the LDW is located inside $T_{2}$,
we must have $\rho_{2}=\rho_{+}$ for $\epsilon_{1}<x<x_{w}$;
$\rho_{2}=\rho_{-}$ for $x_{w}<x<\epsilon_{1}+\epsilon_{2}$. Applying PNC we
get

$\rho_{\text{HD}}\epsilon_{1}+(x_{w}-\epsilon_{1})\rho_{2+}+(\epsilon_{1}+\epsilon_{2}-x_{w})\rho_{2-}+\rho_{\text{LD}}(1-\epsilon_{1}-\epsilon_{2})=n.$
(30)

Solving (30), we get

$\displaystyle x_{w}$ $\displaystyle=$
$\displaystyle\frac{1}{(\rho_{2+}-\rho_{2-})}\bigg{[}n+\rho_{2+}\epsilon_{1}-\rho_{\text{HD}}\epsilon_{1}$
(31) $\displaystyle-$
$\displaystyle(\epsilon_{1}+\epsilon_{2})\rho_{2-}-\rho_{\text{LD}}(1-\epsilon_{1}-\epsilon_{2})\bigg{]}$

giving a unique position of the LDW in $T_{2}$. See Fig. 5 for an LDW in
$T_{2}$ ($p=0.2,n=0.5,q=0.65,L=9000$). The overall density profile, consisting
of HD phase in $T_{1}$ and LD phase in $T_{3}$, with an intervening LDW in
$T_{2}$ takes the form of a step-like structure. This is specifically
attributed to $T_{2}$ being an extended defect with a hopping rate $q<1$.
Setting $x_{w}=\epsilon_{1}$ gives the boundary between the DW-LD-LD and HD-
DW-LD phases (28). Next, if we set $x_{w}=\epsilon_{1}+\epsilon_{2}$ in (30),
we get the HD-HD-LD phase, which gives the boundary between the HD-DW-LD and
HD-HD-DW phases:

$\displaystyle\frac{\epsilon_{1}}{1+p}+\frac{\epsilon_{2}}{2}\left[1+\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]$
(32) $\displaystyle+$
$\displaystyle\frac{p}{1+p}(1-\epsilon_{1}-\epsilon_{2})=n=n_{p2}.$

For the specific choice of equal-sized $T_{1},T_{2},T_{3}$,
$\epsilon_{1}=\epsilon_{2}=1/3$, we get

$\frac{1}{3}+\frac{1}{6}\left[1+\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]=n_{p2}.$
(33)

Finally, consider an LDW in $T_{3}$. Thus,
$\rho_{3}(x)=\rho_{\text{HD}},\;\epsilon_{1}+\epsilon_{2}<x<x_{w}$,
$\rho_{3}(x)=\rho_{\text{LD}},\;x_{w}<x<1$. Further,
$\rho_{1}(x)=\rho_{\text{HD}}$, $\rho_{2}(x)=\rho_{+}$, since both $T_{1}$ and
$T_{2}$ are in their HD phases. Then applying PNC, we obtain

$\displaystyle\frac{\epsilon_{1}}{1+p}+\frac{\epsilon_{2}}{2}\left[1+\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]$
$\displaystyle+(x_{w}-\epsilon_{1}-\epsilon_{2})\frac{1}{1+p}+\frac{(1-x_{w})p}{1+p}=n.$
(34)

Solving, we find

$x_{w}=\frac{1+p}{1-p}\left[n+\frac{\epsilon_{2}}{1+p}-\frac{p}{1+p}-\frac{\epsilon_{2}}{2}\left[1+\sqrt{1-\frac{4p}{q(1+p)^{2}}}\right]\right]$
(35)

as the LDW position in $T_{3}$. See Fig. 6 for an LDW in $T_{3}$, where both
MFT and MCS results are shown ($p=0.2,n=0.65,q=0.65,L=9000$). Setting
$x_{w}=\epsilon_{1}+\epsilon_{2}$ gives HD-HD-LD phase, which exists on the
boundary (32) between the HD-DW-LD and HD-HD-DW phases. On the other hand, if
we set $x_{w}=1$, the LDW ends and the system is in its boundary between the
HD-HD-DW phase and HD-HD-HD phase. We find

$\displaystyle\frac{\epsilon_{1}}{1+p}+\frac{\epsilon_{2}}{2}\left[1+\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]$
(36) $\displaystyle+$
$\displaystyle(1-\epsilon_{1}-\epsilon_{2})\frac{1}{1+p}=n=n_{cU}^{p},$

an upper threshold on $n$, such that as $n$ exceeds $n_{cU}^{p}$, identical
phase solutions reappear. For equal-sized $T_{1},T_{2},T_{3}$, we get

$\frac{1}{3}+\frac{1}{6}\left[1+\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]=n_{cU}^{p}.$
(37)

Further, if we set $\epsilon_{2}=0$, i.e., a vanishing extended defect, we get
$n_{cU}^{p}=1/(1+p)$, as found in Ref. niladri-tasep .

