Driven diff usive systems with disorder

12 Pages·0.192 MB·English

by  Mustansir Barma

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PhysicaA](]]]])]]]–]]]
www.elsevier.com/locate/physa
Driven diffusive systems with disorder
(cid:1)
Mustansir Barma
DepartmentofTheoreticalPhysics,TataInstituteofFundamentalResearch,Mumbai400005,India
Abstract
We discuss recent work on the static and dynamical properties of the asymmetric exclusion process, generalized to
include the effect of disorder. We study in turn, random disorder in the properties of particles; disorder in the spatial
distributionoftransitionrates,bothwithasingleeasydirectionandwithrandomreversalsoftheeasydirection;dynamical
disorder, where particles move in a disordered landscape which itself evolves in time. In every case, the system exhibits
phase separation; in some cases, it is of an unusual sort. The time-dependent properties of density fluctuations are in
accord with the kinematic wave criterion that the dynamical universality class is unaffected by disorder if the kinematic
wave velocity isnonzero.
r2006Elsevier B.V.All rightsreserved.
Keywords:Drivensystems;Quencheddisorder;Phaseseparation
1. Introduction
Simple models of particles moving on a lattice have shed a good deal of light on the general question of
collective effects in transport. Even the simplest sorts of interactions between particles can induce collective
phenomena involving large numbers of particles, ranging from the motion of density fluctuations as a
kinematicwave,totheoccurrenceofphasetransitionstostateswithmacroscopicallyinhomogeneousdensity
(jams). Models which display these phenomena include the asymmetric simple exclusion process (ASEP), the
simplestmodel whichincorporatesdirectedmotionandmutual exclusion[1–3],anditsgeneralizationswhich
include more realistic features which mimic vehicular traffic [4].
Here we are concerned with the effects of extensive disorder on interacting-particle transport in one
dimension. As in equilibrium systems, disorder has a strong effect on the properties of driven systems which
reachanonequilibriumsteadystate.However,incontrasttoequilibriumsystems,relativelylittleisknown,in
a general way, about the effects of quenched disorder on nonequilibrium systems, as the absence of detailed
balance, together with the breaking of translational invariance, makes even the determination of the steady
state weights difficult in general. In this backdrop, the study of simple models of disordered nonequilibrium
systemshasprovedvaluable,asitrevealshowinteractions,driveanddisordercombinetoproducenewtypes
of states; the simplicity of the models allows for a detailed characterization and understanding.
(cid:1)
Tel.:[phone];fax:[phone].
E-mailaddress:[email protected].
0378-4371/$-seefrontmatterr2006ElsevierB.V.Allrightsreserved.
doi:10.1016/j.physa.2006.05.002
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Inthispaper,wewillreviewsomeresultsobtainedforadisorderedsystemofparticlesmovingstochastically
on a one-dimensional lattice. We restrict ourselves to the disordered ASEP, in which interactions between
particles enter through rules for the hopping rates—the hard-core constraint is modelled by forbidding any
movewhichwouldleadtomorethanoneparticlepersite.Wewillconsiderthreetypesofdisorder.Thefirstis
particle-wise quenched disorder, in which different particles have different properties which do not change in
time.Secondly,wewillconsiderspace-wisequencheddisorder,inwhichparticlesdonotdifferfromeachother
but are subject to hopping rates which are randomly distributed in space. Finally, we will turn to dynamical
disorder where particles move in a disorder-induced landscape, which is itself evolving in time. Interestingly,
there are inter-relations between the effects of the three types of disorder, which we discuss below.
It is clear that extensive disorder at the microscopic scale would induce inhomogeneities in the particle
density on a similar scale. Furthermore, disorder can also induce variations of the density on a macroscopic
scale, associated with phase separation [5] . The models considered here share this feature, but in some cases
thephaseseparatedstatehasquiteunusualproperties.Wealsodiscussthetime-dependentpropertiesofthese
systems, both in the steady state and while approaching it.
