The Totally Asymmetric Simple Exclusion Process with Langmuir Kinetics

Abstract

We discuss a class of driven lattice gas obtained by coupling the one-dimensional totally asymmetric simple exclusion process to Langmuir kinetics. In the limit where these dynamics are competing, the resulting nonconserved flow of particles on the lattice leads to stationary regimes for large but finite systems. We observe unexpected properties such as localized boundaries (domain walls) that separate coexisting regions of low and high density of particles (phase coexistence). A rich phase diagram, with high and low density phases, two and three phase coexistence regions, and a boundary independent "Meissner" phase is found. We rationalize the average density and current profiles obtained from simulations within a mean-field approach in the continuum limit. The ensuing analytic solution is expressed in terms of Lambert W functions. It allows one to fully describe the phase diagram and extract unusual mean-field exponents that characterize critical properties of the domain wall. Based on the same approach, we provide an explanation of the localization phenomenon. Finally, we elucidate phenomena that go beyond mean-field such as the scaling properties of the domain wall.

Full text

arXiv:cond-mat/0408034v2 [cond-mat.stat-mech] 22 Oct 2004
The Totally Asymmetric Simple Exclusion Process with Langmuir Kinetics
A. Parmeggiani
1,∗
, T. Franosch
1,2
, and E. Frey
1,2
1
Hahn-Meitner Institut, Abteilung Theorie, Glienicker Str. 100, D-14109 Berlin, Germany
2
Fachbereich Physik, Freie Universit¨at Berlin, Arnimallee 14, D-14195 Berlin, Germany
(Dated: February 2, 2008)
Abstract
We discuss a new class of driven lattice gas obtained by coupling the one-dimensional totally asymmetric simple exclusion
process to Langmuir k inetics. In the limit where these dynamics are competing, the resulting non-conserved flow of particles on
the lattice leads to stationary regimes for large but finite systems. We observe unexpected properties such as localized boundaries
(domain walls) that separate coexisting regions of low and high density of particles (phase coexistence). A rich phase diagram,
with high an low density phases, two and three phase coexistence regions and a boundary independent “Meissner” phase is
found. We rationalize the average density and current profiles obtained from simulations within a mean-field app roach in the
continuum limit. The en su ing analytic solution is expressed in terms of Lambert W -functions. It allows to fully describe the
phase diagram and extract unusual mean-field exponents that characterize critical properties of the domain wall. Based on the
same approach, we provide an explanation of the localization phenomenon. Finally, we elucidate phenomena that go beyond
mean-field such as the scaling properties of the domain wall.
PACS numbers: 02.50.Ey, 05.40.-a, 64.60.-i, 72.70.+m
I. INTRODUCTION
Many natural phenomena driven by some external
field or containing self-pr opelled particles evolve into sta-
tionary states carrying a steady current. Such states
are characterized by a constant gain or loss of energy,
which distinguishes them from ther mal equilibria. Exam-
ples range from biolo gical systems like ribosomes moving
along m-RNA or motor molecules “walking” along molec-
ular tra cks to ions diffusing along narrow channels, or
even cars proceeding on highways. In order to elucidate
the nature of such non-equilibrium steady states a vari-
ety of driven lattice gas models have been introduced and
studied extensively [1]. Here we focus on one-dimensional
(1D) models, where particles preferentially move in one
direction. In this context, the Totally Asymmetric Sim-
ple Ex c lus ion Process (TASEP) has become one of the
paradigms of non-equilibrium physics (for a review see
Refs. [2, 3, 4, 5]). In this model a single species of parti-
cles is hopping unidirectionally and with a uniform rate
along a 1D lattice. The only interaction between the par-
ticles is hard-core repulsion, which prevents more than
one particle from occupying the same site on the lattice;
see Fig. 1.
It has been found that the nature of the non-
equilibrium steady state of the TASEP depends sensi-
tively on the boundar y conditions. For periodic bound-
ary conditions the s ystem reaches a steady state of c on-
stant density. Interestingly, density fluctuations are
found to spr e ad faster than diffusively [6]. This can be
∗
New address: Laboratoire de Dynamique Mol´eculaire des Interac-
tions Membranaires, UMR 5539 CNRS/Universit´e de Montpelli er
2, Place Eug`ene Bataill on, 34095 Montpellier Cedex 5, France.
email: parmeggiani@univ-montp2.fr, parmeggiani@hmi.de.
understood by an exact mapping [7] to a growing in-
terface model, whose dynamics in the continuum limit
is described in terms of the KPZ equation [8] and its
cousin, the noisy Burgers equation [9]. In contrast to
such r ing systems, open systems with particle reservoirs
at the ends exhibit phase transitions upon varying the
boundary conditions [10]. This is genuinely different from
thermal equilibrium systems where boundary effects usu-
ally do not affect the bulk behavior and become negligible
if the system is large enough. In addition, general the-
orems do not even allow equilibrium phase transitions
in one-dimensional systems at finite temperatures (if the
interactions are not too long-range) [11].
α
β
/
particle
vacancy
/
FIG. 1: Illustration of the Totally Asymmetric Simple Exclu-
sion Process with open boundaries. The entrance and exit
rates at the left and right end of the one-dimensional lattice
are given by α and β, respectively.
Yet another difference between equilibrium and non-
equilibrium processes can be clearly seen on the level
of its dynamics. If transition rates between microscopic
configurations are obeying detailed balance the sy stem is
guaranteed to evolve into thermal equilibrium [61]. Sys-
tems lacking detailed balance may still reach a steady
state, but at present there are no universal concepts like
the Boltzmann-Gibbs ensemble theory for characterizing
such non-equilibrium steady states. In most instances
one has to resort to solving nothing les s than its full dy-
namics. It is only recently, that exact (non-local) free
energy functionals for driven diffusive sys tems have been
1

derived [12, 13].
This has to be contrasted with dynamic processes such
as the adsorption-desorption kinetics of particles on a lat-
tice co upled to a bulk rese rvoir (“Langmuir Kinetics”,
LK), see Fig. 2. Here, particles adsorb at an empty site
or desorb from a n occupied one. Microscopic reversibility
demands that the corresponding kinetic rates obey de-
tailed balance such that the system evolves into an equi-
librium steady state, which is well described within stan-
dard concepts of equilibrium statistical mechanics. If in-
teractions between the particles other than the hard-core
repulsion are neglected, the equilibrium density is solely
determined by the ratio of the two kinetic rates [14], as
given by the Gibbs ens e mble.
ω
D
ω
A
particle
vacancy
FIG. 2: Illustration of Langmuir kinetics. ω
A
and ω
D
denote
the local attachment and detachment rates.
The TASEP and LK can be considered as two of the
simplest paradigms which contrast equilibrium and non-
equilibrium dynamics and stationary states. Langmuir
kinetics evolves into a steady state well described in terms
of standard concepts of equilibrium statistical mechanics.
Driven lattice gases such as the TASEP evolve into a sta-
tionary non-equilibrium state ca rrying a finite c onserved
current. Whereas such non-equilibrium steady states are
quite sensitive to changes in the boundary conditions,
equilibrium steady states are very robust to such changes
and dominated by the bulk dynamics. In the TASEP the
number of particles is conse rved in the bulk of the one-
dimensional lattice. It is only through the particle reser-
voirs at the system boundaries that particles can e nter or
leave the system. In LK particle number is not conserved
in the bulk. Particles can enter or leave the system at
any site. Depending on whether we consider a canonical
or grand canonical ensemble the lattice is connected to
a finite or infinite particle reservoir. Unlike the steady
state of the TASEP, the equilibrium s teady state of LK
does not have any spatial corr elations.
Combining b oth of these processes may at first sight
seem a trivial exercise since one might expect bulk effects
to b e predominant in the thermodynamic limit. This is
indeed the case for attachment and detachment rates,
ω
A
and ω
D
, which are independent of system size N.
For large but finite systems interesting effects can only
be expected if the kinetics from the TASEP and LK com-
pete. Then, as we have shown recently [15], novel behav-
ior different from both LK and the TASEP appears. In
particular, one observes phase separation into a high and
low density domain for an extended region in parameter
space.
When should one expect competition between bulk
dynamics (LK) and boundary induced non-equilibrium
effects (TASEP)? Let’s consider the following heuristic
argument. A given particle will typically spend a time
τ ∼ 1/ω
D
on the lattice before detaching. During this
“residence” time the number of sites n explored by the
particle is of the order of n ∼ τ. Hence, for fixed ω
D
,
the fraction n/N ∼ 1/(ω
D
N) of sites visited by a par-
ticle during its walk o n the lattice would go to zero as
N → ∞. Only if we introduce a “total” detachment rate
by Ω
D
= N ω
D
and keep it constant instead of ω
D
as
N → ∞ will the particle travel a finite fraction of the
total lattice size. Similar arguments show that a vacancy
visits an extensive number of sites until it is filled by at-
tachment of a particle if ω
A
scales to zero as ω
A
= Ω
A
/N
with a fixed “total” attachment rate Ω
A
. In other words,
competition will be e xpected only if the particles live long
enough such that their internal dynamics or the externa l
driving fo rce tra ns ports them a finite fraction a long the
lattice before detaching. Then, particles spend enough
time on the lattice to “feel” their mutual interaction and,
eventually, produce collective effects. In summary, com-
petition between bulk and boundary dynamics in large
systems (N ≫ 1) is expected if the kinetic r ates ω
A
and
ω
D
decrease with increasing system size N such that the
total rates Ω
A
and Ω
D
with
Ω
A
= ω
A
N , Ω
D
= ω
D
N (1)
are kept c onstant with N.
The competition between boundary and bulk dynamics
is a physical process that has, to our knowledge, not yet
been studied in the context of driven diffusive systems.
In previous models emphasis was put on the analysis of
boundary induced phenomena in driven gases of mono- or
multi-sp ecies of particles [4, 5, 16, 17, 18, 19], in presence
of interactions (see e.g. Ref. [20, 21]), disorder [22, 23] or
local inhomogeneities [24, 25], particles with sizes lar ger
than the lattice spacing [26, 27], lattices with different
geometries (e.g. multi-lanes lattice gases [28]), or systems
in presence of several conservation laws (for a review see
e.g. Ref. [19]).
In this manuscript we explo re the consequences of par-
ticle ex change with a r eservoir along the track (LK) on
the stationary density and curr ent profiles and the en-
suing phase diagram of the TASEP. A short account of
our ideas has been given r e c ently [15], wher e we have
introduced the model and have shown how our Monte
Carlo results can be rationalized on the basis of a mean-
field theory, which we also solved analytically. The pur-
pose of the present manuscript is to give a complete and
comprehensive discussion of the topic. We will prese nt
results from Monte-Carlo results for the full parameter
range of the model including the particular case where
on- and off-rates equal each other, which were left out in
our s hort contribution [15] due to the lack of space. In
addition, we will give the full reasoning for the derivation
and ana lytical solutions of our mean-field theory. Here,
additional insight is gained by identifying a branching
point that explains all the features of the density pro-
files and phase diagram analytically. In particula r, we
2