Our MFT and MCS results on LDWs in $T_{1},\,T_{2},\,T_{3}$ are shown in Fig.
4, Fig. 5 and Fig. 6 respectively. We find reasonable agreement between our
MFT and MCS results.

Figure 4: Plot of an LDW in $T_{1}$ in the point defect dominated regime with
$p=0.2,n=0.35,q=0.65,L=9000,\epsilon_{1}=\epsilon_{2}=1/3$. MFT (blue lines)
and MCS (red points) results are shown (see text). Figure 5: Plot of an LDW in
$T_{2}$ in the point defect dominated regime with
$p=0.2,n=0.5,q=0.65,L=9000,\epsilon_{1}=\epsilon_{2}=1/3$. MFT (blue lines)
and MCS (red points) results are shown (see text). Figure 6: Plot of an LDW in
$T_{3}$ in the point defect dominated regime with
$p=0.2,n=0.65,q=0.65,L=9000,\epsilon_{1}=\epsilon_{2}=1/3$. MFT (blue lines)
and MCS (red points) results are shown (see text).

Extended defect dominated steady states:- So far in the above, we have studied
the situation where the point defect dominates by assuming that by adding
particles $n$ reaches a threshold $n_{cL}^{p}$ above which the point defect
enforces macroscopically nonuniform steady states. Alternatively, a rising $n$
can reach a different lower threshold, denoted $n_{cL}^{q}$ below, at which
the extended defect enforces a DW. As $n$ rises, both $\rho_{1}=\rho_{3}$ and
$\rho_{2}$ rise while being in their LD phases. Eventually, as $n$ reaches
$n_{cL}^{q}$, $T_{2}$ reaches its MC phase with $\rho_{2}=1/2$ in the bulk,
and $\rho_{1}=\rho_{3}<1/2$. As $n$ is further increased beyond $n_{c}^{q}$, a
DW is formed in the system, just behind the bottleneck, which in the present
case is $T_{2}$. This means a DW is first formed at
$x=\epsilon_{1}+\epsilon_{2}$ in $T_{3}$ as soon as $n$ reaches $n_{c}^{q}$.
As $n$ rises further, the DW starts moving along $T_{3}$, crossing over to
$T_{1}$ at $x=0$, and then finally reaching $x=\epsilon_{1}$ at an upper
density threshold $n_{cU}^{q}$, when the DW ends. Any further increase in $n$
will push the system to the HD-HD-HD phase. Thus, the DW never enters $T_{2}$,
which remains in its MC phase. This is a fundamental difference with the LDW
formed due to the point defect. This physical description is presented in a
movie (Movie 2) in SM sm1 . Below we provide a quantitative analysis that
corroborates this intuitive picture. We now work out the corresponding
quantitative details that corroborates this intuitive picture.

Since $T_{2}$ is in its MC phase, we must have $\rho_{2}=1/2$ and $J_{2}=q/4$.
Current conservation then yields

$\rho(1-\rho)=\frac{q}{4},$ (38)

where $\rho$ is $\rho_{1}$ or $\rho_{3}$. Solving,

$\rho=\frac{1}{2}\left[1\pm\sqrt{1-{q}}\right]=\rho_{+}(>1/2),\rho_{-}(<1/2),$
(39)

giving the densities of the high density and low density parts of the DW,
which meet at $x_{w}$. As before, $x_{w}$ can be calculated by using PNC.
Assume an LDW in $T_{3}$. This is the LD-MC-DW phase. Then PNC gives

$\rho_{-}\epsilon_{1}+\frac{\epsilon_{2}}{2}+(1-x_{w})\rho_{-}+(x_{w}-\epsilon_{1}-\epsilon_{2})\rho_{+}=n.$
(40)

Setting $x_{w}=\epsilon_{1}+\epsilon_{2}$ gives the condition for transition
from the LD-LD-LD phase to LD-MC-DW phase, when the DW is due to the extended
defect. We get

$\epsilon_{1}\rho_{-}+\frac{\epsilon_{2}}{2}+(1-\epsilon_{1}-\epsilon_{2})\rho_{-}=n.$
(41)

This is independent of $p$, but depends upon $q$ through the dependence of
$\rho_{-}$ on $q$. Using (39), we find

