# Pinned or moving: states of a single shock in a ring

Parna Roy<EMAIL_ADDRESS>Condensed Matter Physics Division, Saha
Institute of Nuclear Physics, Calcutta 700064, West Bengal, India  Anjan Kumar
Chandra<EMAIL_ADDRESS>Malda College, Malda 732101, West Bengal, India
Abhik Basu<EMAIL_ADDRESS>Condensed Matter Physics Division, Saha
Institute of Nuclear Physics, Calcutta 700064, West Bengal, India

###### Abstract

Totally asymmetric exclusion processes (TASEP) with open boundaries are known
to exhibit moving shocks or delocalised domain walls (DDW) for sufficiently
small equal injection and extraction rates. In contrast TASEPs in an
inhomogeneous ring have been shown to display pinned shocks or localised
domain walls (LDW) under similar conditions [see, e.g., H. Hinsch and E. Frey,
Phys. Rev. Lett. 97, 095701 (2006)]. By studying periodic exclusion processes
composed of a driven (TASEP) and a diffusive segments, we uncover smooth
transitions between LDW and DDW; the latter mimics DDWs in an open TASEP,
controlled essentially by the fluctuations in the diffusive segment. Mean-
field theory together with Monte Carlo simulations are employed to
characterize the emerging nonequilibrium steady states. Our studies provide an
explicit route to control the degree of shock fluctuations in periodic
systems, and should be relevant in cell biological transport where the
availability of molecular motors is the rate limiting constraint.

##  I Introduction

Totally asymmetric simple exclusion process (TASEP) with open boundaries was
originally proposed as a simple model for the motion of molecular motors in
eukaryotic cells pipkin . Subsequently, it was reinvented as a paradigmatic
one-dimensional model for nonequilibrium statistical mechanics, that shows
boundary induced phase transitions characterised by $\alpha$ and $\beta$, the
entry and exit rates derrida . Extensive Monte-Carlo simulations (MCS)
supplemented by mean field theories (MFT) reveal that for $\alpha=\beta<1/2$,
the steady state density profile shows a moving shock or delocalised domain
wall (DDW) that moves randomly and unrestrictedly along the whole TASEP tasep-
basic . This is usually explained in terms of completely uncorrelated entry
and exit events.

Different situations can emerge in TASEPs in closed rings. For TASEP on a
homogeneous ring, translational invariance ensures a uniform mean density
along the ring. Macroscopically nonuniform steady state densities can emerge
only when translational invariance is explicitly broken. This can happen in a
variety of ways. For instance, TASEP in a ring with a single bottleneck or a
defect site with a lower hopping rate than the remaining system can show a
pinned shock or localised domain wall (LDW) for moderate average densities. In
yet another manifestation of breakdown of translational invariance, a closed
system composed of two segments of equal size - a TASEP and a diffusive lane
with exclusion, executing what is known as symmetric exclusion process or SEP
- also shows an LDW in the steady states for moderate average densities hinsch
. In contrast, inhomogeneous TASEP in a ring with strict particle number
conservation can display two or more DDWs only when there are more than one
bottlenecks of equal strength niladri1 ; tirtha1 . To our knowledge, a single
DDW in a closed heterogeneous TASEP has never been observed till the date. It
is thus pertinent to ask: can a single LDW observed in closed TASEPs in
heterogeneous rings be converted into a single DDW, resembling open TASEPs? If
so, under what situations and how - is it a smooth or sudden transition?

In the present work, we address this issue systematically by studying a class
of conceptual models constructed by us. We principally focus on the nature of
a single shock in a periodic system with strict number conservation and study
inter-conversions between an LDW and a DDW. To this end, we propose and
construct three different but related periodic model, each composed of a TASEP
part and a diffusive part and study the ensuing nonequilibrium steady states
(NESS). All of these models fall in the class of a TASEP connected to a
reservoir with finite resources (i.e., a given total number of particles),
with the reservoir having its own internal dynamics modeled as one-dimensional
(1D) diffusion. We show that in all the models studied here (i) a single DDW
can be formed in a periodic system with large average particle content in the
diffusive segment; the latter can be used to control interconversion of LDW
and DDW, (ii) this interconversion between LDW and DDW is continuous, i.e.,
the span of a DDW can be continuously shrunk to zero converting it into an
LDW. All the models that we construct and study here bear out this physical
picture, underscoring the robustness of the mechanism for deconfinement of an
LDW elucidated in this article. The remaining part of this article is
structured as follows: In Sec. II.2 we discuss construction of our models,
which are (i) Model IA and Model IB, where the dynamics in the diffusive
segment does not respect any exclusion principle, and (ii) Model II, which
enforces exclusion in the diffusive channel. In Sec. III.1 the steady state
density profiles of Model I is discussed and in Sec. III.2 the steady state
density profiles of Model II is discussed. We summarise and discuss our
results in Sec. IV. Some calculational details including phase diagrams are
given in Appendices at the end.

##  II Models for periodic exclusion processes

We consider a periodic 1D model composed of a driven TASEP (${\cal T}$) and a
diffusive (${\cal S}$) parts. In ${\cal T}$, hopping of particle is
unidirectional at unit rate that is subject to exclusion, where as in $\cal S$
particles jump independently and randomly to the neighbouring site at rate $D$
with equal probability to left and right. Particle dynamics in $\cal S$ may or
may not be subject to exclusion. We consider both the possibilities for $\cal
S$. Particle exchanges between $\cal T$ and $\cal S$ are allowed at the
junction with dynamical rules governed by the originating site. Depending upon
the detailed dynamical rules for $\cal S$ we consider three different models
as described below.