show that a new critical point o rganizes the topology of
the diagram and leads to unexpected phenomena already
briefly discussed in Ref. [15]. In a recent work, Evans et
al. [56] have rephrased the mea n-field analysis first given
in Ref. [15] and reproduced some of our results. The
mean-field equations are, however, left in their implicit
form and thus miss the interesting feature s we will obtain
from the identification of a bra nching point.
These effects differ from those known in reference mod-
els of e quilibrium and non-equilibrium s tatistical me-
chanics like LK and the TASEP. Indeed, the coupling be-
tween the TASEP and LK, as is was introduced above,
produces new phenomena a nd extends the interest to-
ward systems which break conservation law in a non-
trivial way. As we shall see in the next Section, these
features emerge already at level of properties of the mi-
croscopic dynamics in configuration space described by
the master equation.
Recently a variant of our model has been suggested
by Popkov et al. [60]. Upon supplementing the Katz-
Leb owitz-Spohn model by Langmuir kinetics and ana-
lyzing it within the mean-field approach similar to [15],
an e ven richer scenario for the stationary density pro-
file is obtained that includes the emer gence of localized
downward domain walls a nd the a ppearance of several
’shocks’ separ ating three distinct phases. It is a lso noted
in Ref. [60] that in general it may be important to r e -
place the mean-field current by the ex act current in the
stationary state.
In addition to its fundamental importance for non-
equilibrium physics in general, competition between bulk
dynamics and boundary effects are ubiquitous in nature,
in particular biological phenomena. The TASEP has ac-
tually been introduced in the biophysical literature as
a model mimicking the dynamics o f ribosomes moving
along a messenger RNA chain [29]; for generalizations of
these studies, see the recent work in Refs. [27, 30]. Also
some aspects of intracellular transport show close resem-
blance to our model. For example, pr ocessive molecular
motors advance along cytoskeletal filaments while attach-
ment and detachment of motors between the cytoplasm
and the filament occur [31]. Typically kinetic rates are
such that these motors walk a finite fraction along the
molecular track before detaching. This falls well into
the regime where we expect novel stationary s tates. Re-
cently, it has been shown that such dynamics can be rel-
evant for modeling the filopod growth in e ukaryotic cells
produced by motor proteins interacting within actin fil-
aments [32]. Finally, our model could also be relevant
for studies of surface-adsorption and growth in pres e nce
of biased diffusion or for traffic models with bulk on-off
ramps [33].
Since our paper contains a rather comprehensive dis-
cussion of the topic, we will give a deta iled outline to
provide the reader with s ome g uidance thro ugh the anal-
ysis. In Section II we define the model by its dynamic
rules and make a connection to its stochastic dynamics
on a network. Though the relation between stochastic
dynamics and networks is interesting to fully understand
the peculiar features of the model introduced by the com-
bination of conserved dynamics and on/off kinetics, it
may be skipped for the first reading. We then present
the problem in terms of a Fock space formulation and
discuss the symmetries of the model, both key features
for the subsequent formulation of the mean-field theory.
In Section III we briefly discuss some technical details
of the Monte-Carlo simulation. Then follows a key sec -
tion of the manuscript, a detailed development of the
mean-field approximation and the r e sulting “Burgers”-
like equations in the continuum limit. Here we also dis-
cuss a series of features of these equations which will turn
out to be crucial for the understanding of the ensuing
density and curr e nt profiles.
In Sect. IV an analytic solution of the continuum equa-
tions is derived and compared to simulation results. We
start the discussion for the special case that on- and off-
rates are identical. Though simpler to analyze, this case
is somewhat artificial as it requires a fine-tuning of the
on- and off-rates. Generically, one expects on- and off-
rates to differ. Then the mathematical analysis becomes
significantly more complex. We ar e still a ble to give an
explicit analytical solution in terms of so-called Lambert
functions, which allows us to identify a branching point
that explains all the features of the density profiles and
phase diagram analytically. In pa rticular, we find a spe-
cial point that organizes the topology of the diagram.
In Sect. V we discuss the properties of the doma in wall
characterizing the phase coexistence upon changes of the
model parameters. In par ticula r, we show that in the
vicinity of the special point mentioned above the domain
wall exhibits non-analytic behavior similar to a critical
point in continuous phase transitions. We derive the crit-
ical exponents and the scaling related to the amplitude
and position of the domain wall. A conclusion, Sect.
VI, summarizes our re sults and provides additional ar-
guments on the phenomenon of pha se coexistence. Last,
we discus s some discrepancies between the mean-field ap-
proach and the simulation results and discuss a possible
reconciliation.
II. THE MODEL
In this Section we are going to describe the model in
some detail. We will also put it into the context of net-
work theories. This will help us to pinpoint the differ-
ences between the TASEP and LK dynamics and show
how a model combing both aspects will lead to novel
phenomena. Finally, we briefly review the key ideas of
the Fock space formulation of stochastic particle dynam-
ics. In la ter chapters this formulation will be used for an
analytic discussion of the model.
3

A. Definition of the dynami c rules
In the microscopic model we consider a finite one-
dimensional lattice with sites labeled i = 1, ..., N (see
Fig. 3) and lattice spacing a = L/N, where L is the total
length of the lattice. The site i = 1 (i = N) defines the
left (r ight) b oundary, while the collection i = 2, ..., N − 1
is referred to as the bulk.
The microscopic state of the system is characterized
by a distribution of identical particles on the lattice, i.e.
by configurations C = {n
i=1,..N
}, where each of the oc-
cupation numbers n
i
is equal either to zero (vacancy) or
one (particle). We impose a hard core repulsion between
the particles, which implies that a double or higher oc-
cupancy of sites is forbidden in the model. The full state
space then consists of 2
N
configurations.
The statistical properties of the model are given in
terms of the probabilities P(C, t) to find a particular con-
figuration C = {n
i
} at time t. We consider the evolution
of the probabilities P described by a master equation:
dP(C, t)
dt
=
X
C
′
6=C
h
W
C
′
→C
P(C
′
, t) − W
C→C
′
P(C, t)
i
. (2)
Here, W
C→C
′
is a non-neg ative transition rate from con-
figuration C to C
′
. As usual, master equations con-
serve probabilities. The microscopic processes connect-
ing two subsequent c onfigurations are local in configura-
tion space. Out of the possible 2
N
× 2
N
transitions, we
consider only the following ele mentary steps connecting
neighboring configurations:
A) at the site i=2, .., N − 1 a particle can jump to site
i + 1 if unoccupied with unit rate;
B) at the site i = 1 a particle can enter the lattice with
rate α if unoccupied;
C) at the site i = N a particle can leave the la ttice
with rate β if occupied.
Additionally, in the bulk we assume that a particle:
D) can leave the lattice with site-independent detach-
ment rate ω
D
;
E) can fill the site (if empty) with a r ate ω
A
by attach-
ment.
Processes A) to C) constitute a totally asymmetric sim-
ple exclusion process with open boundaries [3, 4, 5], while
processes D) and E) define a Langmuir kinetics [14]. We
have taken the attachment and detachment rates to be
independent of the particle concentration in the reser-
voir, i.e. we have assumed that the Langmuir kinetics
on the lattice is reaction and not diffusion limited. The
effect of diffusion in confined geometry has been studied
in Ref. [34]. A schematic graphical representation of the
resulting totally asymmetric e xclusion model with Lang-
muir kinetics [15] is given in Fig. 3.
α
β
ω
D
ω
A
/
particle
vacancy
/
FIG. 3: Schematic drawing of the totally asymmetric simple
exclusion process with bulk attachment and detachment [15].
The entrance and exit rates at th e left and right end of the
one-dimensional lattice are given by α and β, respectively; ω
A
and ω
D
denote the local attachment and detachment rates.
Once we know the dynamic rules of the s tochastic pro-
cess, one may introduce the notion of neighboring con-
figurations for C and C
′
, if they differ only by a small
fraction (O(1/N)) of the co rresponding oc cupation num-
bers. This naturally leads us to a reinterpretation of the
dynamics in terms of networks as described in the follow-
ing subs e c tion.
B. Stochastic dynamics and networks
The Markovian dy namics of the system can be repre-
sented in terms of a network (graph), where the configu-
rations of the stochastic process correspond to the nodes
(vertices) o f the network. Each transition allowed by the
dynamics is repres e nted as a direc ted link (edge), and
weighted by the corresponding transition r ate which can
be read off from the dynamic rules A) − E). Due to the
local dynamics the network is very dilute. A given node
in the network is connected to a maximum number O(N)
of nearby configurations. Nevertheless, any configuration
can still be reached fr om any point within the network.
In other words the network is connected and does not
break into disjunct pieces. In addition, every node has
at least one ingoing and one outgoing link. This guaran-
tees that the system is ergodic, at lea st as long as N is
finite, and all states are recurrent [35].
On such a network a distance between two different
configurations can be defined as the minimal number of
steps required to connect them. Note that the “architec-
ture” of the network corresponding to a pure TASEP is
very different from a pure L K; see Fig. 4 fo r a n illustra-
tion.
The TASEP network is characterized by large fluctu-
ations in the connectivity. Take for exa mple the com-
pletely filled configuration. This state can only be left
if the particle at the right end of the lattice is ejected
from the s ystem. Similarly, a config uration described by
a step function n
i
= Θ(x
i
− x
0
) with a completely filled
lattice to the left and a completely empty lattice to the
right of x
0
can only be left by a single process where the
rightmost particle is hopping forward. We call such and
similar states “periphery states” since they are linked to
the rest of the network by a s ingle o r only a few outgoing
and ingoing links. This is to be contrasted with “typi-
4

periphery node
typical node
O(N) O(N)
O(1)
O(1)
2
TASEP network
O(N)
O(N)
O(N)
LK network
O(N )
FIG. 4: Illustration of the network architecture corresponding
to the totally asymmetric simple exclusion p rocess (TASEP)
and Langmuir kinetics (LK).
cal states” for a given density, where particles are more
or less r andomly distributed over the lattice. Then, the
conditional probability that an empty site is in front of a
filled site will be finite. In other words, there will be an
extensive number of pairs (1, 0) on the lattice. This im-
plies that a typical sta te will be connected with an exten-
sive number O(N ) of directed ingoing and o utg oing links
to other nodes in the network. Similarly, the shortest
path connecting two non-neighboring configurations has
a broad length distribution. Given two randomly chosen
sequences of occupation numbers n
i
= 0 and n
i
= 1 (i.e.
nodes) one has to ask, how many local moves of the typ e
A) to C ) (i.e. links) are needed to transform one sequence
into the other. In ge neral, there will be a distribution of
paths connecting these nodes. The shortest connection
may be only a few links, if local rearrangements of par-
ticles are sufficient fo r matching the microscopic config-
urations. It seems plausible that this is the case for such
microscopic co nfigurations, whose coarse-grained density
profiles are identical or at least very s imila r. If the s pa-
tial profiles of the coarse-grained densities corresponding
to the two configurations differ significantly, one e xpects
O(N
2
) local re arrangements to be necessary for matching
the microscopic configuratio ns . This is simply a conse-
quence of particle conservation in the bulk. For e xample,
to completely empty a totally filled state obviously r e -
quires O(N
2
) steps. In addition, distances betwee n two
configurations in a TASEP network can also be highly
asymmetric. Consider a configuration C corresponding
to a node at the periphery of the network connected to
a configuration C
′
. Then the corresp onding reverse step
does not exist, and in order return to the configuration
C one has to make a large loop in configuration space.
In summary, a network corresponding to TASEP con-
tains only directed links. A characteristic feature is its
heterogeneity in the connectivity of nodes and distances
between nodes. The network contains loops, many of
which may be very long due to the conservatio n law in
the bulk.
This has to be contrasted with the architecture of a
network co rresponding to LK. Here, the connectivity of
all nodes is independent of the particular configuration.
Since each occupation numbe r n
i
at a given site i can
be independently changed, the number of links outgoing
from a node is simply N . To each outgoing link there is
an ingoing link with weights related by detailed balance.
Moreover, any two configurations can be reached by a t
most N transitions. Since there is no conservation law,
only local moves (particle attachment or detachment) are
necessary. The distance of two configurations (alo ng the
shortest path) in a LK network is d(C, C
′
) =
P
N−1
i=2
|n
i
−
n
′
i
| [62]. Since the order of the necessary attachment and
detachment processes is irrelevant the number of such
shortest paths is highly degenerate, and depends only
on the distance as d!. In summary, the LK netwo rk is
not dire c ted, very homogeneous, highly connected and
contains many loops of all size.
An important distinction between LK and the TASEP
can be clearly seen if one compares the nature o f the cor-
responding stationary states. Langmuir kinetics has a
solution described in terms of the thermodynamic equi-
librium distribution:
P(C) =
K
|C|
(K + 1)
N−2
. (3)
Here |C| ≡
P
N−1
i=2
n
i
is the number of occ upied sites in
the bulk and K = ω
A
/ω
D
is the binding constant . Note
that the equilibrium distribution of LK can be charac-
terized by a Boltzma nn weight upon introducing an ef-
fective Hamiltonian H = −k
B
T
P
N−1
i=2
n
i
ln K. The case
K = 1 has an interesting topological interpretation since
the links in the LK network loose their directiona lity and
the effective Hamiltonian H evaluates to 0.
In contrast, the totally asymmetric exclusion process
does not satisfy the detailed balance condition
W
C
′
→C
P(C
′
) = W
C→C
′
P(C) ,
and evolves into a non-equilibrium steady state. Actu-
ally, if one would assume detailed bala nce along a closed
directed loop in the TASEP network, one would be lead
to the conclusion that all probabilities along the path
have to be zero. This , in turn, would contradict the er-
godicity of the finite system.
The network analogy discussed above can be used to
understand why a stochastic dynamics combining the to-
tally asymmetric exclusion process and Langmuir kinet-
ics is interesting and show a range of novel features not
contained in the TASEP or LK alone. We have seen
that the number of links necessary to connect two non-
neighboring states in the TASEP (O(N
2
)) is much larger
than in LK (O(N)). Then, if we take both the weights fo r
hopping and the weights for attachment and detachment
to scale the same way, LK dynamics will dominate due
to its higher connectivity. In order to have competition,
the weight of ea ch LK link has to be decreased as pre-
scribed in the Introduction such that the weighted path
5