$(1-\epsilon_{2})\frac{1}{2}\left[1-\sqrt{1-{q}}\right]+\frac{\epsilon_{2}}{2}=n\equiv
n_{cL}^{q},$ (42)

defining a critical density $n_{c}^{q}$ above which an LDW due to the extended
defect appears. For the particular case of $\epsilon_{1}=\epsilon_{2}=1/3$, we
get

$\frac{1}{3}\left[1-\sqrt{1-{q}}\right]+\frac{1}{6}=n_{cL}^{q}.$ (43)

If we set $\epsilon_{2}=1/2$, i.e., the extended defect covers half of the
ring, as case discussed in Refs. mustansir ; tirtha-niladri  without any point
defect (i.e., $p=1$), we get

$\frac{1}{4}\left[1-\sqrt{1-{q}}\right]+\frac{1}{4}=n_{cL}^{q}.$ (44)

If $x_{w}=1$, the system is at the boundary between its LD-MC-HD phase and DW-
MC-HD phase, which is also the location of the (assumed subdominant in this
Section) point defect. Now setting $x_{w}=1$ together with
$\epsilon_{1}=1/4,\,\epsilon_{2}=1/2$ in (40), we find $n=1/2$, which is
independent of $q$. We thus conclude that so long as the condition for an LDW
due to the extended defect is satisfied, in the half-filled limit ($n=1/2$)
with $\epsilon_{1}=1/4,\,\epsilon_{2}=1/2$, there is always an LDW at the
location of the point defect, independent of the value of $q$ mustansir .

Now consider an LDW in $T_{1}$. In this case, $T_{3}$ is in the HD phase. PNC
gives

$x_{w}\rho_{+}+\frac{\epsilon_{2}}{2}+(1-\epsilon_{1}-\epsilon_{2})\rho_{+}+(\epsilon_{1}-x_{w})\rho_{-}=n.$
(45)

Setting $x_{w}=\epsilon_{1}$ produces the boundary between the DW-MC-HD phase
and HD-HD-HD phase:

$(1-\epsilon_{2})\rho_{+}+\frac{\epsilon_{2}}{2}=n.$ (46)

Substituting for $\rho_{+}$, we get

$(1-\epsilon_{2})\frac{1}{2}\left[1+\sqrt{1-{q}}\right]+\frac{\epsilon_{2}}{2}=n\equiv
n_{cU}^{q},$ (47)

defining an upper threshold on $n$, such that for $n>n_{cU}^{q}$, identical
phase solutions are predicted. In the special case of $\epsilon_{2}=1/3$, we
get

$\frac{1}{3}\left[1+\sqrt{1-q}\right]+\frac{1}{6}=n_{cU}^{q}.$ (48)

Furthermore, in the case where $\epsilon_{2}=1/2$, we find

$\frac{1}{4}\left[1+\sqrt{1-{q}}\right]+\frac{1}{4}=n_{cU}^{q},$ (49)

see also Refs. mustansir ; tirtha-niladri . Our MFT and MCS results on LDWs in
$T_{3}$ and $T_{1}$ are shown in Fig. 7 and Fig. 8 respectively. We find
reasonable agreement between our MFT and MCS results.

Figure 7: Plot of an LDW in $T_{3}$ in the extended defect dominated regime
with $p=0.2,n=0.35,q=0.55,L=9000,\epsilon_{1}=\epsilon_{2}=1/3$. MFT (blue
lines) and MCS (red points) results are shown (see text). Figure 8: Plot of an
LDW in $T_{1}$ in the extended defect dominated regime with
$p=0.2,n=0.65,q=0.55,L=9000,\epsilon_{1}=\epsilon_{2}=1/3$. MFT (blue lines)
and MCS (red points) results are shown (see text).

####  III.1.3 Minimum current principle and stationary state selections

We have discussed the possibilities of nonuniform steady steady states that
can borne out of two distinct mechanisms - being controlled by the point
defect or by the extended defect. We must then be able to specify which one
will be observed for a given set of $p,q$. To do this end, we invoke a minimum
current principle: among the possible steady state solutions, the one with the
lowest stationary current should actually be observed.

To proceed further, we note that stationary current $J_{\text{PD}}$
corresponding to a domain wall due to the point defect is

$J_{\text{PD}}=\frac{p}{(1+p)^{2}},$ (50)

which depends only on $p$. Similarly, the stationary current $J_{\text{ED}}$
corresponding corresponding to a domain wall due to the extended defect is

$J_{\text{ED}}=\frac{q}{4}.$ (51)

Minimum current principle stipulates that when

$J_{\text{PD}}<(>)J_{\text{ED}},$ (52)

the nonuniform steady states are controlled by the point defect (extended
defect).

