Phase diagram of one-dimensional driven lattice gases with open boundaries

Abstract

We consider the asymmetric simple exclusion process (ASEP) with open boundaries and other driven stochastic lattice gases of particles entering, hopping and leaving a one-dimensional lattice. The long-term system dynamics, stationary states, and the nature of phase transitions between steady states can be understood in terms of the interplay of two characteristic velocities, the collective velocity and the shock (domain wall) velocity. This interplay results in two distinct types of domain walls whose dynamics is computed. We conclude that the phase diagram of the ASEP is generic for one-component driven lattice gases with a single maximum in the current-density relation.

Full text

J. Phys. A: Math. Gen. 31 (1998) 6911–6919. Printed in the UK PII: S0305-4470(98)92448-9
Phase diagram of one-dimensional driven lattice gases with
open boundaries
Anatoly B Kolomeisky†, Gunter M Sch¨
utz‡, Eugene B Kolomeisky§and
Joseph P Straleyk
†Department of Chemistry, Baker Laboratory, Cornell University, Ithaca, NY 14853-1301, USA
‡IFF, Forschungszentrum J¨
ulich, 52425 J¨
ulich, Germany
§Department of Physics, University of Virginia, Charlottesville, VA 22901, USA
kDepartment of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055,
USA
Received 13 March 1998
Abstract. We consider the asymmetric simple exclusion process (ASEP) with open boundaries
and other driven stochastic lattice gases of particles entering, hopping and leaving a one-
dimensional lattice. The long-term system dynamics, stationary states, and the nature of phase
transitions between steady states can be understood in terms of the interplay of two characteristic
velocities, the collective velocity and the shock (domain wall) velocity. This interplay results in
two distinct types of domain walls whose dynamics is computed. We conclude that the phase
diagram of the ASEP is generic for one-component driven lattice gases with a single maximum
in the current-density relation.
1. Introduction
Driven lattice gases with open boundaries are stochastic lattice models of particles which
hop preferentially in one direction and which are connected at their boundaries to particle
reservoirs of constant densities. As a result of this coupling, a stationary particle current
is maintained and the system never reaches an equilibrium state. A well known example,
which has become one of the standard models of nonequilibrium statistical mechanics, is
the asymmetric simple exclusion process (ASEP) where particles interact through hard-core
exclusion. This model is relevant to a variety of phenomena in physics and beyond [1]. The
one-dimensional version of this model was first introduced in 1968 to provide a qualitative
understanding of the kinetics of protein synthesis on RNA templates in terms of a ‘traffic
jam’ of ribosomes along the RNA [2, 3]. Other open systems in which transport of hard-
core particles through narrow (i.e. essentially one-dimensional) channels has been observed
in more recent experiments include certain types of zeolites [4]. Also recently, generalized
ASEPs have been succesfully used in traffic flow modelling [5]. On the mathematical side
[6, 7], the system is of interest in the theory of interacting particle systems because of its
tractability in conjunction with its rich and nontrivial behaviour.
The one-dimensional ASEP with open boundaries shows a complex behaviour exhibiting
nonequilibrium phase transitions that have no analogues in thermal equilibrium [1, 8] (see
section 2). There are intuitive arguments and evidence from the study of related problems
[7, 9, 10] that domain walls (shocks) and the nature of the boundary conditions imposed on
the system play a central role in understanding the physics of the system. The quantitative
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6912 A B Kolomeisky et al
clarification and development of the domain wall picture for the system with open boundaries
[11] constitute the primary goal of this paper. The totally asymmetric exclusion process
(TASEP) is the simplest case and we shall use this model to develop a theory of boundary-
induced phase transitions (section 3). The central ingredient of this theory is an interplay
of two characteristic shock velocities which can be derived from the (bulk) current density
relation j(ρ). We are lead to the observation of two different types of domain walls
whose structure and motion can be characterized to a large extent. The system dynamics,
stationary states, the nature and the location of the phase transitions can then be understood
quantitatively in terms of the domain wall picture. The basic mechanisms which generate
the phase transitions in this model will be shown to be universal and will thus eventually
lead us to an understanding of more generic cases of driven diffusive systems which have
a single maximum in the current density relation j(ρ). Moreover, we shall obtain—with
somewhat stronger assumptions on the coarse-grained domain wall dynamics—quantitative
results on the stationary density profile close to the phase transition lines (section 4). Some
further conclusions are drawn in section 5 where we also summarize our main results.
2. The TASEP with open boundaries
In the TASEP particles enter an N-site lattice from the left with rate αwhenever the first
site is empty. In the bulk particles can jump to the neighbouring right-hand site with
constant rate 1 provided that this site is empty (hard-core exclusion). On site Nparticles
exit the system with rate β. This choice of rates corresponds to a coupling of the system to
reservoirs of constant densities αand 1 −β, respectively. The steady states in this system
(figure 1) have been determined exactly [11, 12]. For self-containedness we summarize the
main results.
The hard-core exclusion implies that the steady state particle current jis related to the
bulk density ρby j=ρ(1−ρ) [13]. When particles are supplied at the left end with the
rate α>β, and removed at the right end not too fast, β<1
2
, there results a high-density
Figure 1. Phase diagram of the ASEP with stationary values of the particle density ρ, current
j, and allowed types of domain walls: low-density/high-density (0|1), maximal-current/high-
density (m|1), low-density/maximal-current (0|m).