lengths of the TASEP and LK are comparable. Yet an-
other way to generate competition would be to only allow
a finite (non-extensive) number of sites to cause attach-
ment and deta chment with a system size independent
rate [36]. The network structure of the totally asymmet-
ric exclusion process with Langmuir kinetics also indi-
cates why standard matrix product ansatz methods could
be rather difficult to implement.
C. Fock space formulation of stochastic dynamics
It is sometimes convenient to formulate problems in
stochastic particle dynamics in terms of a quantum
Hamiltonian representation instead of a master equation.
This formalism was developed already some time ago by
several groups [37, 38, 39]. In the meantime it has found
a broad range of applications (see, e.g. Ref. [40]). We re-
fer the reader for details to various review articles [5, 40]
and lecture notes [41, 42].
In o ur case, the occupation number s n
i
(C) constitute in a
natural way state space functions by measuring whether
site i is occupied (n
i
= 1 ) or not (n
i
= 0 ) in configura-
tion C. The corresponding Heisenberg equations for ˆn
i
(t)
then read
d
dt
ˆn
i
(t) = ˆn
i−1
(t)
h
1 − ˆn
i
(t)
i
− ˆn
i
(t)
h
1 − ˆn
i+1
(t)
i
+
+ ω
A
h
1 − ˆn
i
(t)
i
− ω
D
ˆn
i
(t) (4a)
for any site in the bulk, while for sites at the boundaries
one obtains
d
dt
ˆn
1
(t) = α
h
1 − ˆn
1
(t)
i
− ˆn
1
(t)
h
1 − ˆn
2
(t)
i
,
d
dt
ˆn
N
(t) = ˆn
N−1
(t)
h
1 − ˆn
N
(t)
i
− βˆn
N
(t) .
(4b)
The first line of Eq. (4a) is the usual contribution due to
the TASEP. Introducing the current operator
ˆ
j
i
(t) = ˆn
i
(t)
h
1 − ˆn
i+1
(t)
i
,
one can rewrite the right hand site of this line as
ˆ
j
i−1
−
ˆ
j
i
,
which is a discrete form of the divergence of the cur rent.
This pa rt defines a dynamics which satisfies particle num-
ber conservation. The se c ond line of Eq. (4a) represents
the additional Langmuir kinetics, which acts as source
and sink terms in the bulk.
These equations can now be understood as e quations
of motions for a q uantum many body problem. There
are different routes to arrive at a solution. For one-
dimensional problems there are many instances where
exact methods are applicable [5]. Cohere nt state path
integrals are useful to explore the scaling behavior at
critical points [40, 41, 43]. One can also try to ana-
lyze the equations of motion directly [44, 45]. By taking
averages of Eqs. (4) in order to compute the time evo-
lution of hˆn
i
(t)i one needs the corresponding averages of
two-point correlations such as hˆn
i−1
(t)(1 − ˆn
i
(t))i. This
two-point correlation function obeys itself an equation of
motion connecting it to three-point and four-point corre-
lation functions. Thus we are lead to an infinite hierar-
chy of equations of motion, as is quite generally the case
for quantum many body systems [44, 45]. To proceed
one can then utilize standard approximation schemes of
many body theory.
D. Symmetries
The system exhibits a particle-hole symmetry in the
following sense. A jump of a particle to the right corre-
sp onds to a vacancy move by one step to the left. Sim-
ilarly, a particle entering the system at the left bound-
ary can be interpreted as a vacancy leaving the lattice,
and vice versa for the right boundary. Attachment and
detachment of pa rticles in the bulk is mapp e d to detach-
ment and attachment of vacancies, resp e c tively. There-
fore, one can easily verify that the transforma tio n
ˆn
i
(t) ↔ 1 − ˆn
N−i
(t) (5a)
α ↔ β (5b)
ω
A
↔ ω
D
(5c)
leaves Eqs. (4) invariant. Due to this property we can re -
strict the discussion to the cases ω
A
> ω
D
and ω
A
= ω
D
,
i.e. to K > 1 and K = 1, r e spectively. Eventually, for
ω
A
= ω
D
= 0, one arr ives ba ck at the TASEP respecting
the same particle-hole symmetry described above.
III. SIMULATIONS, MEAN-FIELD APPROXI-
MATION AND CONTINUUM LIMIT
In this Section we describe the Monte-Carlo simula-
tions (MCS) and the mean-field approximation (MFA)
we have used to compute the stationary average profile
hˆn
i
i and the average current h
ˆ
j
i
i = hˆn
i
(1 − ˆn
i+1
)i.
A. Simulations
We have performed Monte-Carlo simulations with ran-
dom sequential updating using the dynamical rules A) −
E) and evaluated both time and sample averages. The
resulting profiles coincide in both averaging procedures
for given pa rameters and different system sizes. In the
simulations, stationary profiles have been obtained ei-
ther over 10
5
time averages (with a typical time interval
≥ 10 N between each step of average) or over the same
number of samples (in the case of sample ave rages).
6

B. Mean-field approximation and continuum limit
Averaging Eqs. (4) over the stationary ensemble re-
lates the mean occupation number to higher order corre-
lation functions. The mean-field approximation consists
in neglecting these correla tio ns (random phase approxi-
mation) [44, 45]:
hˆn
i
(t)ˆn
i+1
(t)i = hˆn
i
(t)i hˆn
i+1
(t)i . (6)
Here, averages in the stationary state h·i are actually time
independent and correspond to either sample or time av-
erages due to the ergodicity property of the finite system.
In this approximation the average current is given by
h
ˆ
j
i
i = hˆn
i
(t)i(1 − hˆn
i+1
(t)i) .
Once we have defined the average density at site i as
ρ
i
= hˆn
i
(t)i, Eq. (4a) results in:
ρ
i−1
(1− ρ
i
)−ρ
i
(1− ρ
i+1
)+ ω
A
(1− ρ
i
)−ω
D
ρ
i
= 0, (7a)
while at the boundaries, Eqs. (4b), one obtains:
α(1 − ρ
1
) − ρ
1
(1 − ρ
2
) = 0 ,
ρ
N−1
(1 − ρ
N
) − βρ
N
= 0 . (7b)
Note that the average density is a real number with 0 ≤
ρ
i
≤ 1, and Eqs. (7) form a s et of N real algebraic non-
linear relations, which can be solved numerically.
An explicit solution of the previous equations can be
obtained by coarse-graining the discrete lattice with lat-
tice constant ε = L/N to a continuum, i.e. considering a
continuum limit. To simplify notation, we fix the total
length to unity, L = 1. For large systems N ≫ 1, ε ≪ 1,
the rescaled position variable x ≡ i/N, 0 ≤ x ≤ 1, is
quasi-continuous. An expansion of the average density
ρ(x) ≡ ρ
i
in powers of ε yie lds:
ρ(x ± ε) = ρ(x) ± ε∂
x
ρ(x) +
1
2
ε
2
∂
2
x
ρ(x) + O(ε
3
) . (8)
Taking the scaling of the Langmuir rates, Eq. (1 ), into
account, Eqs. (7) are to leading order in ε equivalent
to the following non-linear differential equation for the
average profile at the s tationary state [15]:
ε
2
∂
2
x
ρ + (2ρ − 1)∂
x
ρ + Ω
A
(1 − ρ) − Ω
D
ρ = 0 . (9)
Equations (7b) now translate into boundary conditions
for the density field, ρ(0) = α and ρ(1) = 1 − β. This can
be interpreted as if the system at both ends is in con-
tact with particle reservoirs of respective fixed densities
α and 1 − β. Note that the binding constant K remains
unchanged in this limit.
For finite ε, the average current is written j = −
ε
2
∂
x
ρ+
ρ(1 − ρ). In the continuum limit ε → 0
+
, this sug-
gests that j = ρ(1 − ρ) and that the current is bounded,
j ≤ 1/4. However, this bound holds only if the density is
a smooth function of the position x. We shall show that
density discontinuities can arise in the continuum limit.
Then, for small ε, these discontinuities would appear as
rapid crossover regions where one cannot neglect the first
order derivative term in the current definition so that the
relation j ≤ 1/4 needs not to be satisfied. The inequality
can be violated also by the additional c ontribution aris-
ing from current fluctuations neglected in the mean-field
approximation; see e.g. Fig. 9 at the system boundaries.
The equations obtained in mean-field approximation
and the subsequent continuum limit still respect the
particle-hole symmetry mentioned above. In terms of
the continuous averaged density ρ, the symmetry now
reads ρ(x) 7→ 1 − ρ(1 − x), α ↔ β, Ω
A
↔ Ω
D
. Note that
a numerical solution of the differential equation above
necessarily uses a discretization. Using a standard al-
gorithm for integrating differential equations, one would
merely recover the original mean-field equations (7).
Equation (9) has mathematical similarities to the sta-
tionary case of a viscous Burgers equation [46, 47, 48]
∂
t
ρ −
ε
2
∂
2
x
ρ + (∂
ρ
j)∂
x
ρ = F
A
− F
D
. (1 0)
In the Burgers equation ρ is identified with the fluid ve-
locity and j is related to this veloc ity via j = ρ
2
/2. In
our case, the hard-core interaction between particles im-
plies a non-linear current-density relationship. As shown
above, one finds in the continuum limit a parabolic rela-
tion j = ρ (1 − ρ). Dissipation is due to the term ε∂
2
x
ρ,
while the sources represent fluxes from a nd to the bulk
reservoir F
A
= Ω
A
(1−ρ) and F
D
= Ω
D
ρ. The net source
term F
A
−F
D
= (K +1)Ω
D
(ρ
l
−ρ) is positive or negative
depending on whether the density ρ is below or above the
Langmuir isotherm, ρ
l
= K/(K + 1), expressed in terms
of the binding constant K = Ω
A
/Ω
D
. In conjunction
with the non-linear current-density relation this implies
that the density of the Langmuir is otherm w ill act like
an “attractor” or “repellor”. If the slope of the current-
density relation is positive, ∂
ρ
j > 0, and the density at
the left end falls below the L angmuir isotherm the bulk
reservoir will feed particles into the system. As result,
the density grows towards ρ
l
as one moves away from
the boundary. In contrast, for a negative slope ∂
ρ
j < 0,
i.e. for densities larger than 1/2 , the density profiles are
“repelled” from the Langmuir isotherm. The latter case
can a lso be understood as an “attraction” by the Lang-
muir isotherm if read starting from the right end of the
system. Then, depending on whether the density at the
right boundary is larg e r or smaller than ρ
l
there is a loss
or gain of particles from the reservoir as one moves away
from the right boundary into the bulk. This will turn
out to be an important principle for the disc us sion of the
density profiles in later sections; see e.g. Section IV B.
From the analogy to fluid dynamics problems [49] one
exp ects singularities such as shocks in the density ρ to
appear in the inviscid or non-dissipative limit ε → 0
+
.
This conclusion can also be inferred by a direct inspection
of the non-linear differential equation (9) in the limit ε =
7