###  II.1 Domain walls in closed heterogeneous TASEPs

Before we embark upon our studies, it is useful to recall the existing studies
and results on domain walls in closed inhomogeneous TASEP. Some of these model
studies have a single heterogeneity or a bottleneck in the form of a point
lebo  or extended gautam  bottleneck, or an intervening diffusive segment
hinsch . All these display a single LDW for appropriate choices of the model
parameters and particle number densities. Other studies include multiple
bottlenecks, point niladri1  or extended tirtha1 , and can display not just an
LDW, but multiple DDWs as well in some regions of the phase space. The DDWs in
these models always appear more than one in number. These contrasting
behaviours are by now well-understood within MFT. For instance, in the models
of Refs. hinsch ; lebo ; gautam , where only a single LDW may be observed
under appropriate conditions, strict particle number conservation ensures a
unique solution for the domain wall position within the MFT that is consistent
with an LDW. In contrast, for models in Refs. niladri1 ; tirtha1 , where
multiple DDWs can be found for specific choices of the parameters, particle
number conservation does not yield unique solutions for the individual domain
wall positions, but rather gives only relations between them leaving them
overall undetermined. Thus there would be many solutions of the positions of
the domain walls, all of which maintain the strict particle number
conservation. Since all these solutions are equally probable and visited by
the systems in course of time due to the inherent stochasticity of the
underlying dynamics, long time averages yield envelops of these all possible
domain wall solutions which are nothing but DDWs. Thus these DDWs necessarily
appear more than one in number. All these are in contrast to an open TASEP,
where a single system-spanning DDW and not an LDW can be observed. The latter
is ascribed to the lack of strict particle number and uncorrelated entry and
exit events in open TASEP. It is thus tempting to conclude that in a closed
heterogeneous TASEP there cannot be a transition between a conventional single
LDW to a single DDW that resembles a single DDW in an open TASEP. We however
argue below that the existing arguments that favour a single LDW in a closed
heterogeneous TASEP do not consider the particle number fluctuations in the
TASEP segment of the heterogeneous ring (which is non-zero even though the
total particle number remains conserved). When the latter is accounted for,
possibilities of delocalisation of an LDW opens up. Indeed for sufficiently
large fluctuations in the TASEP segment - that is controlled by the
fluctuations in the non-TASEP part of the ring due to overall particle number
conservation - an LDW fully delocalise to take the form of a single DDW that
spans the whole length of the TASEP segment. In order to establish this
behaviour, we propose and study a class of conceptual models below.

###  II.2 Models

We study two related conceptual models, each of which consists a diffusive
part $\cal S$ and a driven part $\cal T$; see Fig. 1 for a generic schematic
diagram for all the two models we consider here.

Figure 1: Schematic diagram of the models. Particles in segment $\cal S$
diffuse, where as in $\cal T$ they undergo dynamics controlled by asymmetric
exclusion processes.

The asymmetric segment $\cal T$ has $N$ sites, where as the diffusive segment
$\cal S$ has $\tilde{N}=rN$ sites with $r$ can be smaller or larger than
unity. In order to define the models formally, we denote the location of the
lattice sites and occupation numbers by $x_{i}\in[1,N]$ and $n_{i}$ for $\cal
T$ and $\tilde{x_{i}}\in[1,rN],\,r\geq 1$ and $\tilde{n_{i}}$ for $\cal S$
respectively. Total number of particles

$N_{p}=\sum_{i=1}^{N_{1}}n_{i}+\sum_{i=1}^{N_{2}}\tilde{n_{i}}$ (1)

is conserved.

The driven segment $\cal T$ is identical in both the models. These two models
are however distinguished by their respective diffusive segments $\cal S$. In
details:

(i) In Model I, Unlike in $\cal T$, we do not impose any condition of
exclusion in $\cal S$, i.e., a site in $\cal S$ is here allowed to accommodate
any number of particles without restrictions. As a result, the over all
particle density $n_{p}=\frac{N_{p}}{N(1+r)}$ is not restricted to $[0,1]$.
Furthermore, particles can exit $\cal T$ at a given rate $\beta$ to enter into
$\cal S$ that is completely unaffected by the occupation of the site
$\tilde{i}=1$ in $\cal S$. Thus, $\beta$ is a tuning parameter for Model I.
Thus, the NESS of Model I are parametrised by three parameters: $D,n_{p}$ and
$\beta$.

(ii) In contrast in Model II, we impose exclusion in $\cal S$. Thus the
dynamics in the whole ring in Model II is subject to exclusion - each site in
$\cal T$ or $\cal S$ can be occupied by at most one particle. Hence, the
particle density $n_{p}=\frac{N_{p}}{N(1+r)}\in[0,1]$ necessarily, since there
could be at most one particle per site. This directly generalises the model
studied in Ref. hinsch . Notice that the rate at which particles can exit
$\cal T$ and move to $\cal S$ does depend on the occupation at $\tilde{i}=1$
\- a direct consequence of exclusion in $\cal S$. Thus Model II is a two-
parameter model - $n_{p}$ and $D$. For reasons similar to those discussed in
Ref. hinsch , we let diffusivity $D$ scales with the system size that ensures
a non-zero steady state current in the system (see also below).

All the models are shown to display similar deconfinement of LDWs as the total
particle content of $\cal S$ rises, a feature that is attributed to enhanced
fluctuations in $\cal S$ (see below).

##  III Steady state density profiles

We use MFT together with extensive Monte-Carlo simulation (MCS) of our model
to obtain the steady state density profiles. In MFT approaches, the system is
considered as a collection of one TASEP ($\cal T$) with open boundaries having
effective entry and exit rates and a diffusive lane ($\cal S$) again with open
boundaries having its own effective entry and exit rates hinsch . These
effective rates are determined from the condition of particle current
conservation in the steady states. We then use them in conjunction with the
known results for TASEP and diffusive lane with open boundaries to obtain the
density profiles here. An isolated open TASEP in steady state can be in three
different states, the low density (LD), high density (HD) and maximal current
(MC) phases; we expect to find analogues of these phases for $\cal T$.

We denote the steady state density at a particular site $m$ in $\cal T$ as
$\rho_{m}=\langle n_{m}\rangle$ and in $\cal S$ as
$\tilde{\rho}_{m}=\langle\tilde{n}_{m}\rangle$, where $\langle..\rangle$
represent time-averages in the NESS. MF analysis entails taking continuum
limit with $\rho(x)$ and $\tilde{\rho}(\tilde{x})$ as the densities in $\cal
T$ and $\cal S$, where $x=i/N$ and $\tilde{x}=\tilde{i}/N$. In the
thermodynamic limit (TL), $L\gg 1$ and consequently $x$ and $\tilde{x}$ vary
effectively continuously with $0\leq x\leq 1,\,0\leq\tilde{x}\leq r$. We
further introduce the following notations for the stationary densities at the
junctions B and A respectively; $\rho(x=0)=\alpha$ and $\rho(x=1)=(1-\beta)$
at $\cal T$ according to the standard TASEP convention and
$\tilde{\rho}(\tilde{x}=0)=\gamma$ and $\tilde{\rho}({\tilde{x}=r})=\delta$
for $\cal S$. In what follows below, we use the continuum labeling $x$ for the
densities in NESS. The MFT analysis is complemented by extensive MCS studies
using random sequential updates.