2. Particle-wise disorder
Consider asystemofcars onasingle-laneroad,too narrowforovertakingtobepossible. Each carhasan
intrinsic maximum speed,which however may not be achieved due to a slowly moving car in front of it. It is
thenevidentthattheslowestcarinthesystemwillbetrailedbyanumberofintrinsicallyfastercars,andwill
seealargerthanaverageheadwayinfrontofit.Thequestionarises:Inthelimitofaninfinitelylongroadwith
a finite density of cars, will this headway be finite or infinite? Is there a phase transition between these two
possible types of behaviour as the density is changed?
This question motivates the study of a collection of particles with a different intrinsic rate of motion for
each.Inasimplelatticemodel,studiedinRefs.[6,7],Nparticlesresideonthesitesofaone-dimensionalring
with L sites. The dynamics involves random sequential updating of configurations. An elementary move
consistsofanattemptedrightwardhopofaparticle,saythekth,withaparticle-dependenthoppingrateuðkÞ;
the move is actually implemented only if the site immediately to the right is unoccupied. A configuration is
specifiedbythesetofparticlelocationsfy g.Onemayinsteadconsiderthesetfm g,wherem ¼y (cid:1)y (cid:1)1
k k k kþ1 k
isthe headwayor gap between the kth particle and theone ahead ofit. Ona periodic ring, atypicalset fm g
k
represents N equivalent configurations which are translational shifts of each other. Evidently the overall
density r¼N=L is conserved by the dynamics.
Disorder enters through the selection of hopping rates uðkÞ, each of which is drawn independently from a
distribution ProbðuÞ. Especially interesting is the case with a power law variation near a minimum cutoff
speed: ProbðuÞ¼½ðnþ1Þ=ð1(cid:1)u Þnþ1(cid:2)ðu(cid:1)u Þn with u in the range ½u ;1(cid:2). Here n is an index which
0 0 0
characterizesthedecayofthedistributionfunctionnearthecutoff.Asweshallseebelow,thissystemshowsa
phasetransitionasthedensityisdecreased,fromastatewithfiniteheadwaysaheadofeachparticle,toonein
which the headway ahead of the slowest particle is infinitely long, or more accurately, a finite fraction of the
lattice size L.
2.1. Mapping to zero range process
The analysis of this system is aided by exactly mapping this system with particle-wise disorder to another
systemwithspace-wisedisorder[6,7].Viewtheparticleindexkaslabellingurnsarrangedinaone-dimensional
sequence(Fig.1).Theoccupancyofurnkistakentobethelengthofheadwaym infrontofthekthparticle
k
intheoriginalparticleproblem.Now,asinglerightwardhopofaparticlereducestheheadwayinfrontofit,
and augmentsit behind. In theurnrepresentationthis corresponds toan elementary moveofa particle from
urn k to urn k(cid:1)1 (Fig. 1).
Thismodelisaspecialcaseofthezerorangeprocess(ZRP)whichdescribesthemotionofparticlesbetween
wells or urns, with (in general), occupancy dependent, site-dependent hopping rates uðk;m Þ [1,8]. The ZRP
k
has the virtue that the steady state measure P ðCÞ is known. In our case, where the rates depend on sites but
s
not on occupancies, the steady state weights are particularly simple. In a grand canonical formulation of the
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u (k)
k
u (k)
k
Fig.1. Particle-wisedisorderedmodelandthezerorangeprocess.
problem we have
1 YN (cid:1) z (cid:2)mk
P ðCÞ¼ . (1)
s Z uðkÞ
k¼1
Here Z is the grand partition function, m the mass at site k, and z the fugacity which is determined by
k
P
requiring m ¼M. For a given realization of disorder fuðkÞg, the density r~ ¼M=N is given by
k k
N(cid:1)1P z=ðuðkÞ(cid:1)zÞ. As r~ is increased, the model undergoes a phase transition [6,7,9]. We use the self-
k
averaging property to replace the sum over k by an average over ProbðuÞ, separating out the lowest u:
1 z Z 1 z
r~ ¼ þ du fðuÞ, (2)
Nu (cid:1)z u(cid:1)z
min u0
where u is the lowest realized value of uðkÞ; u approaches u in the thermodynamic limit. The fugacity z
min min 0
increaseswithr~ untilonereachesacriticaldensityr~ ¼u ðnþ1Þ=nð1(cid:1)u Þbeyondwhichzgetspinnedtou .
c 0 0 0
Beyondthispoint,thesystementersahigh-densityphaseinwhichtheexcessmassM(cid:1)r~ N is‘condensed’at
c
thesitewiththelowesthoppingrateintheZRP.ThetransitionisanalogoustoBose–Einsteincondensationin
the ideal Bose gas or a system of bosons in a random potential [10].