0. It reduces to a first order differential equation,
(2ρ − 1)∂
x
ρ + Ω
A
(1 − ρ) − Ω
D
ρ = 0 , (11)
instead of a s e c ond order one, while the solution still has
to satisfy two boundary c onditions. Such a boundary
value problem is apparently over-determined. However,
we can define solutions of Eq. (11) respecting only one
of the boundary c onditions. Depending on whether they
obey the boundary conditions on the left or right end
of the lattice we call them the left solution ρ
α
and the
right solution ρ
β
, respectively. Then, for 0 < ε ≪ 1 the
full solution of Eq. (9) will be close to ρ
α
for positions
on the left side of the system a nd similarly to ρ
β
on the
right side. In general, we can not expect both so lutions
to match continuously at some point in the bulk of the
lattice. Instead, for a large but finite system, the solu-
tion of Eq. (9) will exhibit a rapid crossover from the left
to the right solution. In the limit ε → 0
+
this c rossover
regime decreases in width and eventually leads to a dis-
continuity of the average density profile at some position
x
w
. Note that the discontinuity shows up only on the
scale of the system size, i.e. in the rescaled variable x,
whereas on the s c ale of the lattice spacing the crossover
region always covers a large number of lattice sites.
To loc ate the position of the discontinuity x
w
in the
limit of large system sizes N ≫ 1, i.e. ε → 0
+
, it is
useful to derive a continuity equation for the current j
and the sourc es F
A
, F
D
. Consider Eq. (9) in the form
∂
x
j = F
A
−F
D
, where j = −
ε
2
∂
x
ρ+ρ(1−ρ). Integrating
over a small region of width 2 δx close to x
w
, one obtains
j(x
w
+ δx) − j(x
w
− δx) =
R
x
w
+δx
x
w
−δx
(F
A
− F
D
) dx ≡ S
ε
. In
the limit ε → 0
+
the relation simplifies to j
α
(x
w
+ δx) −
j
β
(x
w
− δx) = S
0
, where we have defined the left current
j
α
= ρ
α
(1 − ρ
α
) and similarly for the right current j
β
.
Now, for δx → 0
+
, the contribution due to the sources
S
0
is of order δx yielding the matching condition in terms
of the left and right currents
j
α
(x
w
) = j
β
(x
w
) . (12)
The equivalent condition for the densities reads
ρ
α
(x
w
) = 1 − ρ
β
(x
w
) . (13)
A discontinuity of the density profile such as a domain
wall can appear in the system depending on whether the
previous condition is fulfilled fo r 0 ≤ x
w
≤ 1. Relation
(13), there fore, defines implicitly where a do main wall is
located in the system. It allows to compute the domain
wall position x
w
as well as its height ∆
w
= ρ
β
(x
w
) −
ρ
α
(x
w
). The domain wall separates regions of low (ρ <
1/2) and high density (ρ > 1/2). In the ensuing phase
diagram this will lead to an extended regime of phase
coexistence.
We shall see that in addition to domain walls, there
may appear als o discontinuities in the current [63], which
are located at the boundary of the system. We refer to
them as boundary layers.
IV. ANALYTIC SOLUTION OF THE CONTIN-
UUM EQUATION
In this Section, we will show in detail how one c an
treat the continuum equations, Eq. (9), analytica lly in
the limit ε → 0
+
. We shall compare these results with
numerical solutions of Eq. (9) for finite ε [64] , and
with corresponding profiles obtained from Monte-Carlo
simulations. For the Monte-Carlo simulation the plots
will s how the average density hˆn
i
i and the average cur-
rent h
ˆ
j
i
i = hˆn
i
(1 − ˆn
i+1
)i. The densities and currents
obtained from the numerical integration of the mean-
field equations at finite ε will be indicated as ρ
ε
and
j
ε
= −ε/2∂
x
ρ
ε
+ ρ
ε
(1 − ρ
ε
) in the figures, respectively.
This discussion will result in a classification of the
possible solutions as a function of the entry and exit
rates α and β, the binding constant K = Ω
A
/Ω
D
and
the detachment rate Ω
D
(phase diagram). Due to the
particle-hole symmetry we can restrict ourselves to val-
ues K ≥ 1. Then, there are two cases to distinguish:
K = 1 and K > 1. For K = 1 the constant density
profile, ρ
l
= K/(K + 1), given by the Langmuir kinet-
ics coincides with a point of particula r symmetry of the
TASEP. Indeed, for a density of ρ = 1/2 the system is
dual under particle-hole exchange, the non-linear term in
Eq. (9) vanishes, and it correspo nds to a point of maxi-
mal current [65]. It will turn out that K = 1 introduces
particular fea tur e s and requires a specific treatment and
discussion. Since it is technically simpler we discuss this
case first.
A. The sy mmetric case: K = 1
The mathematical analysis is simplified by the fact
that the attachment and detachment rates are equal,
Ω
A
= Ω
D
≡ Ω. Then Eq. (11) factorizes to:
(2ρ − 1)(∂
x
ρ − Ω) = 0 . (14)
The boundary conditions read ρ(0) = α and ρ(1) = 1−β.
Note that this equation is symmetric with respect to
particle-hole exchange. Indeed, except for the bound-
aries, the equation is invariant under the transformation
ρ(x) 7→ 1−ρ(1−x). This has impor tant consequences for
the density pro files, as w ill become clear in the following.
1. The density and current profiles
Equation (14) has only two basic solutions. A con-
stant density ρ
l
(x) = 1/2 identica l to the stoichiometry
in Langmuir kinetics and also the density in the maxi-
mal current phase of the TASEP. The other solution is
a linear profile ρ = Ωx + C. The value of the integra-
tion constant C depends on the boundary c ondition. One
finds C
α
= α and C
β
= 1 − β − Ω for solutions, ρ
α
(x)
and ρ
β
(x), ma tching the density at the left and the r ight
8

boundary, r e spectively. Depending on how the three so-
lutions ρ
α
(x), ρ
β
(x) and ρ
l
(x) can be matched, different
scenarios arise for the full density profile ρ(x). In the
following we discuss the characteristic features of the so-
lution in each quadrant of the α–β phase diagram for
fixed Ω.
a. Lower left quadrant: α, β ≤ 1/2. In this case the
boundary conditions enforce a density less than 1/2 a nd
greater that 1/2 at the left and right boundaries, respec-
tively. This allows for a continuous density pro file, where
a constant density of ρ
l
= 1/2 intervenes between the two
linear solutions emerging fro m the left and right bound-
aries. The corresponding positions separating the low
density from the max imal current phase, ρ
α
(x
α
) = 1/2,
and the maximal current phase from the high density
phase, ρ
β
(x
β
) = 1/2, are given by x
α
= (1 − 2α)/2Ω > 0
and x
β
= (2β + 2Ω − 1)/2Ω < 1, respectively. The phase
boundary x
α
→ 0
+
move s to the left upon increasing
the entry rate α → 1/2
−
and similarly x
β
→ 1
−
for the
exit-rate β → 1/2
−
. Hence, depending on the values of
the points x
α
and x
β
, one can classify the po ssible so-
lutions according to the relative or dering of the phase
boundaries: (i) x
α
< x
β
, (ii) x
α
= x
β
, and (iii) x
α
> x
β
.
(i) The density profile is continuous and piecewise lin-
ear and given by
ρ(x) =



Ωx + α for 0 ≤ x ≤ x
α
,
1/2 for x
α
≤ x ≤ x
β
,
Ω(x − 1) + 1 − β for x
β
≤ x ≤ 1 .
(15)
One observes a region of 3-phase coexistence: a low
density phase (LD) with ρ(x) < 1/ 2 and j(x) < 1/4
for 0 ≤ x ≤ x
α
, a maxima l current phase (MC) with
ρ(x) = 1/2 and j(x) = 1/4 for x
α
≤ x ≤ x
β
and high
density phase (HD) with ρ(x) > 1/2 and j(x) < 1/4 for
x
β
≤ x ≤ 1. For a plo t of the densities and currents see
Fig. 5.
(ii) For x
α
= x
β
the width of the intermediate max imal
current phase vanishes and the solution becomes a simple
linear profile, continuously matching the densities of the
LD and HD phase.
(iii) Upon further increasing x
α
over x
β
, the interven-
ing maximal cur rent phase is lost and it is no longer
possible to continuously concatenate the linear dens ity
profiles of the low and high density phase. There is nec-
essarily a density discontinuity, located at a point x
w
where the currents corresp onding to the r ight and left
solutions match, j
α
(x
w
) = j
β
(x
w
). The position of the
ensuing domain wall may be in or outside of the system.
This leads us to further distinguish between the following
three subcases:
(iii
1
) If x
w
< 0 the density profile in the bulk is
above 1/2, i.e. in a HD phase. The profile is entirely
described by the solution ρ
β
(x) up to a boundary layer
at the le ft end. One observes that the boundary layer
corresponds to a discontinuity in the current. The bulk
current j
β
(x → 0
+
) does in general not match the in-
coming particle flux α(1 − α) at the left boundary (see
Fig. 6).
0
0.4
0.8
0.16
0.2
0.24
0 0.2 0.4 0.6 0.8 1
ρ
ρ
α
ρ
β
j
x
j
β
j
α
(a)
(b)
FIG. 5: Average density ρ(x) (a) and current j(x) (b) for
parameters α = 0.4, β = 0.4, Ω = 0.3 and K = 1. In
this parameter range one observes a 3-phase coexistence: a
maximal current phase is intervening between a low and high
density phase. The profiles are computed analytically in the
inviscid limit (dashed lines) and numerically for ε = 10
−3
within a mean-field approximation (solid smooth line), and
from Monte-Carlo simulations (solid wiggly line). Note that,
within the resolution of the figures, the Monte-Carlo results
and t he numerical mean-field results can not be distinguished.
The analytic density profile is shown for the solutions respect-
ing the left and right boundaries conditions, ρ
α
and ρ
β
; we
also show the Langmuir isotherm ρ
l
= 1/2.
(iii
2
) For 0 < x
w
< 1 the domain wall is within the
system boundaries. Then the density profile connects a
LD profile to a HD profile via a domain wall at position
x
w
= (Ω − α + β)/2Ω [66]. The density profile is given
by:
ρ(x) =

Ωx + α for 0 ≤ x ≤ x
w
,
Ω(x − 1) + 1 − β for x
w
≤ x ≤ 1 .
(16)
Here we c an already illustrate an important feature of
our model. As one can infer from Fig. 7, the current
forms a cusp at the position of the domain wall, with
j
α
(x) and j
β
(x) being monotonically increasing and de-
creasing functions of x, respectively. This follows directly
from the continuum equation, Eq. (10), and the density
dependence of the source term F
A
− F
D
= 2Ω(1/2 − ρ),
which is positive or negative depending on whether the
density is sma ller or lar ger than 1/2. Hence, the domain
wall is located at a maximum of the current. In addi-
tion, the strict monotonicity o f the current also implies
that the domain wall is localized. A displacement of the
domain wall to the right of x
w
would result in a current
j
α
> j
β
. This in turn would incre ase the influx of par-
ticles at the left boundary, which will drive the domain
9

0
0.1
0.2
0 0.2 0.4 0.6 0.8 1
0
0.4
0.8
ρ
j
x
(a)
(b)
ρ
ε
FIG. 6: Average den sity ρ(x) (a) and current j(x) (b) for
parameters α = 0.4, β = 0.1, Ω = 0.3 and K = 1. We use
the same legend as in Fig. 5. The bulk profile is almost com-
pletely described by the solution ρ
β
matching only the right
boundary condition. At the left end, the bu lk density does
not match the boundary condition. As a result, a boundary
layer appears. Only there does one find a noticeable differ-
ence between the profiles of the Monte-Carlo simulation, the
numerical computation at finite ε and the analytic profile for
vanishing ε.
wall back to its original position x
w
[67].
(iii
3
) The s olution for x
w
> 1 can be inferred by
particle-hole s ymmetry from case (iii
1
). The low density
profile is given by the solution ρ
α
(x) up to a boundary
layer at the right end.
b. Lower right quadrant: α > 1/2, β < 1/2. Here
the density at both left and right boundaries is larger
than ρ
l
= 1/2. Two different scenarios are possible. In
the first scenario, the slope Ω of the density profile ρ
β
(x)
(matching the density at the rig ht boundary) is so small
that ρ
β
(x) is always larger than ρ
l
= 1/2; this requires
Ω < 1/2 − β. Then, the bulk of the system is in the HD
phase with a boundar y layer on the left. This scenario
is identical to the previous case (iii
1
), s uch that there is
no qua lita tive change in the bulk upon crossing the line
α = 1/2. In other words, there is no phase boundary
and the system remains in the HD phase. In the second
scenario, the slope Ω > 1/2−β such that we have a phase
boundary between a high density and a maximal current
phase. This solution can also be viewed as a limit of
the 3-phase coexistence region, where for α → 1/2
−
the
phase boundary x
α
leaves the system through the left
end and a boundary layer is created replacing the LD
region (see Figs. 8(a,b)).
0
0.1
0.2
0 0.2 0.4 0.6 0.8 1
0
0.4
0.8
ρ
β
ρ
α
j
α
j
β
j
ρ
x
(a)
(b)
ρ
ε
FIG. 7: Average density ρ(x) (a) and current j(x) (b) for
parameters α = 0.2, β = 0.1, Ω = 0.3 and K = 1. We
use the same legend as in Fig. 5. Only in proximity to the
domain wall the results from the mean-field approximation
show deviations from the density profile obtained by Monte-
Carlo simulation.
0.16
0.2
0.24
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
x
ρ
β
j
β
ρ
j
(a)
(b)
ρ
ε
0
0.2 0.4 0.6 0.8 1
x
ρ
α
j
α
(c)
(d)
ρ
ε
FIG. 8: (a)-(b) Average density ρ(x) and current j( x) for
α = 0.8, β = 0.35, Ω = 0.3 and K = 1. We use the same
legend as in Fig. 5. Except for the left boundary layer, in fig.
(a) t he analytic solution is described by the Langmuir den sity
ρ
l
= 1/2 and the density ρ
β
matching the right boundary
condition. (c)-(d) Average density ρ(x) and current j(x) for
α = 0.35, β = 0.8 and the same Ω and K as before. Note that
the curves map to those of (a)-(b) by particle-hole symmetry.
10