When $J_{\text{PD}}=J_{\text{ED}}$, the two defects compete, as we illustrate
below, a new kind of states - a pair of delocalised domain walls (DDW)
emerges. The condition $J_{\text{PD}}=J_{\text{ED}}$ gives a unique relation
between $p$ and $q$, which in the MFT is

$\frac{p}{(1+p)^{2}}=\frac{q}{4},$ (53)

\- a line in the $p-q$ plane, on which the two defects compete and
consequently a pair of DDWs emerge on this line in the $p-q$ plane. The
numerical analogue of the line (53) has been measured in our MCS simulations
by determining the emergence of a pair of DDWs for a particular $p,q$ pair.
Both the MFT line (53) and the corresponding MCS results are shown in Fig. 9.
We notice a small disagreement between the two, which we believe is due to the
fluctuation effects ignored in the MFT.

Figure 9: Plot of Eq. (53) in the $p-q$ plane (continuous line) and the
corresponding MCS results (points) (see text).

###  III.2 Domain walls and the extended defect size

In the previous Section, we have discussed the minimum current principle in
selecting the steady states. This analysis implicitly assumes selecting
between an LDW due to the point defect and an LDW due to the extended defect.
This holds when the density $n$ is above both the critical densities
$n_{cL}^{p}$ and $n_{cL}^{q}$ for a domain wall due to the point defect and
extended defect. However, $n_{cL}^{p}$ and $n_{cL}^{q}$ are in general
unequal. Consider now a situation with $J_{\text{PD}}<J_{\text{ED}}$, i.e.,
the point defect dominates according to our previous analysis. Now consider

$\displaystyle
n_{cL}^{q}-n_{cL}^{p}=(1-\epsilon_{2})\frac{1}{2}\left[1+\sqrt{1-{q}}\right]+\frac{\epsilon_{2}}{2}$
(54) $\displaystyle-$
$\displaystyle\frac{p}{1+p}(1-\epsilon_{2})-\frac{\epsilon_{2}}{2}\left[1-\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right].$

Now consider the extreme limit of $\epsilon_{2}\rightarrow 1$, which means the
size of $T_{2}$ is much greater than those of $T_{1}$ or $T_{3}$. In that
limit,

$n_{cL}^{q}-n_{cL}^{p}=\frac{1}{2}\sqrt{1-\frac{4p}{q(1+p)^{2}}}>0,$ (55)

so long as $\frac{q}{4}-\frac{p}{(1+p)^{2}}>0$, which is equivalent to
$J_{\text{PD}}<J_{\text{ED}}$, which in turn is what we have assumed. On the
other hand, in the opposite limit of $\epsilon_{2}\rightarrow 0$, i.e., when
$T_{1}$ or $T_{3}$ (or both) is much larger than $T_{2}$,

$n_{cL}^{q}-n_{cL}^{p}<0$ (56)

if

$\frac{q}{4}<\frac{p}{(1+p)^{2}},$ (57)

which contradicts the condition $J_{\text{PD}}<J_{\text{ED}}$. Therefore,
$n_{cL}^{q}-n_{cL}^{p}>0$ necessarily for $\epsilon_{2}\rightarrow 0$, just as
it is for $\epsilon_{2}\rightarrow 1$. Can $n_{cL}^{q}-n_{cL}^{p}$ be negative
for intermediate values of $\epsilon_{2}$, i.e., for $0<\epsilon_{2}<1$? For
this to happen, $n_{cL}^{q}-n_{cL}^{p}$ must flip sign for even number of
times (at least twice) for intermediate values of $\epsilon_{2}$. Since (54)
is linear in $\epsilon_{2}$, such a possibility is ruled out. Therefore,
$n_{cL}^{q}-n_{cL}^{p}>0$ for all admissible values of $\epsilon_{2}$. This
then means that with $J_{\text{PD}}<J_{\text{ED}}$, $n_{cL}^{q}-n_{cL}^{p}>0$
necessarily. Hence, as $n$ rises from a small value, it will first meet
$n_{cL}^{p}$, creating an LDW due to the point defect.

###  III.3 Phase diagram for $\epsilon_{1}=1/3=\epsilon_{2}$

With $\epsilon_{1}=1/3=\epsilon_{2}$, the phases are determined by the mutual
interplay between the defect strengths $p,\,q$ and the mean density $n$. A
list of the phases is given in Table 1.