Intheparticle-wisedisorderedmodel,thecondensaterepresentsaninfiniteheadwayinfrontoftheslowest
car,implyinganinfinitelylongjambehindit.Suchajamimpliesspatialphaseseparation,andexistsonlyfor
lowenoughdensity;athigherdensities,blockingeffectsbetweencarsreducethecontrastinspeedstosuchan
extent that headways are interspersed throughout the system and there is no phase separation.
2.2. Time-dependent properties
Itisinterestingtoaskaboutthedynamicalpropertiesofthesystem,bothinthesteadystate,andalsodescribing
relaxationtothesteadystate.Insteadystate,itisevidentthatthemeancurrentbetweeneverypairofneighbouring
sitesintheZRPisthesame.Itislessobviousthatthefulldistributionoftimeintervalsbetweenparticletransfer
eventsisalsoidenticaloneachsite.ThispropertyfollowsfromtheapplicationofBurke’stheoremfromqueueing
theory[11,12]totheproblemathand.Thistheoremstatesthatinareversiblebirth–deathprocess,aPoissoninput
ofbirthsresultsinan identicalPoissondistributionofdeaths.IdentifyingtheoccupancyofeachurnoftheZRP
with a population undergoing such a process then implies the result, as the output from one urn constitutes the
input for the next. In the original particle problem, this means that the time history of every particle follows an
identical Poisson distribution despite the fact that each particle has different intrinsic speeds. This holds in the
thermodynamic limit in both the disordered and the phase separated state [13].
Next, let us ask about the behaviour of statistical fluctuations of the density in the steady state. In a
homogeneous driven system with a density-dependent current J~ðr~Þ, density fluctuations are transported
around the system as a kinematic wave with speed c~¼qJ~=qr~, in time N=c~ [3,14]. This wave of fluctuations
decays on a time scale (cid:3)Nzðz41Þ, implying that it circulates several times around a system with periodic
boundary conditions before it dissipates. In the disordered ZRP, there are different sorts of behaviour in
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different phases. For r~or~ and for r~ ¼r~ ; n41, the wave speed c~ is nonzero. There is clear evidence of the
c c
wave in simulation results for the fluctuations in the integrated number of particles which move past a site.
This correlation function shows pronounced oscillations with period N=c~, which ultimately damp down [15].
InRef.[16]ithasbeenarguedthat quencheddisorderdoesnothaveasignificanteffecton thedampingof
fluctuationsindrivensystemsifthekinematicwavespeedc~isnonzero.This‘kinematicwavecriterion’implies
thatthereisthennochangeintheuniversalityclassofthedecayofdensityfluctuations,whichshouldcontinue
tofollowthebehaviourinthe absence of disorder.The reason isthat in sucha case, each density fluctuation
doesnotstaylonginthevicinityofalocalpatchofdisorder.Tothatextent,theeffectofquencheddisorderon
aballisticallymovingdensityfluctuationofspatialextentDxisakintotemporalnoise,andweshouldexpect
the lifetime of the density fluctuation would continue to be (cid:3)ðDxÞz, where z is the dynamical exponentin the
absence of disorder. Monte Carlo simulations [15] confirm that when c~a0, z equals 3, the value for a driven
2
system in the absence of disorder [17]. At r~ ¼r~ , the speed c~ vanishes for np1. In this case, disorder is
c
expectedtohaveastrongeffect,andperhapschangestheuniversalityclassawayfromthatinitsabsence.The
determination of this class remains an open problem.
Forr~4r~ ,thesystemhasdensityr~ inthebulk,withthesurplusmassðM(cid:1)r~NÞbeingontheslowestsite.
c c
Thus, density fluctuations would be expected to move as a kinematic wave in the bulk so long as n41.
However, this wave would be absorbed at the condensate site and not be able to circulate, implying that
correlation functions in this phase would be nonoscillatory.