c. Upper left quadrant: α < 1/2, β > 1/2. This
region in par ameter space is obtained using particle-hole
symmetry from the results for the lower right quadr ant
in the preceding paragraph (see Figs. 8(c,d)).
d. Upper right quadrant: α, β > 1/2. Here two
boundary layers are formed, a nd the bulk of the system
is characterized by a constant density equal to 1/2 (see
Fig. 9). This corresponds to the maximal current phase,
which remains unchanged as compared to the TASEP
without particle on- and off-kinetics. Note again that
due to K = 1 the density with maximal current coin-
cides with the Langmuir isotherm ρ
l
= 1/2.
0.16
0.2
0.24
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
ρ
j
x
(a)
(b)
ρ
ε
ρ
ε
FIG. 9: Average den sity ρ(x) (a) and current j(x) (b) for
parameters α = 0.8, β = 0.8, Ω = 0.3 and K = 1. We use th e
same legend as in Fig. 5. The bulk density profile is given by
the Langmuir density ρ
l
= 1/2 which corresponds also to the
maximal current phase. Due to fluctuations, neglected in th e
mean-field approximation, the current profile obtained from
the simulation exceeds the value 1/4 at each boundary.
2. The phase diagram
The analysis of the current and density profiles allows
to draw cuts of the phase diagram in the (α, β)–plane
for fixed values of Ω. Note that the particle-hole sy mme-
try renders all diagrams symmetric with respect to the
diagonal α = β. Depending on the kinetic rate Ω one
can distinguish three topologies. Topologies of the phase
diagrams change at critical values Ω = 1/2 and Ω = 1;
see Fig. 10.
For 0 < Ω < 1/2, Fig. 10(a), the phase diagram con-
sists of seven phases. A 3-phase coexistence reg ion LD-
MC-HD at the center is surr ounded by three 2-phase co-
existence regions LD-HD, MC-HD and LD-MC. Pure LD,
MC
HD
LD
LD-HD
LD-MC
MC-HD
LD-MC-HD
β
0
1/2 1
Ω
Ω=0.3
1/2-Ω
MC
LD-HD
LD-MC
MC-HD
LD-MC-HD
0
1
1/2
MC
LD-MC
MC-HD
LD-MC-HD
α
0
1/2
1
1/2
1
1/2-Ω
β
α
l , r
r
l
1
r
l
(a)
(b) (c)
x =0
w
x =1
w
α
FIG. 10: Cut of the phase diagram on th e (α, β)–plane in
the mean-field approximation for K = 1 and different val-
ues of Ω: (a) Ω = 0.3, (b) Ω = 0.5, (c) Ω = 1.0. The
cases (a)-(c) correspond to the three different topologies of
phase diagrams discussed in the main text. All lines repre-
sent continuous transitions between different regions in the
(α, β, K = 1, Ω
D
= const.) cut of the 4-dimensional param-
eter space. The line parallel to the anti-diagonal is defined
through the relation α + β + Ω = 1. It represents the bor-
der line where the points x
α
and x
β
(i.e. the points where
the left and right solutions ρ
α
and ρ
β
meet the Langmuir
isotherm ρ
l
= 1/2) coincide, x
α
= x
β
. The phase boundaries
of th e LD-HD coexistence phase, x
w
= 0 and x
w
= 1, corre-
spond to regions in which the domain wall is located at one of
the system boundaries. These lines were computed by using
the matching conditions for the currents: j
α
(1) = β(1 − β)
and j
β
(0) = α(1 − α). In figure (a), we also emphasize the
presence of t he boundary layers at the left or the right end
of the system. These are indicated with the letters ”l” and
”r”, respectively. Such boundary layers remain present in the
same regions as in figure (a) also for increasing Ω.
HD and MC phases are contiguous to the 2-phas e r e -
gions. All lines between different regions represent con-
tinuous changes in the average density ρ. The 3-phase
coexistence region, and two of the 2-phase coexistence
regions (LD-MC and MC-HD) are characterized by con-
11

tinuous density profiles. This is mainly due to the maxi-
mal current phase with density ρ
l
(x) = 1/2. Acting a s a
”buffer”, this phase intervenes b etween the LD a nd HD
phase or connects the LD and HD phases with the right
and left boundary, respectively. Discontinuities only ap-
pear as current and density discontinuities (boundary
layers) at the system boundaries. This ha s to be con-
trasted with the density profile in the coexistence region
between the LD and HD phase. Here, a density disconti-
nuity in the bulk (domain wall), separating both phases,
is formed.
It is also interesting to consider the limit Ω → 0, as one
exp ects to recover the TASEP scenario. Indeed, us ing the
previous results, it is easy to show that for decreasing Ω,
the width of the 2-phase regions , as well as that of the
3-phase region, shrinks to zero. The resulting diagram
reproduces the well known to pology of the pure TASEP
in the mean-field approximation [50].
Upo n increasing Ω up to the value 1/2, we find the first
topology change in the phase diagram. The HD and LD
phases gradually disappear, leaving only the 2-phase or
3-phase coexistence re gions at Ω = 1/2, see Fig. 10(b). If
Ω becomes larg e r than 1, the LD-HD coexistence region
disappears; see Fig. 10(c).
The Langmuir kinetics is approached for Ω → ∞.
Although the topology of the pha se diagram does not
change anymore, the phases become almost indistin-
guishable for large k inetic rates. Here the Langmuir
isotherm ρ
l
= 1/2 occupies most of the bulk, whereas
the LD and HD regions are confined to a vicinity of the
boundaries.
B. The generic case: K > 1
Though simpler to a nalyze, the pr e vious case K = 1 is
somewhat artificial as it requires a fine-tuning of the on-
and off-rates. Generally, one would expect K 6= 1 . Due
to particle-hole symmetry we can restrict ourselves to
K > 1. The analysis becomes significantly more complex
since the continuum equation for the density, Eq. (11),
no longer factorizes into a simple form as for K = 1.
1. The density and current profiles
To proceed, it is convenient to introduce a rescaled
density of the form
σ(x) =
K + 1
K − 1
[2ρ(x) − 1] − 1 , (17)
where σ = 0 correspo nds to the Langmuir isotherm ρ
l
=
K/(K + 1). Since the density ρ(x) is bound within the
interval [0, 1], the rescaled density σ(x) can assume values
within the interval [−2K/(K − 1), 2/(K − 1)]. Then the
continuum equation Eq. (11) simplifies to
∂
x
σ(x) + ∂
x
ln |σ(x)| = Ω
D
(K + 1)
2
K − 1
. (18)
Direct integrations yield
|σ(x)| exp(σ(x)) = Y (x) , (19)
where the function Y (x) is
Y (x) = |σ(x
0
)| exp

Ω
D
(K + 1)
2
K − 1
(x − x
0
) + σ (x
0
)

,
(20)
and σ(x
0
) is the value of the reduced density at the ref-
erence point x
0
. In particular, the o nes that match the
boundary condition o n the left or right end of the system
are wr itten
Y
α
(x) = |σ(0)| exp

Ω
D
(K + 1)
2
K − 1
x + σ(0 )

(21)
Y
β
(x) = |σ(1)| ex p

Ω
D
(K + 1)
2
K − 1
(x − 1) + σ(1)

,
where the boundary va lues σ(0) and σ(1) can be written
in terms of α and β using Eq. (17) and the bo unda ry
conditions ρ(0) = α a nd ρ(1) = 1 − β.
Equations of the form of Eq. (19 ) appear in various
contexts such as enzymology, population growth pro-
cesses and hydrodynamics (see e.g. Ref. [51]). They are
known to have an explicit solution written in terms of a
sp e c ial function called W -function [51]:
σ(x) = W (Y (x)) , σ(x) > 0
σ(x) = W (−Y (x)) , σ(x) < 0 . (22)
The Lambert W -function (see Fig . 11) is a multi-valued
function with two real branches, which we refer to as
W
0
(ξ) and W
−1
(ξ). The branches merge at ξ = −1/e,
where the Lambert W -function takes the value −1.
The first branch, W
0
(ξ), is defined for ξ ≥ −1/e ;
it diverges at infinity sub-logarithmically. The second
branch, W
−1
(ξ), is always negative and defined in the do-
main −1/e ≤ ξ ≤ 0. In the vicinity of the point ξ = −1/e
the function W (ξ) behaves like a square root of ξ since
one gets ∂
ξ
W = W/[(1 + W )ξ] by the definition of the
Lambert W -function, W (ξ) exp(W (ξ)) = ξ.
Using these properties of the Lamb e rt W -function, the
branch of W is selected according to the value of the
rescaled density σ. For σ ∈ [−2K/(K − 1), −1] the re le -
vant solution is W
−1
(−Y ), while for σ ∈ [−1, 0] one ob-
tains W
0
(−Y ). Finally, in the interval σ ∈ [0, 2/(K − 1)]
one finds W
0
(Y ).
The solutions are matched to the boundary conditions
at the left and right ends according to the entry or exit
rates. The left and right solutions, ρ
α
(x) and ρ
β
(x), are
then computed from the express ions in Eqs. (23) upon
using the coordina te transformation given by Eq. (17).
Fig. 12 provides a graphical representation of the possi-
ble set of solutions o f the first order differential equation,
Eq. (18). In order to decide which one of them are ac-
tually physically realized, one needs to go back to the
full equation, either in its discrete form Eq. (7a) or its
12

5
4
3
2
1
0
1
y
0.5
0.5 1 1.5
2
x
W (x)
-1
W (x)
0
branch point
-1/e
FIG. 11: The real b ranches W
0
(ξ) and W
−1
(ξ) of the Lambert
W - function.
0
1
0.2 0.4 0.6 0.8
1
0
1/2
K/(K+1)
W (-Y (x))
-1 α
W (-Y (x))
0 α
W (Y (x))
0 α
x
(a)
α=α
c
1/2
K/(K+1)
W (Y (x))
0 β
W (-Y (x))
0 β
W (-Y (x))
-1 β
β=β =1/2
c
(b)
0
1
FIG. 12: Mathematical solutions for (a) the left density ρ
α
(x)
and (b) the right density ρ
β
(x) for K = 3, Ω
D
= 0.1 and
different values of the entry and exit rate α and β. The so-
lutions which approach the Langmuir isotherm are those for
α, β ≤ 1/2 (thick lines). The solutions where the branch-
ing point coincides with the right boundary are indicated by
α
c
= 0.038532... and β
c
= 1/2.
continuous version Eq. (9). Analogous to the TASEP
a solution matching the density prescribed by the left
boundary condition is stable only if α < 1/2 [68]. Such
solutions are shown as thick lines in Fig. 12(a). They are
monotonically increasing towards the La ngmuir isotherm
ρ
l
= K/(K + 1) > 1/2 . This can be understood as a con-
sequence of the accumulation of particles from the bulk
reservoir via the Langmuir kinetics with increasing dis-
tance from the left boundary. One might now expe c t that
the density will finally approach the Langmuir isotherm.
But, this is not the case. Instead, we find that the den-
sity ρ
α
(x) never increases beyond 1/2, where the current
reaches its largest possible value j
max
= 1/4. Mathemat-
ically, this is a direct consequence of the analytic proper-
ties of the Lambert W -function, which has a branching
point at a density 1/2; see Fig. 12(a). With decreasing α
the site where ρ
α
(x) meets the density 1/2 moves to the
right. At a critical value of the entry rate, α
c
(Ω
D
, K),
the branching point of the left solution ρ
α
touches the
right boundary.
Similarly to the disc ussion in the previous pa ragraph,
solutions matching the r ight boundary condition are sta-
ble only if β ≤ 1 /2. The corresponding density profiles,
shown as thick lines in Fig. 12(b), are always in a high
density regime, i.e. ρ
β
(x) ≥ 1/2. If the density at the
right boundary matches the Langmuir isotherm, the right
solution is flat ρ
β
(x) = ρ
l
. Otherwise, the source terms
do not c ancel, le ading to a net detachment/attachment
flux such that the right density profiles decay monotoni-
cally towa rds the Langmuir isotherm as one moves from
the right boundary to the bulk. As a consequence, the
right density ρ
β
(x) never crosses the Langmuir isotherm.
The density profile for β = 1/2 is an extremal solution
exhibiting the lowest possible density (ρ = 1/2) and high-
est current (j = 1/4) a t the right end, which then also
coincides with the branching point o f the Lambert W -
function.
In conclusion, for the left rescaled solution σ
α
(x), an
entry rate 0 ≤ α ≤ 1/2 implies −2K/(K − 1) ≤ σ ≤ −1.
Hence we have according to the previous analysis:
σ
α
(x) = W
−1
(−Y
α
(x)) < 0 . (23a)
For the right resca le d solution σ
β
(x), one finds c orre-
sp ondingly
σ
β
(x) =