Phase diagram of one-dimensional driven lattice gases 6913
phase (bulk density ρhigh =1−β>1
2
) for which particle extraction is the limiting process.
When the particles are supplied not too fast, α<1
2and removed faster than supplied,
β>α, there results a low-density phase limited by particle supply; the bulk density is
ρlow =α. There is a discontinuous phase transition along the line α=β<1
2
. Here the
average particle density changes linearly across the system from αto 1−α. When particles
are supplied and removed sufficiently rapidly, α>1
2
,β>1
2
, there results a continuous
phase transition into a maximal-current phase for which transport is bulk dominated; the
bulk density is ρm=1
2, and the current takes on its maximal value of 1
4.
Boundary density profiles in the high- and low-density phases decay exponentially with
finite localization length ξ; the localization length diverges as the maximal current phase is
entered. Within the maximal-current phase the decay is algebraic (equivalent to an infinite
correlation length). Remarkably, ξis composed of two individual localization lengths [11]
ξ−1=|ξ
−1
β−ξ
−1
α|(1)
with ξ−1
σ=−ln 4σ(1−σ) for σ<1
2and ξ−1
σ=0 for σ>1
2. A curious implication
of this equation is that the boundary density profile changes form within the high- and
low-density phases (for example, at α=1
2with β<1
2
), yet this is not accompanied by any
nonanalyticity in the current or bulk density. Along the first order transition line α=β< 1
2
the correlation length ξis infinite, but the individual localization lengths ξα,β are finite. The
mean-field solution [2, 14] for the density on this line has the form of a shock front or
domain wall, and the linear stationary state profile that is observed in the exact solution
can be interpreted as a superposition of domain walls present at any position [11, 15, 16].
Excepting the special case of this coexistence line the nature and motion of the domain
wall and the consequences for the phase diagram have not been elucidated generally and
explicitly for the problem with open boundaries.
3. Domain wall dynamics
From the exact solution we realize that understanding the origin and the physical meaning
of the localization lengths ξα,β holds the key to understanding the phase diagram of the
system. We start from two examples that provide insight into the physics of the TASEP.
Since the model has a particle–hole symmetry—it can be reformulated in terms of holes
entering the system from the right end—we only need to study the region α>β.
(1) Let us assume [16] that αand βare very small (αN 1,βN 1)and that the
initial distribution of particles is far from the true stationary state. The particles will travel
to the right end where they get stuck. At late times there will be a low-density region
at the left and a high-density region at the right (which we can present schematically as
00001111), with a domain wall between the low (0)- and high (1)-density segments. The
subsequent late-stage evolution of the system can be interpreted in terms of the motion of
this domain wall.
When a particle exits the system, the remaining particles rearrange themselves so that
the whole filled region shrinks by one lattice unit, and the domain wall moves one step to
the right. The conditions αN 1, βN 1 guarantee that while this rearrangement takes
place, no extra particle enters or leaves the system. Similarly a particle entering the system
from the left causes the domain wall to move one unit to the left. As in the Zel’dovich
theory of kinetics of first-order transitions [17], the domain wall motion can be understood
as diffusion of the ‘size of the high-density segment’. The ‘elementary processes’ that
change the length of the filled region consist of motion of the domain wall to the left at rate