Nowletusturntothekineticsofapproachtothesteadystate,startingfromastatewithuniformdensity.In
the disordered phase ðr~or~ Þ, the kinetics is governed by the same dynamical exponent z that operates in
c
steady state, and the relaxation time is of order Lz, with z¼3. In the condensate phase, however, a different
2
physical process, namely coarsening, governs the approach. Initially, particles hop out of relatively fast sites
quitequickly,andgettrappedatslowsitesintheimmediateneighbourhood.Atmoderatetimes,thereisthusa
finite density ofnoticeablylarge aggregates atsuch slow sites, which thenrelax by releasing their mass toyet
slower sites [18]. Thus the mass at all such slow sites, except for the very slowest, shows a nonmonotonic
variationintime.Themassattheslowest(condensate)sitegrowsasM (cid:3)tb,withb¼ðnþ1Þ=ðnþ2Þ,aresult
1
firstderivedforamodelwithdeterministicdynamics[19],butwhichremainstrueeveninthestochasticmodel
under consideration [5,6,15]. This translates into a statement about the growth of headway lengths x in the
particle model
xðtÞ(cid:3)tðnþ1Þ=ðnþ2Þ. (3)
A similar coarsening description for headways is also found to hold in a lattice gas model of traffic, which
incorporates features of acceleration and slowing down, and in which different drivers have different
propensities for random braking [20].
Furthermore, the problem has also been studied using open boundary conditions, with injection and
ejection rates specified [21]. In the absence of disorder, it is known that the open system exhibits phase
transitions between a low-density phase, a high-density phase, and a high-current phase, as the injection and
ejection rates arevaried[22,23].Withparticle-wisedisorder,there isashiftofthephaseboundariesowingto
the breaking of particle–hole symmetry. Within the low-density phase, there is a regime in which the drift
velocity of all particles is determined by the slowest [21], as with periodic boundary conditions.
Finally,wediscusssomerecentworkontwo-waytraffic[24,25].Considerparticle-wisedisorderinwhichthe
majority of particles move preferentially rightward, while a minority (a fraction f) move preferentially
leftward.Foreachparticle,theratioofthelargerhoppingratetothesmallerone,whichisameasureofbias,
is taken to be fixed, but the direction of hopping is random. The problem was mapped onto a ZRP, which
inherits the random easy-direction property. This system has much in common with the spatially disordered
ASEP with bidirectional bonds, to be discussed in the next section. In particular, as in that case, the current
decays as an inverse power of the size, with the power depending continuously on both f and the bias.
3. Space-wise disorder
Consider the transport of interacting particles which are driven through a randomly disordered one-
dimensional path. To address the question of what sort of macroscopic states result, we restrict ourselves to
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thedisorderedASEP.Disorderentersthroughthetransitionrates,whichareassignedrandomlytothebonds
betweenneighbouringsites,e.g.u describestherateofattemptedhoppingfromsiteitoiþ1,whileu is
i;iþ1 iþ1;i
the attempt rate in the reverse direction. A hop to a site actually occurs only if the site in question is
unoccupied. Results obtained for a particular disorder realization R(cid:4)fu ;u g are then to be averaged
i;iþ1 iþ1;i
over R using a pre-specified probability distribution, taken to be the product of identical and independent
distributions across all bonds. The larger of the two rates on each bond defines the easy direction on that
bond. Below we will distinguish between two situations: unidirectional, in which the easy direction of every
bond is the same, but the strength is a random variable ((a) and (b) in Fig. 2) and bidirectional, in which the
easy direction is itself a random variable ((c) and (d) in Fig. 2).
In the unidirectional case, we will further specialize to only the case of no backward hopping ðu ¼0Þ:
iþ1;i
Probðu Þ¼ð1(cid:1)fÞdðu (cid:1)u Þþfdðu (cid:1)u Þ (4)
i;iþ1 i;iþ1 1 i;iþ1 2
with u2ou1. The fraction f and relative strength r¼u2=u1 of weak bonds specify the extent of disorder.