###  III.1 Domain walls in Model I

In Model I, multiple occupancy in each site of $\cal S$ is allowed. In NESS,
$\tilde{\rho}(\tilde{x})$ is given by a linear profile in MFT:

$\tilde{\rho}(\tilde{x})=\delta+(\gamma-\delta)\tilde{x}/r.$ (2)

The corresponding current in $\cal S$ is given by

$J_{\cal S}=(\gamma-\delta)D/Nr.$ (3)

This must be equal to $J_{\cal T}\sim{\cal O}(1)$ in NESS. Now $J_{\cal S}$,
as given in (3), can be ${\cal O}(1)$, e.g., when (i)
$\gamma,\;\delta\sim{\cal O}(N)$, so that the difference
$\gamma-\delta\sim{\cal O}(N)$ together with $D\sim{\cal O}(1)$. We call this
Model IA or (ii) $D\sim{\cal O}(N)$, but $\gamma,\;\delta\sim{\cal O}(1)$. We
call this Model IB. This is significantly different from Model IA foot11 .

It is instructive to first consider the phases and the ensuing phase diagrams
of Model IA and Model IB in qualitative terms. Comparing with an open TASEP,
LD and HD phases are to be found for $\alpha<\beta,\;\alpha<1/2$ and
$\beta<\alpha,\;\beta<1/2$, respectively, where as, for $\alpha,\;\beta\geq
1/2$ MC phase ensues. Further, with $\alpha=\beta\leq 1/2$ one DDW that spans
the whole of an open TASEP is found. Unlike in an open TASEP, where both
$\alpha$ and $\beta$ are free parameters that can be tuned, in Model IA or IB
$\alpha$ is to be determined from the various conditions available (see
below), while $\beta$ remains free, the other free parameters being $n_{p}$
and $D$. Consider the phases for $\beta<1/2$ in Model IA. By tuning $n_{p},D$
$\alpha$ may be varied, as shown below. Thus in the $n_{p}-D$ plane, regions
with $\alpha<\beta<1/2$ correspond to LD phase; the remaining regions where
$\alpha>\beta<1/2$ correspond to HD phase. However, there is no MC phase for
$\beta<1/2$. Further, regions correspond to $\alpha=\beta<1/2$ imply domain
walls, investigation of whose nature is a primary goal of this work. In
contrast when $\beta>1/2$, in the $n_{p}-D$ plane one obtains LD phase in the
region with $\alpha<1/2$. In the remaining region for which
$\alpha>1/2,\beta>1/2$, MC phase is found. Thus, there are LD and MC phase
with no HD phase. This physical picture remains unchanged in Model IB. In the
main text, we focus on the nature of the domain wall and set $\beta<1/2$. The
detailed phase diagrams for Model IA and Model IB obtained from MFT and MCS
studies are given in the Appendices.

We assume a domain wall with a mean position $x_{w}$. Thus steady state
density in $\cal T$

$\rho(x)=\beta+\Theta(x-x_{w})(1-2\beta),$ (4)

where we have used $\alpha=\beta$, which is known since $\beta$ is a given
external tuning parameter; here $\Theta(x)$ is the Heaviside $\Theta$-function
at $x$. The corresponding steady state current in $\cal T$ is given by

$J_{\cal T}=\alpha(1-\alpha)=\beta(1-\beta).$ (5)

The incoming current to site $x=0$ at $\cal T$ is $\delta D(1-\alpha)$. By
using the current conservation in the steady state, we obtain

$\delta D(1-\alpha)=\beta(1-\beta).$ (6)

Since multiple occupancy is allowed in $\cal S$, the mean particle number
$N_{s}$ in $\cal S$ in the NESS can be both larger or smaller than
$\tilde{N}=Nr$. Again due to possible multiple occupancy, both $\gamma$ and
$\delta$ can be ${\cal O}(N)$. As long as there is a particle in the last site
$i_{N}$ of $\cal T$, it jumps to first site $\tilde{i}_{1}$ of $\cal S$ with
probability $\beta$, independent of how many particles are already there in
$\tilde{i}_{1}$. Thus, $\beta$ is independent of $\gamma$.

####  III.1.1 Domain walls in Model IA

Consider a domain wall in Model IA. Here, $D\sim{\cal O}(1)$. Domain wall
position $x_{w}$ can be calculated from the conservation of total particle
number. This requires solving for $\tilde{\rho}(\tilde{x})$, which in turn
requires knowledge of $\gamma$ and $\delta$ in terms of the model parameters.
This can be done by using the conservation of the current in the system in the
steady state, i.e., by equating $J_{\cal S}$ in (3) with $J_{\cal T}$ given by
(5). Straight forward algebra, whose details interested readers can find in
Appendix A, gives for $x_{w}$

$x_{w}(2\beta-1)+(1-\beta)+\frac{Nr^{2}}{2D}[\beta(1-\beta)+\frac{\beta}{Nr}]+\frac{\beta
r}{2D}=n_{p}(1+r).$ (7)

Equation (7) clearly yields $x_{w}$ uniquely in terms of $\beta,n_{p},r,D$.
Since $0<x_{w}<1$, $n_{p}$ must be small enough to make (7) valid. Since the
mean position $x_{w}$ is fixed, this implies an LDW within our MFT. In Fig.
(2) we have plotted the stationary density for $N=400$, $r=1$ and $n_{p}=0.6$,
from both MFT and MCS, both of which clearly show an LDW, which are in good
agreement with each other. The picture is dramatically different in Fig. (3),
where we have plotted the stationary density in $\cal T$ for $N=400$, $r=1$
and $n_{p}=20.0$ from MCS studies.The MCS study unexpectedly shows a DDW,
where as MFT results continue to imply an LDW. The delocalised nature of the
domain wall in MCS study can be confirmed from the corresponding kymograph
(Fig.4) that clearly shows that the spatial extent of the domain wall
movements due to its fluctuations spans the entire length of $\cal T$. This is
indeed surprising, since all previous studies on closed inhomogeneous TASEP
reported that when there is only one imhomogeneity - point or extended - at
most one domain wall in the form of an LDW can be observed. Before we analyse
and explain this paradoxical behaviour, below we first consider Model IB to
check the robustness of a single DDW.