6914 A B Kolomeisky et al
αor to the right at rate β, so that the domain wall does a biased random walk with drift
velocity V=β−αand diffusion coefficient D=(α +β)/2. As a result, three physically
different situations are possible.
If α<β, then the domain wall is drifting to the right and will eventually reach the
end of the system; thereafter the system is in the low-density stationary state. If α>β,
the domain wall is travelling to the left, leading to the high-density stationary state. When
α=β, the domain wall position fluctuates with no net drift, and its rms displacement
increases with time as (Dt )1/2. Hence, at large times it can be anywhere in the system,
resulting in a linearly increasing stationary density profile as suggested earlier for finite rates
α, β on an intuitive basis. We note that following the dynamics of the wall also explains
why the phase transition between the cases α>βand α<βis discontinuous.
(2) Let us now assume that α>1
2and that only βis very small. We start from an
empty lattice. After a while, but before the true high-density stationary state is reached, the
system consists of two visually different segments which can be presented schematically as
mmmm1111. Near the left end of the system the high entering rate causes the formation
of a region closely resembling the maximal-current phase (m), while on the right there
is a high-density region (1) dominated by the small exit rate β. The expansion of the
high-density segment is again a biased random walk with some drift velocity and diffusion
coefficient, determined below.
In equilibrium phenomena a domain wall is a localized region where the order parameter
interpolates between degenerate ground states. In this nonequilibrium system a domain wall
is an object connecting two possible stationary states of the system. The domain wall picture
exhibited in the two examples is not specific to the case that one of the boundary rates is
small; we argue that everywhere in the low-/high-density phases there are two kinds of
domain walls: the (0|1)wall (for α< 1
2) connecting the stable high-density stationary state
to the metastable low-density state (as in example (1)), and the (m|1)wall [for α>1/2] that
connects the stable high-density stationary state to the maximal-current phase (example (2)).
This second, distinct type of domain wall is a new notion that we need to introduce here
for a full understanding of the system. The bulk densities far to the left and right (ρlow and
ρhigh, respectively) of the (0|1)domain wall are reached exponentially fast with length scale
ξ. As we increase the entering rate α(holding the exit rate β<1
2fixed), the localization
length ξαcharacterizing the low-density behaviour of the domain wall increases, going to
infinity [11, 12] at α=1
2: The (0|1)wall undergoes a continuous phase transition into the
maximal-current/high-density domain wall (m|1)described in our second example. Because
the maximal current phase is algebraic, the stationary density profile for α>1
2approaches
its bulk value not purely exponentially, but with an algebraic correction. Thus the domain
wall transition explains the non-analytical change of the stationary density profile within
the high-density phase at α=1
2for any value of β<1
2
.
We turn now to a quantitative analysis of the physical origin of these observations in
terms of the drift velocity of the domain wall and of the collective velocity of the lattice
gas, defined below. To this end we will now show how the late-stage dynamics of the
system and the approach to the true high-density stationary state is governed by the motion
of the two types of domain walls.
In the introductory examples the domain wall was easy to visualize because the wall
is sharp when the entering and exit rates are small. To compute the drift velocity in the
general case we first note that in any lattice gas (with conserved total particle number) the
local particle density ρ(x,t) =hn
x
(t) isatisfies a lattice continuity equation
d
dtρ(x,t) =jx−1(t) −jx(t ) (2)