W
0
(Y
β
(x)) > 0 , 0 ≤ β < 1 − ρ
l
0 , β = 1 − ρ
l
W
0
(−Y
β
(x)) < 0 , 1 − ρ
l
< β ≤ 1/2 ,
(23b)
where ρ
l
= K/(K + 1) is the constant density of the
Langmuir isotherm. After the coordinate change (17),
the general solution of the continuum mean-field equa-
tion at ε → 0
+
, Eq. (11), is obtained by matching left
and right solutions ρ
α
and ρ
β
. The remaining task is now
to identify the different scenarios where domain walls and
boundary layers appear. Such analytic results are con-
firmed by the numerical computation at finite ε.
a. Lower left quadrant: α, β ≤ 1/2. This is the only
case where there are solutions that approach the Lang-
13

muir isother m in the bulk and match both boundary con-
ditions. The full density profile is obtained by finding the
position x
w
where the left and right currents coincide, i.e.
ρ
α
(x
w
) = 1 − ρ
β
(x
w
). One has to consider three cases:
(i) 0 < x
w
< 1, (ii) x
w
< 0, and (iii) x
w
> 1.
In case (i), a domain wall is formed s eparating a region
of low density on the left with a region of high density on
the r ight. Depe nding on whether 1 − β is above or below
ρ
l
, different profiles are observed, see Figs. (13)(a),(c).
In the case β = 1 − ρ
l
, one obtains a flat profile of ρ
β
matching the value of the Langmuir isotherm ρ
l
. We
note again that the left and right solutions approach the
Langmuir isotherm in the bulk. In analogy with the case
K = 1, the doma in wall is stabilized by the current pro -
files controlled by the boundary conditions.
0
0.1
0.2
0 0.2 0.4 0.6 0.8
1
0
0.2
0.4
0.6
0.8
ρ
j
x
ρ
α
ρ
β
j
β
j
α
(a)
(b)
0 0.2 0.4 0.6 0.8 1
j
α
j
β
ρ
β
ρ
α
x
(c)
(d)
ρ
ε
ρ
ε
FIG. 13: Average density ρ(x) (a)-(c) and corresponding cur-
rent j(x) (b)-(d) for α, β ≤
1
2
in a parameter regime showing
phase separation. We have chosen Ω
D
= 0.1, K = 2 and (a)-
(b) α = 0.1, β = 0.1 or (c)-(d) α = 0.3, β = 0.4. Solid lines
correspond to the numerical solution of the mean-field theory
with ε = 10
−3
. Monte-Carlo simulations are shown as solid
wiggly line. The flat dashed line represents the Langmuir
isotherm, ρ
l
= K/(K + 1). The other dashed lines represent
the analytic solutions given by the branches of the Lambert
W - functions matching the boundary conditions on the right
and left end, respectively. For both cases (a) and (c), the
solution matching the left boundary condition ρ
α
is given by
the branch of the Lambert W -function W
−1
(−Y
α
). For the
solution matching th e right boundary condition, ρ
β
, one has
to consider the branch W
0
. For (a) the branch of W has the
argument Y
β
(x), while for (c) the argument is −Y
β
(x) (see as
illustration also Fig. 12).
In cases (ii) and (iii), one of the two phases is confined
to the boundary. Explicitly, for (ii) the bulk is charac-
terized by a HD with a b oundary layer at the left end,
see Fig. 14(a). Correspondingly, fo r (iii) the so lution ex-
hibits a LD bulk phase accompanied by a b oundary layer
on the right end side of the system, see Fig. 15(a).
b. The upper left quadrant, α < 1/2, β > 1/2. As
discussed above, for β > 1/2 the solutions of the first
0
0.1
0.2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
ρ
j
x
(a)
(b)
FIG. 14: Average density ρ(x) (a) and current j(x) (b) for
α = 0.3, β = 0.1, Ω
D
= 0.1 and K = 2. We use the same
legend as in Fig. 13. Except the left boundary layer, the bulk
density profile is given by the Lambert W -function, ρ
β
=
W
0
(Y
β
(x)).
0
0.1
0.2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
ρ
j
x
(a)
(b)
FIG. 15: Average density ρ(x) (a) and current j(x) (b) for
α = 0.1, β = 0.4, Ω
D
= 0.1 and K = 2. We use the same
legend as in Fig. 13. Except the right boundary layer, the
bulk density profile is given by the Lambert W -function, ρ
α
=
W
−1
(Y
α
(x)).
14

order differential equation, Eq. (11), matching the right
boundary condition are physically unstable. Instead, the
actual density profile at the right boundary approaches
the extr e mal solution W
0
(−Y
β=1/2
) o f the first order dif-
ferential equation. The density difference to the bound-
ary value is bridged by a boundary layer, which vanishes
in the limit ε → 0
+
.
For the discussion of the density pro files in the up-
per left quadrant we can simply parallel the arguments
used for the lower left quadra nt, once the right solution
has bee n substituted with the extremal one. Depend-
ing on the matching of the current, one finds again three
cases, a LD phase, a 2-phase LD-HD coexistence and a
HD phase. We conclude that the phases of the lower left
quadrant extend to β > 1/2 with phase b oundaries which
are independent of the exit rate β, i.e. parallel to the β
axis. The HD phase for β > 1/2 has some interesting
features which are genuinely distinct from the HD phase
for β < 1/2. The density pro file in the bulk is indepen-
dent of the entrance and exit rates, α and β, at the left
and right boundaries; it is given by the extremal solution
W
0
(−Y
β=1/2
). The density approaches ρ(L) = 1/2 and
hence the current the maximal possible value j
max
= 1/4
at the right boundary. These features are reminiscent
of the maximal current phase for the TASEP. The only
difference seems to be tha t here current and density are
spatially varying along the system while they ar e con-
stant for the TASEP. The essential characteristic in both
cases is that the behavior of the system is determined
by the bulk and not the boundaries. One is reminded
of similar behavior of the Meissner phase in supercon-
ducting materials. In the ensuing phase diagram we will
hence indicate this regime as the “Meissner” (M) pha se
to distinguish it from the HD phase with boundary dom-
inated density profiles [69]. Note also that the parameter
range for the M phase is broadened as compared to the
maximal current phase of the TASEP.
c. The remaining quadrants, α > 1 /2. At α = 1/2,
the system is already in the high density phase where the
bulk profile does no t match the entry rate. Increasing α
beyond the value 1/2, therefor e merely a ffects the bound-
ary layer at the left end. The dens ity profile is given by
the rig ht solution ρ
β
for β < 1/2 o r the extremal one for
β ≥ 1/2 as before; for an illustration compare Fig. 16.
For β ≥ 1/2 the same conclusion apply as in the pre c e d-
ing paragraph, resulting in a “Meissner” phase for the
upper right quadrant.
Let us conclude this subse c tion with some additional
comments on boundary layers. Boundary layers arise
from a mismatch between the bulk profile and the bound-
ary conditions. They ca n bend either upwards or down-
wards depe nding on w hether the left or right boundary
rates are above or below the values of the bulk solution at
the ends. For example, in the right lower quadrant of the
HD phase, a change from a depletion to an accumulation
layer at the left end of the system occurs at α = ρ
β
(0)
for β < 1 /2.
0
0.1
0.2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
ρ
j
x
ρ
β
(a)
(b)
ρ
ε
FIG. 16: Average density ρ(x) (a) and current j(x) (b) for
Ω
D
= 0.1, K = 2, α = 0.75 and β = 0.75. We use the same
legend as in Fig. 13. Except for th e left and right boundary
layers, the bulk profile obtained from the analytic mean-field
result is given by the branch ρ
β
(x) = W
0
(−Y
β
(x)) of the
Lambert W -function computed for β = 1/2.
2. The phase diagram
We discuss the to pology of the phase diagram with re-
sp e c t to cuts in the (α, β)–plane for different values of K
and Ω
D
. We first consider the situation in which Ω
D
is
fixed and K increases, starting from values slightly larger
than unity. Figure 17(a) s hows the phase diagram for
K = 1.1. A low density (LD) phas e occupies the upper
left of the plane, while a high density (HD) and a Meiss-
ner (M) phase are located on the right. In between there
is a 2-phase coexistence region (LD-HD). In the coexis-
tence phase a domain wall is localized at the point x
w
in
the bulk. The boundaries of the coexistence region in the
phase diagram are deter mined by those parameters where
the domain wall hits either the entrance, i.e. x
w
= 0, or
the exit of the system, x
w
= 1. For β > 1/2, the density
profile only develops a boundary layer at the right end,
but remains unchanged in the bulk. Since the domain
wall position becomes indepe ndent of β, the boundaries
of the 2-phase coexistence region become parallel to the
axis α = 0. It is important to remark that from the an-
alytic results the left solution ρ
α
is strictly smaller than
1/2, except for the special point C in the pha se dia gram
where ρ
α
(1) = β = 1/2. We shall see in Section V B
that in the vicinity of this point the doma in wall exhibits
critical properties.
Upo n increasing K, the LD phase pro gressively shrinks
to a region close to the β-axis, while the size of the two
other phases increases; see Figs. 17(b) and 17(c). A
15