Figure 2: Plot of density profiles with DWs of the active part of model IA for
$N=400$, $r=1$ and $n_{p}=0.6$ for different $\beta$ and $d=28.8$. These show
LDW. Figure 3: Plot of density profiles with DWs of the active part of model
IA for $N=400$, $r=2$ and $n_{p}=20.0$ for different $\beta$ and $d=0.0522$
from MCS and MFT studied. MCS studies show DDW, whereas MFT studies predict
LDWs.  Figure 4: Kymograph for Model IA for $N=400$, $r=2$, $d=0.715$,
$\beta=0.0052$ and $n_{p}=20.0$, obtained from MCS studies.

####  III.1.2 Model IB

Let us now study Model IB, when $D$ scales with system size $N$; we set $D=dN$
with $d\sim{\cal O}(1)$. Following the logic outlined for Model IA above and
using the overall particle number conservation, we obtain

$x_{w}(2\beta-1)+(1-\beta)+\frac{r^{2}}{2d}[\beta(1-\beta)]=n_{p}(1+r).$ (8)

Since $\beta$ is a fixed model parameter, $x_{w}$ is uniquely determined,
implying an LDW. In Fig. 5, we have plotted $\rho(x)$ versus $x$ for $N=400$,
$r=1$ and $n_{p}=0.4$ for different $\beta$ and $d=0.4$, from both MCS and MFT
studies. Unsurprisingly, we find an LDW; MFT and MCS results agree with each
other well.

Figure 5: Plot of density profiles for the DW in $\cal T$ in Model IB for
$N=400$, $r=1$ and $n_{p}=0.4$ for different $\beta$ and $d=0.4$, from both
MCS and MFT studies. Both studies reveal LDWs.

Consider now Fig. 6, where we have plotted $\rho(x)$ versus $x$ for $N=400$,
$r=2$ and $n_{p}=10.0$ for different $d$ and $\beta=0.1$. Unexpectedly, we
obtain a single DDW from the MCS study, where as the MFT result still predicts
an LDW. That $\rho(x)$ in Fig. 6 is indeed a DDW can be confirmed from the
corresponding kymograph in Fig. 7 that clearly shows that the fluctuation of
the domain spans the entire length of $\cal T$, confirming a DDW. Therefore,
the existence of a single DDW, being independent of the precise dynamics in
$\cal S$, is fairly robust.

Figure 6: Plot of density profiles for DW in $\cal T$ of Model IB for $N=400$,
$r=2$ and $n_{p}=10.0$ for different $d$ and $\beta=0.1$.MCS studies exhibit
DDWs. In contrast, MFT yields LDWs. Figure 7: Kymograph for Model IB for
$N=400$, $r=2$, $d=0.00632$, $\beta=0.1$ and $n_{p}=10.0$, obtained from MCS
studies.

####  III.1.3 Pinned or moving shocks?

This conundrum between LDW and DDW can be explained when fluctuations are
taken into account. We discuss this in details now. Strict particle number
conservation in Model IA and IB ensures unique determination of $x_{w}$ that
must imply an LDW. On the other hand, in an open TASEP, total particle number
is conserved only on “average”. Since for any $x_{w}$ between 0 and 1 $J_{\cal
T}$ is same, the domain wall position can move anywhere in the system, keeping
the current in the TASEP unchanged. Furthermore, since the domain wall
position fluctuations cover the entire length $N$ of the TASEP, the size of
the associated number fluctuations is ${\cal O}(N)$. In order for a single DDW
to exist in $\cal T$, fluctuations of $x_{w}$ is to occur over a scale
comparable to $N$, which in turn means fluctuating particles in $\cal T$.
Since total number of particles is constant, fluctuations in $\cal T$ is
bounded by fluctuations in the mean particle number $N_{s}$ in $\cal S$ with
$N_{s}$ given by

$N_{s}=\int_{0}^{r}\,\tilde{\rho}(x)\,d\tilde{x}=\int_{0}^{r}\,[\delta+(\gamma-\delta)\tilde{x}/r]\,d\tilde{x}.$
(9)

With values of $\gamma$ and $\delta$ already obtained in MFT above, $N_{s}$
can be obtained for Model IA and Model IB. For example in Model IA $N_{s}$ is
given by

$N_{s}=N\left[\frac{Nr^{2}}{2D}[\alpha(1-\alpha)+\frac{\alpha}{Nr}]+\frac{r\alpha}{2D}\right].$
(10)

For Model IB since $D$ scales with system size in the TL $D\to\infty$. Total
particle in $\cal S$, $N_{s}$ is given by

$N_{s}=N\left[\frac{Nr^{2}}{2D}\alpha(1-\alpha)\right]=N\left[\frac{r^{2}}{2d}\alpha(1-\alpha)\right].$
(11)

Thus, for large $N$ and large $r$, $N_{s}\sim N^{2}r^{2}$ in Model IA, and
$N_{s}\sim Nr^{2}$ in Model IB. Now, for a DDW to exist in $\cal T$ of a span
of size ${\cal O}(N)$, the associated particle number fluctuations in $\cal T$
are $\Delta\sim{\cal O}(N)$. In order to maintain overall particle number
conservation, $\cal S$ must also have particle number fluctuations of size
$\Delta\sim{\cal O}(N_{s})$. That the variance of the particle number in $\cal
T$ rises monotonically with the variance of the particle number in ${\cal S}$
in the steady states is easily seen in Fig. 21 and Fig. 22, respectively, for
Model IA and Model IB. In fact, within our numerical accuracy, these two are
found to be equal. Assuming non-interacting particles, fluctuations $\Delta$
in the total number of particles in $\cal S$ should scale with
$\Delta\sim\sqrt{N_{s}}$. Clearly, in the limit $r<<1$, $\Delta<<1$,
fluctuations in $x_{w}$ is negligible; hence an LDW is observed. In contrast
for $\tilde{r}>>1$, $\Delta>>1$, fluctuations in $x_{w}$ is large; hence a DDW
ensues.