Phase diagram of one-dimensional driven lattice gases 6915
with the local particle current jx(t). In the continuum limit this turns into the usual continuity
equation ∂ρ/∂ t +∂j /∂x =0. With a travelling wave solution of the form ρ(x −Vt), and
integrating between minus and plus infinity, we find the domain wall velocity
V=j+−j−
ρ+−ρ−
.(3)
In a finite macroscopic system the assumption ρ(x,t) =ρ(x −Vt) breaks down near
the boundaries, and therefore the parameters j+,−and ρ+,−(given in figure 1) should be
understood as bulk stationary state values of the current and density in the far left (−)and
far right (+) parts of the domain wall.
For the low-density/high-density domain wall (0|1)one has j+=β(1−β),ρ+=1−β,
and j−=α(1−α),ρ−=α(figure 1). Substituting these in (3) we find the drift velocity
Vfor the TASEP
V=β−α(4)
which we expect to be valid for all α, β < 1
2and which is a well-established result for the
infinite system without boundaries [7, 13]. One realizes that the domain velocity changes
sign at α=β<1
2
, leading to the same scenario as in example 1 for small α, β.
For the maximal-current/high-density domain wall (m|1)we have j+=β(1−β),
ρ+=1−β, and j−=1
4,ρ−=1
2. Hence for an initially empty lattice
V=β−1
2.(5)
The expression for Vchanges its functional form at α=1
2because the wall has a different
form beyond the phase transition (0|1)→(m|1).
To understand why the transition takes place at α=1
2we consider the collective velocity
Vcoll ≡d
dt
Pxxhnx(t)(n0(0)−ρ)i
Pxhnx(n0(0)−ρ)i(6)
of a lattice gas, averaged over a translationally invariant grand-canonical stationary
distribution of density ρ. The collective velocity measures the drift of the centre of mass of
a momentary local perturbation of the stationary distribution. For any lattice gas satisfying
a continuity equation (2) we obtain by taking the thermodynamic limit of a finite system
the exact nonequilibrium fluctuation-dissipation theorem
Vcoll =∂
∂ρ j(ρ). (7)
For the TASEP with j=ρ(1−ρ) one finds Vcoll =1−2ρwhich changes sign at ρ=1
2
where the current takes its maximal value j=1
4. As in traffic flow a small perturbation (e.g.
an additional car which has just entered the road) will move with positive velocity (in the
direction of the flow) if the overall density is sufficienty low. However, in a high-density
regime, such a perturbation causes incoming particles to pile up behind the perturbation
(traffic jam) and thus leads to a negative collective velocity of the centre of mass of the
perturbation.
To appreciate the significance of the collective velocity for the phase diagram of the
TASEP consider first the low-density phase along a line with fixed β>1
2
. For a left
boundary density α<1
2a small perturbation of the stationary state (corresponding to a
fluctuation in the injection of particles) travels with positive speed into the bulk where
it will eventually dissipate. However, if the perturbation is maintained, i.e. the constant
left boundary density is increased by a small amount, the perturbation will continuously
penetrate into the bulk and lead to an increase of the bulk density. This happens until

6916 A B Kolomeisky et al
Figure 2. An ASEP with a domain wall present. The diagonal lines represent the instantaneous
position x(measured in lattice spacings) of every 10th particle at various times t(measured in
Monte Carlo steps per site). (a)α=β=0.2; (b)α=0.3, β=0.2; (c)α=0.7, β=0.2.
α=1
2. Further increase of the left boundary density results in a negative collective velocity
and the perturbation does not spread into the bulk. The system has entered the maximal-
current phase where it remains even if the left boundary density is further increased [8].
This phenomenon explains what may be intuitively described as the onset of an overfeeding
with particles [11].
For β<1
2the system does not enter the maximal-current phase (because of the negative
shock velocity), but the overfeeding still occurs for α>1
2and leads to the domain wall
transition (0|1)→(m|1). The overfeeding implies that further increase of the left boundary
density beyond 1
2does not result in any change of the characteristic length scales in the
high-density phase. This is seen in the behaviour of the domain wall velocity V(4), (5)
and also in the divergence of the localization length ξα. The overfeeding originates in the
change of sign in the collective velocity. Particle–hole symmetry can be used to extend our
results to the low-density phase. Thus we can explain both the location of the second-order
phase transition lines in the TASEP and the nonanalytic changes in the density profile within
the low- and high-density phases.
We have performed numerical simulations of the ASEP, shown in figure 2, that give
support to our ideas. We chose N=1000 and on each Monte Carlo step attempted to
advance the particle on a randomly chosen site (if there was one present). The initial
configuration had particles placed independently and randomly, with a density step at the
middle or near one end of the line. Time is measured in units of Monte Carlo steps per
site. We represent the subsequent behaviour of the system by tracing the path of every
10th particle. The figure shows three cases: (a)α=β=0.2, describing a (0|1)domain
wall between phases that are particle–hole equivalents; (b)α=0.3, β=0.2, with a (0|1)
domain wall between inequivalent phases; (c)α=0.7, β=0.2, with a (m|1)domain wall
(choosing any α>0.5 would give exactly the same diagram). We verified that for each of
these, equations (4) or (5) accurately predicted the velocity Vof displacement of the wall.