change of topology occurs when the LD phase collapses
on this axis which happens upon passing a critical value
of K. This critical value depends on Ω
D
and can be
computed using the expressions in Eqs. (13) and (23). A
further increase of K results in a decrease of the extension
of the LD-HD region in the phase diagram, see Figs. 17.
Eventually, fo r very large K the average bulk density in
the HD and M regions approaches saturation ρ
bulk
= 1.
Similarly, increasing Ω
D
at fixed K, the same topology
change occurs, as described above; se e Fig. 18. However,
we note the different limiting behaviors for Ω
D
→ 0
+
and
Ω
D
→ ∞. In the first case, we are considering the limit
of the model to the TASEP for a given binding cons tant
K (although K 6= 1). The 2-phase coexistence region
LD-HD shrinks continuously to the line α = β. In the
same limit, in the upper rig ht quadrant, α, β > 1/2, the
M phase approaches co ntinuously the MC phase of the
TASEP. For a very large detachment rate Ω
D
, the right
boundary of the LD-HD coexistence phase approaches a
straight line at finite entry rate α that can be computed
from the analytic solution as equal to 1 − ρ
l
. In the same
limit, the average density in the bulk reaches asymptoti-
cally the value ρ
bulk
= ρ
l
of the La ngmuir isotherm.
Eventually, one observes that all phase boundaries be-
tween the LD-HD coexistence and the HD phase, i.e.
where the domain is pinned at x
w
= 0 , intersect at the
same point N for any value of the detachment rate Ω
D
.
This nodal point N can be evaluated as α = β = 1− ρ
l
=
1/(K + 1). At this point, indeed, the average density
ρ is given by the flat profile of the Langmuir isotherm
ρ
l
= K/(K + 1) which is obviously independent of Ω
D
.
As a re sult, the domain wall does not move from x
w
= 0
for a ny value of the detachment rate Ω
D
. Interestingly,
one remarks that both points C and N approach contin-
uously the triple p oint of the TASEP α = β = 1 /2 in the
simultaneous limit Ω
D
→ 0
+
and K → 1
+
.
V. DOMAIN WALL PROPERTIES
The knowledge of the analy tic solution in the mean-
field approximation allows for a detailed study of the
behavior of the domain wall height and position upon
a change of the system parameters . While the results for
the symmetric case K = 1 are more or less trivial, novel
properties emerge for K > 1. In this Section, we shall
start from the description of the domain wall behavior on
the (α, β)–plane of the phase diagram along trajectories
of constant entry o r exit rates, respectively.
A. Position and amplitude of the domain wall on
the (α, β)–plane
Figures 19(a,b) show the dependence of the domain
wall position, x
w
, and height, ∆
w
, on the entry rate α
along lines of constant exit rate β. As can be inferred
from the structure of the phase diagram presented in the
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8
LD
HD
0 0.2 0.4 0.6 0.8
HD
0
0.2 0.4 0.6 0.8
1
LD-HD
HD
β
α
α
α
K=1.1
K=3.0
K=6.0
r
l
r , l
0
0.2
0.4
0.6
0.8
1
β
1
(a)
(b)
(c)
1
x =1
x =0
r
w
w
C
LD-HD
LD-HD
M
M
M
FIG. 17: Cuts of the phase diagrams on the (α, β)–plane
obtained by the exact solution of the stationary mean-field
equation (11) in the inviscid limit ε = 0 for Ω
D
= 0.1 and
(a) K = 1.1, (b) K = 3.0, (c) K = 6.0. The two lines,
corresponding to regions in which the domain wall is located
at x
w
= 0 and x
w
= 1, are obtained by using the matching
conditions for the currents: j
α
(1) = β(1 − β) and j
β
(0) =
α(1 − α). In figure (a), we emphasize several features. With
the letters ”l” and ”r” we indicate the presence of boundary
layers in the average density profile, forming at the left or
the right end of the system, respectively. In the lower left
quadrant, the left and right boundary layers form whenever
the domain wall exits the system on the left and right end side.
At the phase boundary between the HD and M phases, for
β = 1/2, a boundary layer forms at the right end. Note that
also in t he M phase ρ
bulk
> 1/2. The presence of boundary
layers in the d ifferent phases of the (α, β)–plane is conserved
upon variation of the binding constant K. The filled black
circle represents the critical point C where the domain wall
exhibits critical behavior; see Sect. V B. This critical point
exits the plane for large values of K, accompanied with a
topological change of the p hase diagram.
preceding Section, for a small enough exit rate β, a do-
main wall can for m in the bulk with a finite amplitude
even for a vanishing entry rate, α = 0. For larger β,
one observes that the do main wall builds up with a finite
16

0
0.2 0.4 0.6 0.8 1
HD
LD-HD
0
0.2
0.4
0.6
0.8
1
0
0.2 0.4 0.6 0.8 1
HD
LD
0
0.2
0.4
0.6
0.8
1
0
0.2 0.4 0.6 0.8 1
HD
LD
Ω
D
=0.2
Ω
D
=0.05
Ω
D
=0.01
β
β
α
α
α
r , l
l
r
(a)
(b)
(c)
x =1
x =0
r
w
w
C
N
LD-HD
HD-LD
M
M
M
FIG. 18: Cuts of the phase diagrams as in Fig. 17 for K = 3
and (a) Ω
D
= 0.01, (b) Ω
D
= 0.05, (c) Ω
D
= 0.2. The white
circle corresponds to a nodal point of the system N defined
by t he condition α = β = 1 − ρ
l
= 1/(K + 1). Every line
x
w
= 0 crosses this point for an increasing Ω
D
.
height on the right boundary only above some specific
value o f α. If one regards the domain wall height as a
kind of order parameter for the coexistence phase such a
behavior can be ter med a first order transition. This has
to be contrasted with the case β = 0.5, where the domain
wall enters the sy stem at x
w
= 1 with infinitesimal height
at a critical e ntry rate α = α
c
. In the sa me terminology
this would then be a second order transition. Indeed, as
we are going to discuss in the next subsection, the do-
main wall exhibits critical properties at this point. In
the phase diagram (Fig. 17(a)) the corresponding critical
point is indicated as C.
In all cases, upon incr e asing α and hence the influx of
particles, the domain wall changes its position continu-
ously from the right to the left end of the sys tem. Then,
at some value α which depends on β, the domain wall
leaves the system with a finite amplitude ∆
w
.
Similar behaviors of the position and height of the do-
main wall is found as a function of β for fixed values of α;
see Fig. 19(c,d). Here one finds that, upon increasing β
0 0.2 0.4 0.6 0.8 1
β
α=0.2
α=α
c
α=0.01
α=0.2
α=α
c
α=0.01
(c)
(d)
0
0.2
0.4
0.6
0.8
0 0.1
0.5
α
α
c
x
w
β=0.5
β=0.4
β=0.2
0
0.2
0.4
0.6
α
c
0.2 0.3
0.4
β=0.5
β=0.4
β=0.2
(a)
(b)
∆
w
FIG. 19: (a)-(b) Domain wall position x
w
and height ∆
w
as
a function of th e entrance α for different values of the exit rate
β at Ω
D
= 0.1 and K = 3. At the critical point α = α
c
and
β = 1/2 a domain wall forms at the right end of the system
with an infinitesimal height ∆
w
. The value of the ”critical”
entry rate is α
c
= 0.038532... and can be written explicitly by
using the analytic solution in the mean-field approximation,
see Eq. (25). (c)-(d) Domain wall position x
w
and height
∆
w
as a function of the exit rate β for different values of the
entrance rate α at Ω
D
= 0.1 and K = 3. For α = α
c
and
β = 1/2 a domain wall forms at the right end of the system
with an infinitesimal height ∆
w
. For exit rates β > 1/2, both
domain wall position x
w
and height ∆
w
become independent
of β. Changes in th e exit rate only affect the size and shape of
the boundary layer on the right end, but not the bulk density
profile.
and hence reducing the out-flux of particles the domain
wall position x
w
move s continuously from the left to the
right boundary . For small α, a domain wall is formed
at a finite position x
w
and β = 0. For larger entry rates,
the domain wall for ms a t x
w
= 0 with a finite amplitude
only for finite values of the exit rate β. As before, the
amplitude of the domain wall ∆
w
vanishes only for the
critical value α = α
c
at β = 1/2. Indeed, when α > α
c
and β > 1/2, one notes that the domain wall position
x
w
remains c onstant upon changing β. As we have ex-
plained above, this corresponds to the situation where
the bulk profile is unaffected by a change in the exit rate
(M phase). Only the magnitude of the boundary layer
changes with increasing β.
B. Critical properties of the domain wall
In this Section, we disc uss the domain wall properties
close to the special point C where the domain wall for ms
with infinitesimal height. The analysis will make use of
the analytic solution in the mean-field approximation.
We show that the do main wall emerges as a consequence
of a bifurcation phenomenon, and calculate the resulting
17

non-analytic behavior of its height and position.
At the point C, the analytic solution of the mean-field
equations is described by a low density profile ρ(x) =
ρ
α
(x) that not only matches the boundary conditions at
the left but a lso the one at the right end; see Fig. 20. This
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1
K=1.1
K=3
x
ρ(x)
α=0.347303...
α=0.038532...
FIG. 20: Average density profiles computed analytically in
the inviscid limit expressed in terms of Lambert W -function
(dashed lines) and numerically for ε = 10
−3
within a mean-
field approximation (solid smooth line). Parameters are: α =
α
c
(see Eq. (25) for the analytic expression), β = 0.5, Ω
D
=
0.1 and different values of K. The profile is entirely given by
the left solution ρ
α
for the value of the entry rate α
c
, defined
by the condition ρ
α
(x = 1) = 1/2, and β = 1/2. Note that
in this case, ρ
α
matches simultaneously the left and the right
boundary conditions.
implies that the left and right currents also match at the
right end of the system. Interestingly, at this position
the current j is maximal [70]. By a small change of the
system parameters in the 2-phase LD-HD coexistence re-
gion the domain wall forms at the right end characterized
by a small height, provided that β ≤ 1/2.
Using the analytic solution (23) o ne can give an explicit
expression of the critical point C as a function of Ω
D
and K. The condition that the left boundary matches
the va lue 1/2 at x = 1 translates to σ
α
(x = 1) = −1 or,
using the properties of the Lambert W -function, as:
Y
α
(x = 1) = 1/e . (24)
From the expression of the function Y
α
, Eq. (21), and
the initial condition σ(0) = (2α − 1)(K + 1)/(K − 1) − 1,
one c omputes the value of the critical entry rate
α
c
=
K
(K + 1)
+
K − 1
2(K + 1)
W
−1

− exp

−Ω
D
(K + 1)
2
(K − 1)
−1

.
(25)
From the discussion of the phase diagram and the general
properties of the domain wall, one already infers that
0 < α < 1/2 for not too large values of Ω
D
and K.
The set
(α = α
c
(K, Ω
D
), β = 1/2, K, Ω
D
) (26)
defines a two dimensional smooth manifold in parameter
space (critical manifold).
In order to study the critical properties close to this
manifold we apply standard methods of bifurcatio n the-
ory [52, 53, 54]. We consider a smooth path in the region
of parameter space close to the critical manifold defined
above. At some point C this path will cross the critical
manifold. The small quantities that describe the behav-
ior of the domain wall clos e to the critical manifold are
the dista nce from the right end side, δx
c
≡ 1 − x
w
, and
the domain wall height, ∆
w
= ρ
β
(x
w
) − ρ
α
(x
w
). These
quantities will be expressed to leading order in terms of
the small deviations from the critica l point δα = α − α
c
,
δβ = β − 1/2, and similarly for δΩ
D
and δK.
The matching condition of the left and right currents,
ρ
α
(x
w
) + ρ
β
(x
w
) = 1, can be rewritten in terms of re-
duced densities σ as σ
α
(x
w
) + σ
β
(x
w
) = −2. As a con-
sequence, the solution close to the critical point writes
as σ
α,β
(x
w
) = −1 ∓ ∆σ, where we have introduced the
reduced domain wall height ∆σ as another small quan-
tity. The relation between δx and ∆σ can be obtained
by e xpanding the equality
σ
β
exp (σ
β
) = −Y
β
(x
w
) , (27)
leading to
δx ∼ (∆σ)
2
, (28)
where the prefac tor can be explicitly c omputed and de-
pends only on the value of the system parameter s at the
critical point C. A second relation connecting ∆σ to
the small distances δα, δβ, δK and δΩ
D
arises from the
definition of the Lambert W -function |σ| exp(σ) = Y by
taking the ratio
σ
β
σ
α
exp (σ
β
− σ
α
) =
Y
β
(x
w
)
Y
α
(x
w
)
. (29)
The important observation is that the right-hand side is
independent of the domain wall position x
w
; see Eq. (21).
Expanding E q. (29) to leading order, one obta ins:
(∆σ )
3
∼ δO = A
α
δα+A
β
(δβ)
2
+A
K
δK+A
Ω
δΩ
D
, (30)
where δO is a distance along a generic path that ends
on the critical manifold. We do not consider the non-
generic case where the the critical manifold is approached
tangentially. Then one finds power laws different fr om
those presented below for the generic case.
As before, the coefficients A can be computed e xplicitly
and shown to depend only on the rates at the critical
point C. Interestingly, the distance δO does not exhibit
a linear term in δβ. This is due to the singular behavior
of the density profile ρ
α
(x) c lose to the right boundary
at the critical point C; see Fig. 20. Combining the two
expansion, one finds the following power laws:
δx ∼ δO
2/3
∆
w
∼ δO
1/3
. (31)
The validity of these e xponents is confirmed numerically
in Figs. 21. We also checked that the amplitudes in
the expansio ns (31) coincide with those obtained by the
numerical data.
18