List of the phases
---
Phases |  Types |  Condition for existence |  Dominant defect
LD-LD-LD |  Identical |  $n<n_{cL}^{p}$ with $J_{\text{PD}}<J_{\text{ED}}$ or $n<n_{cL}^{q}$ with $J_{\text{ED}}<J_{\text{PD}}$ |  None
HD-HD-HD |  Identical |  $n>n_{cU}^{p}$ with $J_{\text{PD}}<J_{\text{ED}}$ or $n>n_{cU}^{q}$ with $J_{\text{ED}}<J_{\text{PD}}$ |  None
DW-LD-LD |  Distinct |  $n_{p1}>n>n_{cL}^{p}$ with $J_{\text{PD}}<J_{\text{ED}}$ |  Point defect
HD-DW-LD |  Distinct |  $n_{p2}>n>n_{p1}$ with $J_{\text{PD}}<J_{\text{ED}}$ |  Point defect
HD-HD-DW |  Distinct |  $n_{cU}^{p}>n>n_{p2}^{p}$ with $J_{\text{PD}}<J_{\text{ED}}$ |  Point defect
LD-MC-DW |  Distinct |  $1/2>n>n_{cL}^{q}$ with $J_{\text{ED}}<J_{\text{PD}}$ |  Extended defect
DW-MC-HD |  Distinct |  $n_{cU}^{q}>n>1/2$ with $J_{\text{ED}}<J_{\text{PD}}$ |  Extended defect
DDW-MC-DDW |  Distinct |  $n_{cU}^{p}=n_{cU}^{q}>n>n_{cL}^{q}=n_{cL}^{p}$ with $J_{\text{ED}}=J_{\text{PD}}$ |  Point and extended defects equally dominant
Table 1: List of the phases on the segments $T_{1},T_{2},T_{3}$.

In this Section, we present a phase diagram of the model in the $p-q-n$ space.
To proceed, we first list the phase boundaries between the different phases.

(i) Transition between the LD-LD-LD to DW-LD-LD phases (controlled by the
point defect):

$\frac{2p}{1+p}+\frac{1}{2}\left[1-\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]=3n.$
(58)

(ii) Transition between the HD-HD-DW to HD-HD-HD phases (controlled by the
point defect):

$\frac{2}{1+p}+\frac{1}{2}\left[1+\sqrt{1-\frac{4p}{q(1+p)^{2}}}\;\right]=3n.$
(59)

(iii) Transition between LD-LD-LD to LD-MC-DW phases (controlled by the
extended defect):

$\sqrt{1-{q}}=3(\frac{1}{2}-n).$ (60)

This is independent of $p$.

(iv) Transition between HD-HD-HD and DW-MC-HD phases (controlled by the
extended defect):

$-\sqrt{1-{q}}=3(\frac{1}{2}-n).$ (61)

Again, this is independent of $p$. The mean-field phase diagram in the $p-q-n$
space is shown in Fig.10

Figure 10: Mean-field phase diagram of the model in the $p-q-n$ space showing
the phases in the model. HD and DW (with one LDW) phases are marked. LD phase
that lies below the DW phase in the figure is not visible. Points on the gray
curved plane lying as a divider of the DW phase have
$J_{\text{PD}}=J_{\text{ED}}$ [see Eq. (53)] satisfied and hence have a pair
of DDWs fully or partially covering $T_{1}$ and $T_{3}$. In the DW phase
region, on the right of this curved surface (i.e., with
$J_{\text{PD}}<J_{\text{ED}}$, there is an LDW due to the point defect in
either $T_{1},T_{2}$ or $T_{3}$; on the left of this plane (i.e., with
$J_{\text{ED}}<J_{\text{PD}}$, one has an LDW due to the extended defect in
$T_{1}$ or $T_{3}$ and correspondingly an MC phase in $T_{2}$. In the HD (LD)
phase region, one has HD-HD-HD (LD-LD-LD) phase in $T_{1},T_{2},T_{3}$ (see
text).

###  III.4 Jump in the LDW position across $J_{\text{PD}}=J_{\text{ED}}$

We focus on the case $\epsilon_{1}=1/3=\epsilon_{2}$. Consider the LD-MC-HD
phase that comes with an LDW at $x_{w}=0$, which is also the location of the
point defect. This LDW is due to the extended defect. Using (41),

$\frac{1}{6}\left[1-\sqrt{1-{q}}\;\right]+\frac{1}{6}+\frac{1}{6}\left[1+\sqrt{1-{q}}\;\right]=n,$
(62)

giving $n=1/2$, independent of $p$ and $q$. Thus, so long as the extended
defect “dominates”, i.e., $J_{\text{ED}}<J_{\text{PD}}$ is satisfied, there is
always an LDW at $x_{w}=0$ with $n=1/2$.