###  III.2 Steady density profiles in Model II

Unlike Model IA and Model IB, sites in $\cal S$ of Model II can accommodate
maximum one particle per site, i.e., exclusion is imposed on $\cal S$ as well.
Nonetheless, the current $J_{\cal S}$ is still given by (3). Due to exclusion,
densities $\gamma$ and $\delta$ cannot exceed $1$. In the steady state, when
$J_{\cal S}=J_{\cal T}$, in order to have a finite current in TL, we must make
$D\propto N$. As in Model IB, we define $D=Nd$, so that the current in $\cal
S$ is written as

$J_{\cal S}=(\gamma-\delta)d/r,$ (12)

as in Model IB. In NESS, the particle current is constant throughout the
system. Considering this at the two junctions of $\cal T$ and $\cal S$, we
find the following two conditions:

$\displaystyle\delta D(1-\alpha)$ $\displaystyle=$
$\displaystyle\alpha(1-\alpha),$ (13) $\displaystyle\gamma(1-\gamma)$
$\displaystyle=$ $\displaystyle(1-\beta)(1-\gamma).$ (14)

The first condition is common with Model IA and Model IB; the second one
appears exclusively here. The latter one - unlike Model IA and IB - fixes
$\beta$, i.e., relates it with $\gamma$. Thus $\beta$ is no longer a free
parameter. Using these conditions one can arrive at the following relations:

$\alpha=D\delta,\gamma=(1-\beta),$ (15)

as found in Ref. hinsch . Thus, $\delta=\alpha/D$ approaches zero in TL since
$D$ scales with $N$ and $\alpha\sim{\cal O}(1)$. In the main part of the
paper, we primarily concern ourselves with the nature of domain walls.
Analysis of the other phases including the phase diagrams are made available
in the Appendix.

For a domain wall, we set $\alpha=\beta$. This implies an active part current
in the bulk $J_{\cal T}=\alpha(1-\alpha)$. Hence Eq. (12) can be rewritten as

$\delta=(1-\alpha)(1-\frac{r\alpha}{d})\rightarrow 0.$ (16)

This then gives $\alpha=\frac{d}{r}$. For a domain wall at $x_{w}$ in $\cal
T$, $\rho(x)$ is given by Eq. (4); the steady density in $\cal S$ is still
given by Eq. (2). Now from particle number conservation,

$n_{p}(1+r)=\int_{0}^{x_{w}}\alpha
dx+\int_{x_{w}}^{1}(1-\beta)dx+\int_{0}^{r}[\delta+(\gamma-\delta)\tilde{x}/r]dx$
(17)

Solving these in the thermodynamic limit the DW position is given by

$x_{w}=\frac{2n_{p}r(1+r)-(r-d)(2+r)}{4d-2r}.$ (18)

MF solution (18) yields the mean position $x_{w}$ of the domain wall that is
parametrised by $d,r,n_{p}$. This matches with the result in Ref. hinsch  for
$r=1$, for which an LDW is obtained, as expected. Surprisingly, for $r\gg 1$
the nature of the domain wall changes drastically, as revealed in Fig. 8
below. Our MF analysis has been complemented by MCS. We have plotted $\rho(x)$
for $N=100$, $r=5$ and $n_{p}=0.48$ for different diffusion coefficient $d$ in
Fig. (8) (top). It is evident from the plot that for small $r$, the domain
wall is pinned, i.e., an LDW is observed. We have also plotted the $\rho(x)$
for $N=100$ with $r=N$ and $r=N/2$ for $n_{p}=0.48$ for different diffusion
coefficient $d$ in Fig. (8) (bottom). For large values of $r$ a moving domain
wall or DDW is found.

Figure 8: Gradual delocalisation of the DW in $\cal T$ in Model II. Steady
state plots of $\rho(x)$ versus $x$ for $N=100$, $r=5$ and $n_{p}=0.48$ (top);
for $N=100$, $r=50$ and $n_{p}=0.48$ (bottom-left) and $r=100$ and
$n_{p}=0.48$ (bottom-right) for different values of $d$.

Figure 9: Kymograph for $N=100$ and $r=50$ for $d=2.0$ and $n_{p}=0.48$ (top)
and $r=100$ for $d=4.4$ and $n_{p}=0.48$ (bottom).

Thus, we notice that as $r$ rises for a fixed $d$ and $n_{p}$, the domain wall
gradually delocalises and we eventually obtain a single DDW for $r\gg 1$. For
$r=1$, we reproduce the results of Ref. hinsch .

Our explanation for the crossover from an LDW to a single DDW as $r$ rises to
a large value runs exactly parallel to our discussions in Sec. III.1.3. As in
Model IA and IB, the existence of a DDW that spans the whole of $\cal T$
implies particles number fluctuations $\Delta$ in $\cal T$ that should scale
with $N$: $\Delta\sim{\cal O}(N)$. Conservation of total particle number
implies equal and compensation particle number fluctuations to take place in
$\cal S$. Assuming non-interacting particles, fluctuations $\Delta$ in the
total number of particles in $\cal S$ should be ${\cal O}\sqrt{rN}$:
$\Delta\sim\sqrt{rN}$. Clearly, in the limit $r\sim{\cal O}(1)$,
$\Delta<<{\cal O}(N)$, fluctuations in $x_{w}$ is negligible; hence an LDW is
observed. In contrast for $r\simeq N$, $\Delta\sim{\cal O}(N)$, i.e., the
fluctuations in $x_{w}$ are large; hence a DDW ensues. The assumption of non-
interacting particles in $\cal S$, however, remains somewhat questionable,
since exclusion certainly introduces a hardcore repulsive interaction among
the particles in $\cal S$.