-12
-8
-4
0
-10 -8 -6 -4 -2 0
-4
-2
0
-6 -4 -2 0
2/3
-6
-4
-2
0
-8 -4 0
-8
-6
-4
-2
0
-12 -10 -8 -6 -4 -2 0
∆
w
1-x
w
(a)
(b)
(c)
(d)
log ( x=α - α )
c
10
-7
-5 -3
-1
-2
-6
-10
log ( x= 1/2 - β )
10
log ( x=K - K )
c
10
log ( x=Ω - Ω )
D
D,c
10
∆
w
1-x
w
1-x
w
1-x
w
∆
w
∆
w
1/3
1/3
2/3
2/3
2/3
4/3
1/3
FIG. 21: Double decimal logarithmic plots of the critical be-
havior of the domain wall height, ∆
w
, and position from the
right end side, 1 − x
w
. We obtained the plot numerically
with the program Maple, release 7, using the analytic mean-
field solution in the vicinity of the critical point C and ap-
plying the matching condition over the left and right cur-
rents, j
α
(x
w
) = j
β
(x
w
). (a) As a function of α starting
from the point C on the critical manifold with coordinates
α
c
= 0.038532..., β = 1/2, Ω
D
= 0.1 and K = 3. (b) As a
function of β starting from the point C on the critical man-
ifold with coordinates α = 0.038532..., β
c
= 1/2, Ω
D
= 0.1
and K = 3. (c) As a function of K from t he point C on
the critical manifold with coordinates α = 0.2, β = 1/2,
Ω
D
= 0.051443... and K
c
= 3. (d) As a function of Ω
D
from the point C on the critical manifold with coordinates
α = 0.2, β = 1/2, Ω
D,c
= 0.051443... and K = 3. The value
of Ω
D,c
can be easily obt ained from Eq. (21) and the initial
condition σ(0). Note the different scaling regime for the exit
rate β.
C. Further properties of the domai n wall position
In this Section we discuss how the position of the do-
main wall x
w
move s upon changing Ω
D
for fixed α and
K > 1 and a set of different values for β. In the fir st
quadrant, α, β < 1/2, and for very small values of Ω
D
the coexistence phase is confined to a narrow strip par-
allel to the diagonal α = β; see Fig. 18(a). It extends to
the quadrant α < 1/2, β > 1/2, where boundary layers
form. On the other hand, for very large Ω
D
the coex-
istence phase corresponds to the reg ion α < 1 − ρ
l
, see
Sect. IV B 2. The interesting features therefore arise in
the region of α < 1−ρ
l
and β < 1/2. We cons ider a path
in the phase diagram with fixed α, β and K and follow
how it intersects the phas e boundaries of the 2-phase co-
existence reg ion LD-HD as Ω
D
is increas e d. From Fig. 22
one c an distinguish three cases.
For α < β the system is in a LD phase for very small
Ω
D
. Then, at a critical value of Ω
D
it enters the LD-HD
region where a domain wall forms at the right b oundary.
A further increase of Ω
D
results in a change of the domain
wall position to the left, as ymptotically reaching the left
boundary for very large values of the detachment rate
Ω
D
.
For β < α, the s ystem is in the HD region for small
Ω
D
. By increasing the detachment rate, it enters the
LD-HD region. Differently from the previous case, the
domain wall now forms at the left boundary, it moves to
the right up to a maximal position x
m
for intermediate
values of Ω
D
, and finally fo r large Ω
D
it moves back to
the left boundary with the same asymptotic behavior as
the previous case.
For β = α, the system remains in the 2-phase coexis-
tence region LD-HD for all values of the detachment rate
Ω
D
. One c an prove, using the analytic solution (23),
that the domain wall position stays finite even in the
limit Ω
D
→ 0
+
and is given by
x
w
=
σ(1)(1 + σ(0))
2(σ(0)σ(1) − 1)
, (32)
where σ(0) and σ(1) are the usual boundary conditions
written in terms the model parameters; see Eq. 17.
Interestingly, the domain wall po sition x
w
obtained for
10
10
10
10
1
10
Ω
β=0.5
0.3
0.201
0.2001
0.20001
0.19999
0.1999
-1
-2
-3
-4
D
10
-5
10
-1
x
w
β=0.1
10
-3
x
m
0.2
FIG. 22: Domain wall position x
w
in logarithmic scale as a
function of Ω
D
at α = 0.2, Ω
D
= .051443... and K = 3 and
different values of β. If β > α the domain wall builds up from
the right boundary, while for β < α from the left boundary.
For α = β the domain wall approaches the position x
m
which
is independent of the d ecreasing detachment rate Ω
D
. At
large Ω
D
the domain wall position x
w
always moves to the
left boundary as 1/Ω
D
.
α = β and vanishing Ω
D
does not reduce to the value
given by the mean-field continuum approximation in a
pure TASEP, i.e. x
w
= 1/2. In order to regain the
TASEP position, the binding constant K has to approach
the unity. Moreover, from Eq. (3 2) one find that in the
limit α = β → 1/2
−
, the position x
m
is a singular func-
tion of the binding constant close to K = 1
+
.
The previous discussion corr oborates the fac t that the
Langmuir Kinetics constitutes a sing ular p e rturbation of
19

the TASEP even in the limit of small rates, yielding new
features that are generated by the competition between
the two dynamics.
VI. CONCLUSIONS
We have presented a detailed study of a model for
driven one-dimensiona l transport introduced rec e ntly in
Ref. [15], where the dynamics of the totally asymmetric
simple exclusion process has been supplemented by Lang-
muir kinetics. T his non-conservative process introduces
competition between boundary and bulk dynamics sug-
gesting a new class of models for non-equilibrium trans-
port. The model is inspired by essential properties of in-
tracellular transport on cytoskeletal filaments driven by
processive motor proteins [31, 55]. These molecular en-
gines move unidirectionally along cytoskeletal filaments,
and simultaneously are subject to a binding/unbinding
kinetics between the filament and the cytoplasm. The
processivity of the motors implies low rates of detach-
ment. Attachment rates can be eas ily tuned by changing
the concentration of motors in the cytoplasm. In partic-
ular one may obtain very low attachment rates using a
low volume concentration of motors.
The non- c onservative dynamics proposed introduces a
non-trivial stationar y state, with features qualitatively
different from both the totally asymmetric simple ex-
clusion process and Langmuir kinetics. The competing
dynamics leads to an unexpected spatial modulation of
the average density profile in the bulk. For extended
regions in parameter space, we find that the density pro-
file exhibits disco ntinuities on the scale of the system
size which is characteristic for phase separation. Fur-
thermore, the coexisting phases manifest themselves by
a domain wall that, contrary to the TASEP, is localized
in the bulk. In c ontrast to previous models [24, 25], the
localization is not induced by local defects, but arises via
a collective phenomenon based on a microscopically ho-
mogeneous bulk dynamics. The resulting phas e diag ram
is topologically distinct from the totally asymmetric ex-
clusion process and exhibits new phases.
An analytic solution for the density profile ha s been de-
rived in the context of a mean-field approximation in the
continuum limit. The properties of the average density
for different kinetic rates are encoded in the peculiar fea-
tures of the Lambert W -function. In particular , the dis-
covery of a branching point is a prerequisite to rational-
izing the behavior of the solution. The analytic solution
has allowed to trace and study in detail the properties of
the phase diagram. We found that the cases of equal and
different binding rates give rise to rather distinct topolo-
gies in the phase diagram. The limiting cases for small or
large kinetic rates have been computed analytically. We
have identified specia l points of the phase diagram which
are the analogue of the ”triple point” (viz. where all three
phase boundaries meet) o f the totally asy mmetric simple
exclusion process. There, a domain wall builds up with
infinitesimal height at the boundary, exhibiting critical
features characterized by unusual mean-field exponents.
Finally, we have discussed some limiting cases in which
the properties of the totally asymmetric simple exclusion
process in the mean-field approximation are recovered.
Let us give some more intuitive arguments on the do-
main wall formation and localization. In the limit of large
system sizes, the corresponding time-dependent version
of Eq. (11) which governs the dynamics of the ”coarse-
grained” density ρ reads
∂
t
ρ + (1 − 2ρ)∂
x
ρ = F
A
− F
D
. (33)
One can easily see that on hydrodynamic scales the
source contribution on the rig ht hand side are negligi-
ble compared to the terms related to the transport dy-
namics. On these scales, the local dynamics is essen-
tially described by mass conservation just as in the totally
asymmetric exclusion process . Neglecting the source con-
tribution, one can give an implicit analytic solution of
Eq. (33) by standard methods of partial differential equa-
tions [47, 48]. From such analysis, o ne can infer the mech-
anism of the formation o f density discontinuities such as
shocks on the hydrodynamic scale, which usually build
up in finite time. An interesting feature of the domain
wall is also its slope S
w
[71] as a function of the sys-
tem size N . In Fig. 23 we show the slope of the domain
wall as obtained from Monte-Carlo simulation for grow-
ing system size N . We find the scaling law S
w
∼ N
η
with the exponent η = 0.50 ± 0.05. This result is fully
1
10
10
10 10 10
N
fit
2
5
4
3
η = 0.51
η=
0.55
η=
0.45
simulation
S
w
FIG. 23: Domain wall slope S
w
estimated from Monte Carlo
simulations as a function of the system size N . Simulations
where performed for α = 0.2, β = 0.6, K = 3 and Ω
D
= 0.1.
compatible with the scaling exponent η = 1/2 computed
for a pure totally asymmetric exclusion process. Note
that the mean-field approximation provides a wrong ex-
ponent η
MF A
= 1. The softening of the slope compared
to mean-field was recently explained in Ref. [56, 60]
on the basis of domain wall fluctuations [57]. In this
20

picture, the fluctuating domain wall performs a random
walk just like in the totally asymmetric e xclusion process.
However, since on global scale there is no mass conser-
vation due to the Langmuir kinetics, the current is s pace
dependent and drives the do main wall to an equilibrium
position corresponding to a cusp in the current profile.
Such domain wall behavior can be rephrased in terms of
a random walk in the presence of a confining potential
[58]. This picture also suggests that the exponent for the
slope of the domain wall is exactly given by η = 1/2.
It would be interesting to study how the case of equal
rates can be obtained by a limiting procedure of the case
with slightly different r ates. The change of topology in
the phase diagrams should be contained in the analytic
solution, however one suspects from Eq. (21) that an es-
sential singularity appears. Furthermore, one would like
to see if possible variants of our model introduce new
features similar to what has been done for a reference
dynamics, i.e. the totally asymmetric exclusion process
[19].
Acknowl edgments
This work was par tially supp orted by the Deutsche
Forschungsgemeinschaft (DFG) under c ontract no.
850/4. A.P. was supported by a Marie-Curie Fellowship
no. HPMF-CT-200 2-01529.
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equations there is a set of potential conditions [59] which
guarantees that the stationary state of the system is de-
scribed by a Gibbs measure.
[62] The left (i = 1) and right (i = N) b oundaries do not
belong to the bulk where the Langmuir kinetics takes
place.
[63] Since we agreed that, for finite ǫ, the current is defined
as j = −
ε
2
∂
x
ρ + ρ(1 − ρ), we have to stress that the
boundary layer becomes a discontinuity in the density
and the current in the continuum limit, i.e. for ε → 0
+
.
[64] The numerical solution of the m ean-field equation for fi-
nite ε > 0, Eq. (9), including t he boundary conditions
have been obtained using standard routines available in
Maple, release 7.
[65] Note that one can write for the non-linear term in Eq. (9)
∂
x
j = ∂
x
ρ∂
ρ
j = (2ρ − 1)∂
x
ρ, which implies ∂
ρ
j = 0 at
density ρ = 1/2.
[66] In terms of the rate constants the condition 0 < x
w
< 1
reads α ≤ β + Ω for 0 < x
w
and α ≥ β − Ω for x
w
< 1.
[67] See also the Section VI.
[68] For a fixed current j the stationary density profile of
the TASEP can be written as a non-linear map with
ρ
i+1
= 1−j/ρ
i
. Consider a boundary value problem with
a prescribed density ρ(L) = 1 − β and β ≤ 1/2. Then the
attractive fixed point of the nonlinear map for forward
iteration is ρ = 1 − β ≥ 1/2. If now one starts with a
density ρ(0) = α ≥ β at the left end it will quickly con-
verge towards the attractive fixed point of the n on-linear
map, ρ = 1−β; we then have a boundary layer at the left
end. The system is in the high density phase with bulk
density and current determined by the right boundary.
Otherwise, i.e. for α ≤ β and α ≤ 1/2, a stable high den-
sity solution is not possible and both bulk current and
density are determined by the left boundary.
[69] Though this behavior was mentioned explicitly in
Ref. [15] we did not indicate it explicitly in the phase dia-
gram. Here we found this useful to emphasize the special
nature of the high density phase above β = 1/2.
[70] Note that in a pure TASEP the point of the phase dia-
gram that satisfies such properties is α = β = 1/2.
[71] By a simple geometrical consideration, t he slope is a mea-
sure for the inverse of th e width.
22