What happens in the other limit, i.e., $J_{\text{ED}}>J_{\text{PD}}$ with
$n=1/2$? In this case, the point defect controls the steady state. Now, if
there is an LDW in $T_{1}$, then

$\frac{1-p}{1+p}x_{w}+\frac{2p}{3(1+p)}+\frac{1}{6}\left[1-\\{1-\frac{4p}{q(1+p)^{2}}\\}^{1/2}\right]=n.$
(63)

Since the LDW is assumed to be confined in $T_{1}$, max$(x_{w})$=1/3. Since
$n$ rises with $x_{w}$ monotonically, we have

$\text{max}({n})=\frac{1}{3}+\frac{1}{6}\left[1-\\{1-\frac{4p}{q(1+p)^{2}}\\}^{1/2}\right]<\frac{1}{2}.$
(64)

On the other hand, if there is an LDW in $T_{3}$,

$\displaystyle\frac{1}{3(1+P)}+\frac{1}{6}\left[1-\\{1-\frac{4p}{q(1+p)^{2}}\\}^{1/2}\right]$
(65) $\displaystyle+$
$\displaystyle(x_{w}-\frac{2}{3})\frac{1}{1+P}+(1-x_{w})\frac{P}{1+P}=n.$

Now, min$(n)$ is when $x_{w}=2/3$. This gives

$\text{min}(n)=\frac{1}{3}+\frac{1}{6}\left[1+\\{1-\frac{4p}{q(1+p)^{2}}\\}^{1/2}\right]>\frac{1}{2}.$
(66)

Therefore if $n=1/2$, the LDW cannot be in $T_{1}$ or $T_{3}$; it must be
somewhere in $T_{2}$. Now, assuming an LDW in $T_{2}$,

$\frac{1}{2}-(x_{w}-\frac{1}{2})\sqrt{1-\frac{4p}{q(1+p)^{2}}}=n.$ (67)

Thus, if $n=1/2$, $x_{w}=1/2$, i.e., the LDW, now formed due to the point
defect, appears at the mid point of the extended defect or $T_{2}$.

We thus note that with $n=1/2$ the positions $x_{w}$ of the LDW are at
diametrically opposite points $x_{w}=0$ and $x_{w}=1/2$, when controlled by
the point defect and extended defect respectively. Let and $n=1/2$ and
$p_{c},\,q_{c}$ satisfy $J_{\text{PD}}=J_{\text{ED}}$. Then if
$p=p_{c}-\delta_{1},\,q=q_{c}+\delta_{2},\,\delta_{1},\,\delta_{2}>0$ (but
$\delta_{1},\,\delta_{2}$ are very small) $J_{\text{PD}}<J_{\text{ED}}$, we
have an LDW at $x_{w}=1/2$ due to the point defect. On the other hand, if
$\delta_{1},\,\delta_{2}<0$, $J_{\text{ED}}<J_{\text{PD}}$ and hence an LDW at
$x_{w}=0$. Thus, as one moves ${\cal
O}(\delta),\,\delta=\delta_{1},\,\delta_{2}$ being vanishingly small, the DW
position changes by an ${\cal O}(1)$ amount - from $x_{w}=0$ to $x_{w}=1/2$.
Thus, the domain wall position as a function of $\delta_{1},\delta_{2}$ (or
$p,q$) shows a jump across the line $J_{\text{PD}}=J_{\text{LD}}$. This may be
described in a more quantitative way. Let $\delta\equiv
2\sqrt{\delta_{1}^{2}+\delta_{2}^{2}}$ be the “distance” between the two
points $(p=p_{c}-\delta_{1},\,q=q_{c}+\delta_{2})$ and
$p=p_{c}+\delta_{1},\,q=q_{c}-\delta_{2}$ in the $p-q$ plane. Then,
$dx_{w}/d\delta$ diverges for small $\delta$. Such divergences can occur only
when the two points in the $p-q$ plane lie on the two sides of the line
determined by the condition $J_{\text{PD}}=J_{\text{ED}}$.

###  III.5 Delocalised DW

Let us now consider the steady states on the critical line in the $p-q$ plane
given by $J_{\text{PD}}=J_{\text{ED}}$. On this line, there should be a domain
wall due to the point defect, so long as $n>n_{cL}^{p}$ for a particular value
of $p$. Let us assume that it is at $x=x_{w}^{\text{PD}}$. On the other hand,
there should be a domain wall, say at $x=x_{w}^{\text{ED}}$, due to the
extended defect, provided $n>n_{cL}^{q}$. Thus, a complete description of the
stationary density profiles require enumeration of both $x_{w}^{\text{PD}}$
and $x_{w}^{\text{ED}}$. However, with just one condition, viz., the PNC at
hand, both $x_{w}^{\text{PD}}$ and $x_{w}^{\text{ED}}$ cannot be uniquely
determined. Rather only a linear relation between the two is obtained. This
then means any pair of ($x_{w}^{\text{PD}}$, $x_{w}^{\text{ED}}$) satisfying
PNC is a valid solution, giving positions of the two LDW’s. Since an LDW due
to the extended defect must be confined to $T_{1}$ and $T_{3}$ only, other LDW
due to the point defect must also be confined to the same (even though an LDW
due to an isolated point defect with $J_{\text{PD}}<J_{\text{ED}}$ can also be
in $T_{2}$). Thus, each of ($x_{w}^{\text{PD}}$, $x_{w}^{\text{ED}}$) must be
confined to $T_{1}$ or $T_{2}$ only. The inherent stochasticity of the
dynamics implies all such valid ($x_{w}^{\text{PD}}$, $x_{w}^{\text{ED}}$)
pairs satisfying PNC will be visited over time. This further means long-time
averages of the densities take the form of inclined straight lines. See sm1
for a related movie (Movie 3) that visually presents this picture, showing a
pair of DDWs. Below we provide a quantitative analysis that corroborates this
intuitive picture.