##  IV Conclusions

We have studied the steady states in periodic heterogeneous exclusion
processes which is composed of one TASEP ($\cal T$) and one diffusive ($\cal
S$) segment. We mainly focus on the nature of the domain wall in the system,
which is usually known to be an LDW in similar closed heterogeneous TASEPs. We
heuristically argue that a sufficiently large particle content in $\cal S$ can
actually delocalise the LDW to a DDW. In order to establish the generality of
our argument, we have constructed three different choices for the dynamics in
$\cal S$, which differ in the presence or absence of exclusion in $\cal S$, or
finite or diverging diffusivity $D$ in the thermodynamic limit. The size of
$\cal S$ relative to $\cal T$ is given by a model parameter $r$. Our MCS
studies reveal a generic depinning of an LDW yielding a single DDW for
sufficiently large $N_{s}$, the average total particle number in $\cal S$ in
the steady states, which we justify by considering the fluctuations. This
feature is shared by all three models we studied. To our knowledge, this is
the first instance of a single DDW in a periodic exclusion process with strict
particle number conservation. Not unexpectedly, this is not borne out by our
MFT. We then argue that this depinning may be explained in terms of the
particle number fluctuations in $\cal S$ that increases monotonically with $r$
and also with particle density $n_{p}$. This mechanism is shown to be quite
robust as all the three models studied here display similar behaviour. In
fact, we expect this mechanism to be generic - a pinned shock in any closed
heterogeneous TASEP is expected to get depinned by this mechanism. For
instance, in the well-known models for TASEPs with finite resources tsep-zia ,
where reservoirs do not have any internal dynamics of their own, a single LDW
in the TASEP is expected to get delocalised by the mechanism illustrated here.
We thus close by remarking that we expect our studies to be potentially
relevant to protein synthesis in a cell. In live eukaryotic cells, the supply
of ribosomes (modeled by particles in TASEP) is finite. For low resources, an
LDW may be observed, where as for larger resources a DDW should follow. Since
a DDW implies larger fluctuations not just in S but also in T (due to the
overall particle number conservation), a DDW can be experimentally detected in
standard ribosome density mapping experiments ribo-profile . We expect our
studies will provide further impetus to future studies along these directions.

##  V Acknowledgement

One of the authors (A.B.) thanks the Alexander von Humboldt Stiftung, Germany
for partial financial support through the Research Group Linkage Programme
(2016).

Figure 10: Plot of density profiles of Model IA for $N=400$, $r=1$ for HD, LD
and MC phase. The first half represents density profile in $\cal T$ and the
second half represents density profile in $\cal S$. Figure 11: Plot of density
profiles of Model IB for $N=400$, $r=1$ for HD, LD and MC phase. Figure 12:
Plot of density profiles in $\cal T$ of Model II for $N=100$, $r=5$ for HD, LD
and MC phase.

##  Appendix A Domain walls in Model IA

We now provide the detailed derivation for the domain wall position $x_{w}$ in
Model IA. Since multiple occupancy in the diffusive channel is allowed,
$\gamma$, $\delta$ are unrestricted and can assume any value. Using (3) for
$J_{\cal S}$ and equality of $J_{\cal S}$ and $J_{\cal T}$ at the two
junctions of $\cal S$ and $\cal T$ implies

$\displaystyle\alpha(1-\alpha)$ $\displaystyle=$
$\displaystyle(\gamma-\delta)D/Nr,$ $\displaystyle D\delta(1-\alpha)$
$\displaystyle=$ $\displaystyle\alpha(1-\alpha)$ (19)

Hence $\delta=\frac{\alpha}{D}\sim O(1)$. Since particles from $\cal T$ are
free to exit at a rate $\beta$ independent of $\gamma$ at the first site of
$\cal S$, there is no connection between $\gamma$ and $\beta$. Using
$\delta=\frac{\alpha}{D}$ we have from Eq. (19),

$\gamma=[\alpha(1-\alpha)+\frac{\alpha}{Nr}]\frac{Nr}{D}.$ (20)

From particle number conservation one can write,

$N_{p}=\int_{0}^{1}\rho(x)Ndx+\int_{0}^{r}\tilde{\rho}(\tilde{x})Nd\tilde{x}$
(21)

Let $x_{w}$ be the position of the DW. Then density distribution in the TASEP
channel $\rho(x)=\alpha+\Theta(x-x_{w})(1-\alpha-\beta)$ and that in the
diffusive channel $\tilde{\rho}(\tilde{x})=\delta+(\gamma-\delta)\tilde{x}/r$.
For a domain wall to exist $\alpha=\beta$. These allow us to write,

$\displaystyle n_{p}(1+r)$ $\displaystyle=$
$\displaystyle\int_{0}^{x_{w}}\alpha
dx+\int_{x_{w}}^{1}(1-\alpha)dx+\int_{0}^{r}[\delta+(\gamma-\delta)x/r]dx$
(22) $\displaystyle=$ $\displaystyle
x_{w}(2\alpha-1)+(1-\alpha)+r\delta/2+r\gamma/2$ $\displaystyle=$
$\displaystyle
x_{w}(2\alpha-1)+(1-\alpha)+\frac{Nr^{2}}{2D}[\alpha(1-\alpha)+\frac{\alpha}{Nr}]+\frac{r\alpha}{2D}.$

Hence,

$x_{w}=\frac{n_{p}(1+r)-(1-\alpha)-\frac{Nr^{2}}{2D}[\alpha(1-\alpha)]-\frac{r\alpha}{D}}{(2\alpha-1)}$
(23)

gives the position of the domain wall in MFT.

###  A.1 Phase diagram of Model IA

As mentioned earlier, the phase space of Model IA is spanned by three
parameters - $n_{p},D$ and $\beta$. As explained above, for $\beta<1/2$, LD
and HD are the only possible phases, with the possibilities of an LD-HD
coexistence phase in the form of a domain wall. The phase boundaries between
the LD, LD-HD and HD phases may be obtained as follows.

The boundary between the LD and the LD-HD coexistence region may be obtained
by setting $x_{w}=1$ in Eq. (23). This gives

$D=\frac{Nr^{2}\beta(1-\beta)+2r\beta}{2[n_{p}(1+r)-\beta]}.$ (24)

Similarly, the phase boundary between HD and LD-HD phase is obtained by
setting $x_{w}=0$ in Eq. (23), giving

$D=\frac{Nr^{2}\beta(1-\beta)+2r\beta}{2[n_{p}(1+r)-(1-\beta)]}.$ (25)

Figure LABEL:phase1b shows the MCS and MFT phase diagram of Model IA in the
($n_{p}-D$) plane for $r=1$ and $\beta=0.1$.

For $\beta>1/2$, it has been argued that only LD and MC phases are possible
with no HD phase. The phase boundary between the LD and MC phases in the
$n_{p}-D$ plane for a given $\beta$ may be obtained as follows.