Phase diagram of one-dimensional driven lattice gases 6917
4. Density profiles
We may go further and check this picture by considering the consequences of the fluctuations
in the domain wall position. The domain wall is a compromise between the particle injection
and extraction processes that attempt to enforce their distinct own stationary states. From
our random walk discussion of the domain wall dynamics we expect that a superposition of
the domain wall localized at the left boundary and uniform bulk density ρhigh =1−βcapture
the physics of the high-density stationary state, in agreement with intuitive arguments [11].
For detailed analysis we adopt a coarse-grained point of view from which the motion of the
domain wall essentially becomes a Poisson process. For the domain wall the combinations
j+,−/(ρ+−ρ−)which determine domain wall velocity (3) can then be interpreted as being
the effective jump rates DR,Lto the right (left). Thus the diffusion constant is given by
D=1
2
j++j−
ρ+−ρ−
(8)
=1
2
β(1−β) +α(1−α)
1−β−αfor α, β < 1
2(9)
=4β(1−β) +1
4(1−2β) for α>1
2,β<1
2
.(10)
We note that equation (8) correctly reproduces the diffusion coefficient of the TASEP in an
infinite system [7, 18] and that on the coexistence line α=β<1
2
, the expression reduces
to D=α(1−α)/(1−2α) proposed earlier on the basis of current-fluctuation arguments
[16]. On approaching the maximal current phase (β→1
2) the drift velocity Vvanishes but
the diffusion coefficient Ddiverges, signalling the failure of the domain wall picture for
β>1
2.
In the TASEP the position of a domain wall is sharp [19]. Thus it is tempting to derive
the localization length which determines the decay of the density profile directly from the
stationary distribution of a biased random walker in a large, but finite system. For a Poisson
process with right and left hopping rates DR,Lthis yields the exact equality ξ=ln (DR/DL).
For the domain wall motion this gives
ξ=|ln (j +/j −)| .(11)
Somewhat surprisingly this simple-minded ansatz is in agreement with the exact expression
(1) in all phases and explains the origin of the two independent length scales ξα=ln j−
and ξβ=ln j+in terms of the domain wall diffusion.
For a generic lattice gas we take again a coarse-grained approach and introduce a
localization time τthat characterizes the length of time the wall spends away from the
boundary. We argue that the drift distance |V|τand the diffusional wandering (Dτ )1/2each
are of the same order of magnitude as the localization length ξitself; then τ∼
=D/V 2and
ξ∼
=D/|V|.(12)
Thus we obtain not only a foundation in terms of the domain wall motion to the
phenomenological derivation of this result by Krug [8] for the transition from the low-
density phase to the maximal-current phase, but also an extension of the validity of (12)
to the other phase transition lines. This domain wall approach is legitimate whenever
the localization length ξis much bigger than the (unit) lattice spacing and than other,
internal bulk correlation lengths which may result from particle interactions in the lattice
gas. Substituting in (12) the expressions (4), (5) and (9), (10) respectively for Vand Dwe
get expressions ξ1,2which are much larger than unity (and thus trustworthy) in the vicinity

6918 A B Kolomeisky et al
of the coexistence line α=β<1
2(case 1) and close to the phase boundary with the
maximal-current phase β=1
2respectively (case 2). In these limits ξcoincides with the
exact expression (11).
Finally we note that by using scaling arguments the power-law decay of the density
profile in the maximal-current phase can be inferred from the superdiffusive spreading [20]
of the fluctuations in the centre-of-mass of a local perturbation [8]. The same result can
also be obtained from a renormalization group analysis of the TASEP with open boundaries
[21].
5. Conclusions
Our domain wall approach to the determination of the phase diagram makes little reference
to the microscopic details of the dynamics. The crucial ingredients, the domain wall velocity
(3) and the collective velocity (7) are generally valid. We conclude that the phase diagram of
the ASEP is universal in the sense that systems with a single maximum jmax in the current-
density relation j(ρ) have a similar phase diagram. Coupling to boundary reservoirs of left
density ρ−=αand right density ρ+gives a low- and a high-density phase separated by a
coexistence line which is determined by j(ρ
−)=j(ρ
+)<j
max. In the low-density phase
(j(ρ
−)<j(ρ
+
)) the bulk density equals the left boundary density ρ−, in the high-density
phase (j(ρ
−)>j(ρ
+
)) the bulk density equals the right boundary density ρ+. Within these
phases there are domain-wall transitions at ρ±=ρ∗which is the density that maximizes
the current and at which the collective velocity changes sign. For ρ+,ρ
−>ρ
∗the system
is in the maximal current phase where the bulk density takes the value ρ∗. The domain
wall velocity Vvanishes at all phase boundaries. Hence, given j(ρ), one finds the domain
wall velocity (3), the collective velocity (7) and thus the location of the phase transition
lines. We argue that also equation (12) is generally valid close to the phase transition lines.
Provided that the motion of the domain wall on a coarse-grained scale is a Poisson process
we can then predict the shape of the density profile from the fluctuations of the domain wall
motion. The diffusion coefficient Dis singular along second-order lines.
This picture is well supported not only by the exact solution of the TASEP, but also by
exact results on other exclusion processes [15, 22–24] for which the current-density relation
and location of phase transition lines is known. Moreover, from the specific form of the
current-density relations for these models we obtain from (3) and (7) new results for the
domain wall velocity and collective velocity of these models. The first-order transition is
consistent with the experimental data obtained for protein synthesis [2]. We note that data
obtained from automobile traffic flow [25] suggest a single maximum in the current-density
relation. Thus our approach may be used to predict the phase diagram of road segments
between junctions where cars can enter and leave the road.
Acknowledgments
ABK has completed this work in the research group of B Widom, and was supported by
the National Science Foundation and the Cornell University Materials Science Center. EBK
was supported by NSF grants DMR-9412561 and DMR-9531430. We thank C L Henley
and B Widom for useful discussions and M H Ernst for sending [24] prior to publication.