Consider an specific example with $n_{cL}^{p},\,n_{cL}^{q}<n<1/2$ together
with $J_{\text{ED}}=J_{\text{PD}}$. We then expect a DW behind the point
defect in $T_{1}$ at, say, $x_{w}^{\text{PD}}$, and another behind the
extended defect in $T_{3}$ at, say, $x_{w}^{\text{ED}}$. The condition
$J_{\text{ED}}=J_{\text{PD}}$ ensures that both the DWs must have the same
height given by $(1-p)/(1+p)$, with
$\rho_{\text{HD}}=1/(1+p),\,\rho_{\text{LD}}=p/(1+p),\,\rho_{2}=1/2$. Applying
PNC,

$\displaystyle\rho_{\text{HD}}x_{w}^{\text{PD}}+\rho_{\text{LD}}(\epsilon_{1}-x_{w}^{\text{PD}})+\frac{\epsilon_{2}}{2}+\frac{x_{w}^{\text{ED}}-\epsilon_{1}-\epsilon_{2}}{1+p}$
$\displaystyle+(1-x_{w}^{\text{ED}})\frac{p}{1+p}=n.$ (68)

Clearly, (68) gives only a linear relations connecting $x_{w}^{\text{PD}}$ and
$x_{w}^{\text{ED}}$, and not each of the positions separately. As a result, a
pair of DDWs is observed. MFT cannot predict the profiles of the DDWs, as it
neglects fluctuations. However, we can employ arguments based on symmetry to
construct the DDW profiles. First consider the special case
$\epsilon_{1}=1/3=\epsilon_{2}$. Equation (68) simplifies to

$\displaystyle\rho_{\text{HD}}x_{w}^{\text{PD}}+\rho_{\text{LD}}(\frac{1}{3}-x_{w}^{\text{PD}})+\frac{1}{6}+\frac{x_{w}^{\text{ED}}-2/3}{1+p}$
$\displaystyle+(1-x_{w}^{\text{ED}})\frac{p}{1+p}=n.$ (69)

Statistically, with equal-sized segments $T_{1},T_{2},T_{3}$ both the DDWs
must symmetric, i.e., the configuration of the long time averaged envelop of
the DDW in $T_{1}$ must be same as the configuration of the long time averaged
envelop of the DDW in $T_{3}$. To know the midpoint for each of them, we
hypothetically replace each of them by an LDW of height $(1-p)/(1+p)$,
connecting $\rho_{\text{HD}}=1/(1+p)$ and $\rho_{\text{LD}}=p/(1+p)$. Such
(hypothetical) replacements evidently satisfy the PNC. Symmetry of the problem
dictates that if $x_{0}$ is the position of the LDW in $T_{1}$, the
corresponding LDW in $T_{3}$ must be located at $x_{0}+2/3$. This ensures that
each of the LDW has HD (or LD) phase parts with the same length, as necessary
from the symmetry considerations. Now in terms of these two LDWs, the PNC
reads

$2x_{0}\frac{1}{1+p}+2(\frac{1}{3}-x_{0})\frac{p}{1+p}+\frac{1}{6}=n.$ (70)

This gives

$2x_{0}\frac{1-p}{1+p}=n-\frac{1}{6}-\frac{2}{3}\left(\frac{p}{1+p}\right).$
(71)

Clearly, if $n=1/2$, $x_{0}=1/6$, which means the midpoint of the LDW is at
the midpoint of $T_{1}$ or $T_{3}$. If $n<1/2$, $x_{0}<1/6$, whereas $n>1/2$,
$x_{0}>1/6$. Our MCS results on DDWs in $T_{1},\,T_{3}$ are shown in Fig. 11,
Fig. 12 and Fig. 13 respectively. The analytically obtained DDW profiles are
also shown.