In LD phase bulk density in $\cal T$ is $\rho(x)=\alpha$ and in MC phase
$\rho(x)=1/2$. In $\cal S$ is
$\tilde{\rho}(\tilde{x})=\delta+(\gamma-\delta)\frac{\tilde{x}}{r}$. From
particle number conservation

$n_{p}(1+r)=\alpha\int_{0}^{1}dx+\int_{0}^{r}[\delta+(\gamma-\delta)\tilde{x}/r]d\tilde{x}$
(26)

Current continuity at the junction A gives $\delta
D(1-\alpha)=\alpha(1-\alpha)$. Using this condition one can write,
$\delta=\frac{\alpha}{D}$. Equality of $J_{\cal T}$ and $J_{S}$ in the NESS
implies $\alpha(1-\alpha)=(\gamma-\delta)D/Nr.$ Hence,
$\gamma=\alpha(1-\alpha)\frac{Nr}{D}+\frac{\alpha}{D}.$ Using these equations
for $\gamma$ and $\delta$ one can obtain from Eq. (26)

$n_{p}(1+r)=\alpha+\alpha\frac{r}{D}+\frac{\alpha(1-\alpha)}{D}\frac{Nr^{2}}{2}$
(27)

For $\beta>1/2$ as long as $\alpha<1/2$ we satisfy the condition for LD phase.
As $\alpha$ exceeds $\frac{1}{2}$, we satisfy the condition for MC phase. Thus
the phase boundary in the $n_{p}-D$ plane between the LD and MC phases is
obtained by substituting $\alpha=\frac{1}{2}$ in Eq. (27) as,

$n_{p}(1+r)=\frac{1}{2}+\frac{r}{2D}+\frac{r^{2}N}{8D}.$ (28)

For $\beta=1/2$, when $\alpha<1/2$ LD phase ensues with a bulk density
$\rho(x)=\alpha$, whereas for $\alpha>1/2$ in a given region of the phase
space in the $n_{p}-D$ plane, the bulk density in $\cal T$ is $\rho=1/2$ with
a boundary layer at $x=0$. This is reminiscent of the density profile in an
open TASEP at the boundary of the MC and HD phases. See phase diagrams in
Figs. (13, 14, 15), our MFT and MCS results agree well.

Figure 13: Solid lines represent the phase diagram obtained using MFT and the
circles represent the same obtained using MCS for Model IA for $\beta=0.1$.
Figure 14: Solid lines represent the phase diagram obtained using MFT and the
circles represent the same obtained using MCS for Model IA for $\beta=0.5$.
Figure 15: Solid lines represent the phase diagram obtained using MFT and the
circles represent the same obtained using MCS for Model IA for $\beta=0.75$.

##  Appendix B Domain walls in Model IB

Now we provide the detailed derivation for the position of the DW $x_{w}$ in
model IB. When $D$ scales with system size $N$ we set $D=dN$ with $d\sim{\cal
O}(1)$. Current continuity at the junction A gives $\delta
D(1-\alpha)=\alpha(1-\alpha)$. Using this condition one can write,
$\delta=\frac{\alpha}{D}\sim{\cal O}(1/N)$. Here $\delta\to 0$ in the TL which
is different from model IA.

Here we impose the exit rate $\beta$ of TASEP from outside. Thus $\beta$ is a
free parameter. In contrast entrance rate $\alpha$ is to be derived from the
conditions of steady state. We focus on the formation of DW for which
$\alpha=\beta$. Equality of $J_{\cal T}$ and $J_{S}$ in the NESS implies

$\alpha(1-\alpha)=(\gamma-\delta)D/Nr.$ (29)

Using $\delta\to 0$ in the TL we have from Eq. (29),

$\gamma=\frac{Nr}{D}[\alpha(1-\alpha)].$ (30)

For a DW to exist, one must have $\alpha=\beta<1/2$.

From particle number conservation one can write,

$N_{p}=\int_{0}^{1}\rho(x)Ndx+\int_{0}^{r}\tilde{\rho}(x)Nd\tilde{x}$ (31)

Let $x_{w}$ be the position of the DW. Now approaching as in Model IA, we have

$\displaystyle n_{p}(1+r)$ $\displaystyle=$
$\displaystyle\int_{0}^{x_{w}}\alpha
dx+\int_{x_{w}}^{1}(1-\alpha)dx+\int_{0}^{r}[\delta+(\gamma-\delta)x/r]dx$
$\displaystyle=$ $\displaystyle
x_{w}(2\alpha-1)+(1-\alpha)+r\delta/2+r\gamma/2$

Here in the TL, $\delta\to 0$. Using the expression of $\gamma$ from eq. (30)
the location $x_{w}$ of the domain wall is given by

$x_{w}=\frac{n_{p}(1+r)-(1-\alpha)-\frac{r^{2}}{2d}[\alpha(1-\alpha)]}{(2\alpha-1)}$
(33)

###  B.1 Phase diagram of Model IB

As mentioned earlier, the phase space of Model IB is spanned by three
parameters - $n_{p},d=D/N$ and $\beta$. As explained above, for $\beta<1/2$,
LD and HD are the only possible phases, with the possibilities of an LD-HD
coexistence phase in the form of a domain wall. The phase boundaries between
the LD, LD-HD and HD phases may be obtained as follows.

For $x_{w}\leq 0$ the DW leave the active part at the left junction resulting
in HD phase having constant density $\rho(x)=1-\beta$, $x_{w}\geq 1$ results
in LD phase having constant density $\rho(x)=\alpha$ and LD-HD phase is
characterised by the presence of a DW inside the system for $0<x_{w}<1$.

Hence the boundary between the HD and LD-HD coexistence region may be obtained
by setting $x_{w}=0$ in Eq. (33). This gives

$d=\frac{r^{2}\beta(1-\beta)}{2[n_{p}(1+r)-(1-\beta)]}.$ (34)

Similarly the boundary between LD and LD-HD coexistence region is obtained by
setting $x_{w}=1$ in Eq. (33) which gives

$d=\frac{r^{2}\beta(1-\beta)}{2[n_{p}(1+r)-\beta]}.$ (35)

Our main results are summarised in the phase diagrams in the ($n_{p}-d$) plane
in Fig. LABEL:phase1a for $r=1$ and $\beta=0.2$. For $\beta>1/2$, only LD and
MC phases are possible with no HD phase. The phase boundary between LD and MC
phases in the $n_{p}$-$d$ plane for a given $\beta$ may be obtained as
follows.