Phase diagram of one-dimensional driven lattice gases 6919
References
[1] Schmittmann B and ZiaRKP1995 Phase Transitions and Critical Phenomena vol 17, ed C Domb and
J Lebowitz (London: Academic)
[2] MacDonald J T, Gibbs J H and Pipkin A C 1968 Biopolymers 61
MacDonald J T and Gibbs J H 1969 Biopolymers 7707
[3] von Heijne G, Blomberg C and Liljenstr¨
om H 1987 J. Theor. Biol. 125 1
Sch¨
utz G M 1997 Int. J. Mod. Phys. B11 197
[4] Kukla V, Kornatowski J, Demuth D, Girnus I, Pfeifer H, Rees L, Schunk S, Unger K and K¨
arger J 1996
Science 272 702
[5] Nagel K and Schreckenberg M 1992 J. Physique I22221
Schreckenberg M, Schadschneider A, Nagel K and Ito N 1995 Phys. Rev. E51 2339
Nagatani T 1995 J. Phys. A: Math. Gen. 28 7079
[6] Ligget T 1985 Interacting Particle Systems (Berlin: Springer)
[7] Spohn H 1991 Large Scale Dynamics of Interacting Particles (New York: Springer)
[8] Krug J 1991 Phys. Rev. Lett. 67 1882
[9] Andjel E D, Bramson M and Ligget T M 1988 Prob. Theor. Relat. Fields 78 231
[10] Janowsky S A and Lebowitz J L 1992 Phys. Rev. A45 618
Sch¨
utz G 1993 J. Stat. Phys. 71 471
[11] Sch¨
utz G and Domany E 1993 J. Stat. Phys. 72 277
[12] Derrida B, Evans M R, Hakim V and Pasquier V 1993 J. Phys. A: Math. Gen. 26 1493
[13] Lebowitz J L, Presutti E and Spohn H 1988 J. Stat. Phys. 51 841 and references therein
[14] Derrida B, Domany E and Mukamel D 1992 J. Stat. Phys. 69 667
[15] Sch¨
utz G 1993 Phys. Rev. E47 4265
[16] Derrida B, Evans M R and Mallick K 1995 J. Stat. Phys. 79 833. For α=βour first example was mentioned
in this reference, however, we go farther in our conclusions.
[17] Landau L D and Lifshitz E M 1981 Physical Kinetics (Course of Theoretical Physics 10) ed E M Lifshitz
and L P Pitaevskii (New York: Pergamon)
[18] Ferrari P A and FontesLRG1994 Prob. Theor. Relat. Fields 99 305
[19] De Masi A, Kipnis C, Presutti E and Saada E 1989 Stoch. Stoch. Rep. 27 151
Ferrari P, Kipnis C and Saada E 1991 Ann. Prob. 19 226
[20] van Beijeren H, Spohn H and Kutner R 1985 Phys. Rev. Lett. 54 2026
van Beijeren H 1991 J. Stat. Phys. 63 47
[21] Janssen H K and Oerding K 1996 Phys. Rev. E53 4544
[22] Sandow S 1994 Phys. Rev. E50 2660
[23] Rajewsky N, Santen L, Schadschneider A and Schreckenberg M 1997 Preprint cond-mat 9710316
[24] Tilstra L G and Ernst M H 1998 J. Phys. A: Math. Gen. 31 5033. The authors have recently obtained the
localization length for the ASEP with parallel update and p=1. For small α, β this result is in agreement
with our predictions and helped us clarify the validity of our expression for the diffusion coefficient.
[25] Hall F L, Brian L A and Gunter M A 1986 Trans. Res. A20 197