Figure 11: Plot of a pair of DDWs in $T_{1}$ and $T_{3}$ with
$n=0.5,\,p=0.2,\,q=0.5933,\,L=9000,\,\epsilon_{1}=\epsilon_{2}=1/3$. Each DDW
spans the entire respective segments $T_{1}$ or $T_{3}$ from our MCS results
(red points). The analytically obtained DDW profiles (blue lines) are also
shown (see text). Figure 12: Plot of a pair of DDWs in $T_{1}$ and $T_{3}$
with $n=0.35,\,p=0.2,\,q=0.5933,\,L=9000,\,\epsilon_{1}=\epsilon_{2}=1/3$.
Each DDW spans the respective segments $T_{1}$ or $T_{3}$ partially with the
remaining part being in the LD phase from our MCS results (red points). The
analytically obtained DDW profiles (blue lines) are also shown (see text).
Figure 13: Plot of a pair of DDWs in $T_{1}$ and $T_{3}$ with
$n=0.65,\,p=0.2,\,q=0.5933,\,L=9000,\,\epsilon_{1}=\epsilon_{2}=1/3$. Each DDW
spans the respective segments $T_{1}$ or $T_{3}$ partially with the remaining
part being in the HD phase from our MCS results (red points). The analytically
obtained DDW profiles (blue lines) are also shown (see text).

###  III.6 MCS algorithm

In our stochastic simulations, we have used sequential updating of the sites
of the TASEP with a periodic boundary condition. We perform stochastic
simulations of the model subject to the update rules (a)-(d) described above
in Sec. II by using a random sequential updating scheme rand . In each
iteration, we choose a site in the TASEP random-sequentially for updating. The
stationary densities are calculated by temporal averaging, once the system
reaches steady states. This naturally produces time-averaged, space-dependent,
steady state density profiles, which are then used to calculate the phase
diagram.

##  IV Summary and outlook

We have thus studied a periodic TASEP with a point and an extended defects, of
strengths $p$ and $q$. We have used a combination of MFT and extensive MCS
studies to systematically explore how the interplay between the trio of the
two defect strengths and mean number density ultimately result into the
stationary density profiles. The MFT and MCS studies results agree reasonably
well. We show that while for small (closed to zero) and high (closed to unity)
mean densities, the TASEP should be in either the LD or HD phases. For mean
densities in intermediate ranges, i.e., when it is neither too high nor too
low, domain walls are observed. In general there is a single domain wall which
is pinned to a particular point in the TASEP lane, i.e., an LDW is formed,
with its height and location being functions of either $p$ or $q$ and $n$, the
mean density. There is a fundamental distinction between an LDW formed by a
point defect, and the one formed by an extended defect: While the former can
be found anywhere in the ring including inside the extended defect segment,
with its position being continuously tuned by $n$, the mean density, the
latter must always be outside of the extended defect segment. We have further
shown that for intermediate values of $n$, controlled by $p$ and $q$, there
can a pair of DDWs instead one LDW, when $p$ and $q$ maintains a specific
relationship. These DDWs must remain outside the extended defect segment.
These DDWs can be partial or full, i.e., they can jointly cover only a part of
the remaining part of the ring, or the whole of the remaining ring (i.e.,
excluding the extended defect). We invoke a minimum current principle to
explain the LDW either due to a point defect or an extended defect, and also
the pair of DDWs. We have argued that the latter appear when certain
conditions on the steady state currents controlled by of the point defect or
the extended defect hold.

In the present work, we have focussed on a periodic TASEP with just one point
defect and one uniform extended defect as a prototype system to explore the
consequence of competitions between defects of different types. In possible
natural realisations of the system, there can be more than one point and
extended defects of different strengths. Furthermore, these extended defect
segments themselves need not be uniform, i.e., can have spatially varying
hopping in the segment. A TASEP with open boundaries or in a ring geometry
having spatially continuously varying hopping rates can already give
surprisingly nontrivial stationary densities atri ; tirtha-qkpz . Now adding
competition between such spatially nonuniform hopping rates with point defects
are expected to introduce complex density profiles and phase diagrams. Yet
another direction of research that follows from our work would be introduce
time-dependent defects, where the either strength or location or both can be
changing. Thus the defects become annealed as opposed to being quenched in the
present study. Such annealed defects can introduce further elements of
competition due to the possible fast to slow defect dynamics relative to the
TASEP hopping rates. It would also be interesting to study the effects of
coupling an inhomogeneous TASEP with diffusive processes on the steady states
atri2 . We hope our work here will stimulate further research along these
directions.

##  V Acknowledgement

S.M. thanks SERB (DST), India for partial financial support through the CRG
scheme [file: CRG/2021/001875].

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