In the LD phase bulk density in $\cal T$ is $\rho(x)=\alpha$ and in MC phase
$\rho(x)=1/2$. In $\cal S$ is
$\tilde{\rho}(\tilde{x})=\delta+(\gamma-\delta)\frac{\tilde{x}}{r}$. From PNC

$n_{p}(1+r)=\alpha\int_{0}^{1}dx+\int_{0}^{r}[\delta+(\gamma-\delta)\tilde{x}/r]d\tilde{x}$
(36)

Using the expressions of $\gamma$ and $\delta$ for Model IB in Eq. (36) one
can obtain

$n_{p}(1+r)=\alpha+\frac{\alpha(1-\alpha)}{D}\frac{Nr^{2}}{2}$ (37)

For $\beta>1/2$ as long as $\alpha<1/2$ we satisfy the condition for LD phase.
As $\alpha$ exceeds $\frac{1}{2}$, we satisfy the condition for MC phase. Thus
the phase boundary in the $n_{p}-D$ plane between the LD and MC phases is
obtained by substituting $\alpha=\frac{1}{2}$ in Eq. (37) as,

$n_{p}(1+r)=\frac{1}{2}+\frac{r^{2}N}{8D}.$ (38)

For $\beta=1/2$, when $\alpha<1/2$ LD phase ensues with a bulk density
$\rho(x)=\alpha$, whereas for $\alpha>1/2$ in a given region of the phase
space in the $n_{p}-D$ plane, the bulk density in $\cal T$ is $\rho=1/2$ with
a boundary layer at $x=0$. This is reminiscent of the density profile in an
open TASEP at the boundary of the MC and HD phases. See phase diagrams in Figs
(16, 17, 18); our MFT and MCS results agree well.

Figure 16: Solid lines represent the phase diagram obtained using MFT and the
circles represent the same obtained using MCS for Model IB for $\beta=0.2$.
Figure 17: Solid lines represent the phase diagram obtained using MFT and the
circles represent the same obtained using MCS for Model IB for $\beta=0.5$.
Figure 18: Solid lines represent the phase diagram obtained using MFT and the
circles represent the same obtained using MCS for Model IB for $\beta=0.75$.

##  Appendix C Domain walls in Model II

From particle number conservation one can write,

$N_{p}=\int_{0}^{1}\rho(x)Ndx+\int_{0}^{r}\tilde{\rho}(x)Ndx$ (39)

Density distribution in the TASEP channel
$\rho(x)=\alpha+\Theta(x-x_{w})(1-\alpha-\beta)$ and that in the diffusive
channel $\tilde{\rho}(x)=\delta+(\gamma-\delta)x/r$. This implies,

$\displaystyle n_{p}(1+r)$ $\displaystyle=$
$\displaystyle\int_{0}^{x_{w}}\alpha
dx+\int_{x_{w}}^{1}(1-\beta)dx+\int_{0}^{r}[\delta+(\gamma-\delta)x/r]dx$ (40)
$\displaystyle=$ $\displaystyle\alpha
x_{w}+(1-\beta)(1-x_{w})+\frac{r\gamma}{2}$

Since $\delta\sim O(1/N)$ in the TL, $\delta\to 0$ and $\alpha\to\frac{d}{r}$.
Also from Eq.(15) $\gamma=1-\beta$. Now from the condition of DW
$\alpha=\beta$. Using these conditions we have from Eq.(40),

$x_{w}=\frac{2r(1+r)n_{p}-(r-d)(2+r)}{4d-2r}$ (41)

###  C.1 Phase diagrams for Model II

For different values of $r$ we can get phase diagram in ($n_{p}$, $d$) space
consisting of different regimes depending on the position of DW. For
$x_{w}\leq 0$ the DW leave the active part at the left junction resulting in
HD phase having constant density $\rho(x)=1-\beta$, $x_{w}\geq 1$ results in
LD phase having constant density $\rho(x)=\alpha$ and LD-HD phase is
characterised by a DW localised inside the system for $0<x_{w}<1$. From Eq.
(41) the phase boundary between HD and LD-HD coexistence is given by

$d=r-\frac{n_{p}2r(1+r)}{2+r}.$ (42)

Phase boundary between LD and LD-HD phase is obtained as,

$d=\frac{2n_{p}r(1+r)-r^{2}}{2-r}$ (43)

Similar to ordinary TASEP the ring system also exhibits maximal current (MC)
phase which is characterised by maximam active part current $J_{MC}=1/4$ and
constant density $\rho=1/2$ and is obtained for $\alpha$, $\beta>1/2$. Hence,
$N/2$ particles have to be present in the active part. Equality of the active
part current and passive part current allows one to write
$\gamma=\delta+\frac{r}{4d}.$

Particle number conservation in MC phase yields,

$(1+r)n_{p}=r(\delta+\frac{r}{8d})+\frac{1}{2}.$ (44)

The constraints $\alpha$, $\beta>1/2$ on active part impose constraints on the
passive part by the use of Eqs. (12) and (15) as

$\delta>\frac{1}{2dN}$ (45)

and

$\delta<\frac{1}{2}-\frac{r}{4d}.$ (46)

Implementing Eq. (45) in Eq. (44) one can get phase boundary between LD and MC
phase in the TL as,

$d=\frac{r^{2}}{8(1+r)n_{p}-4}.$ (47)

Similarly the phase boundary between HD and MC phase can be obtained using Eq.
(46) in Eq. (44) as,

$d=\frac{r^{2}}{4(1+r)(1-2n_{p})}.$ (48)

Our main results are summarised in the phase diagrams in the ($n_{p}-d$) plane
in Fig.19 and 20 for different values of $r$.

Figure 19: Solid lines represent the phase diagram obtained using MFT and the
circles represent the same obtained using MCS for Model II for $r=5$. The
phase diagram exhibits four different phases. Figure 20: Solid lines represent
the phase diagram obtained using MFT and the circles represent the same
obtained using MCS for Model II. Left panel represents phase diagram for
$r=100$ and right panel represents phase diagram for $r=50$. Figure 21: Plot
of variance of TASEP vs SEP for Model 1A. Figure 22: Plot of variance of TASEP
vs SEP for Model 1B. Figure 23: Plot of variance of TASEP vs SEP for Model 2.

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