Inthebidirectionalcase,theeasydirectionofeachbondistakentovaryrandomly,buttherelativehopping
rate l against and along the easy direction on each bond is taken to be the same
(cid:1) (cid:2)
u
Prob b ¼ iþ1;i ¼ð1(cid:1)fÞdðb (cid:1)lÞþfdðb (cid:1)l(cid:1)1Þ. (5)
i;iþ1 u i;iþ1 i;iþ1
i;iþ1
Here f is the fraction of backward pointing bonds.
The large scale behaviours that result depend on the type of disorder as well as on the overall density r.
Fig. 2 shows a schematic depiction of a segment of the disordered lattice, the corresponding variation of the
current J with system size L, and the schematic spatial variation of the mean occupancies r (cid:4)hni in a
i i
particular realization of disorder in different cases. These are discussed below.
(a) Withunidirectionaldisorderandaparticledensitythatissufficientlyfarfrom1,i.e.,jr(cid:1)1j4D,thedensity
2 2
is homogeneous on the macroscopic scale. The value of D depends on the concentration f and relative
strength r¼u =u of the weak bonds. The current J approaches a nonzero value in the thermodynamic
2 1
limit.
(b) With unidirectional disorder and r close to 1, i.e., jr(cid:1)1joD, the state exhibits phase separation; the
2 2
systemischaracterizedbytwodistinctvaluesofthedensity,eachextendingoveramacroscopicregion.As
in(a),Jhasafinitevalueinthethermodynamiclimit.Thisvalueofthecurrentisthesameforallrinthe
regime jr(cid:1)1joD.
2
(c) Withbidirectionaldisorder,butwithanasymmetryinthenumberofforwardandbackwardbondsðfa1Þ,
2
the potential has local minima and an overall tilt. On a macroscopic scale, the system has two distinct
valuesofthedensity,closeto1inoneregionandcloseto0intheother.ThecurrentJfallswithincreasing
system size L as an inverse power of L.
(d) Withbidirectionaldisorderandequalnumbersofforwardandbackwardbondsðf ¼1Þ,thepotentialhas
2
local minima, but no overall tilt. Consequently the mean current is zero. The density profile shows large
sample-to-samplevariations,buteverysamplehasahigh-densityregionthatextendsoverafinitefraction
of the system size.
Let us turn to an explanation of the features described above. Central to the discussion is the observation
thatwhiletherearestrongspatialvariationsinthelocaldensityhni¼r,thesteadystatevalueofthecurrent
i i
J is uniform across all bonds.
3.1. Unidirectional disorder
The time-averaged current in bond ði;iþ1Þ is J ¼u hnð1(cid:1)n Þi. Since the exact steady state
i;iþ1 i;iþ1 i iþ1
weightsofparticleconfigurationsfngarenotknown,onecannotcomputethetwo-pointcorrelationfunction
i
involved.Toproceed,weemployameanfieldapproximationwhichignorescorrelationsbetweensitedensities
in the steady state, and replaces hnni by hnihni¼rr, but does allow for spatial variation of r, which is
i j i j i j i
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J
(cid:1)
i
.........................................................
(a)Unidirectional disorder (Low density) L i
.
J ..................... ..........
(cid:1)i ......
..
..........................
(b)Unidirectional disorder (Intermediate density) L i
.... ....... .
J ..
(cid:1) ..
i .
.
.
..
.
.
...... ...... ..
(c)Bidirectional disorder (Tilted) L i
(cid:1) ... ...............
i
J
... ....
L i
(d)Bidirectional disorder (Untilted)
Fig.2. Space-wisedisorder:depictionofpossiblebehaviours.
crucial.Tofindthemeanfieldsolutionfrgwemayproceedintwoways.(i)ForagivenvalueofJ,iteratethe
i
setofequationsr ¼1(cid:1)J=u r aroundtheperiodicchain,untilconvergenceisachieved.(ii)Foragiven
iþ1 i;iþ1 i
value of the overall density, evolve local densities (in fictitious time) through rðtþ1Þ¼rðtÞþj ðtÞ(cid:1)
i i i(cid:1)1;i
j ðtÞ where the instantaneous currents are given by j ðtÞ¼u rðtÞ½1(cid:1)r ðtÞ(cid:2). The densities then
i;iþ1 i;iþ1 i;iþ1 i iþ1
converge to time-invariant values, and the corresponding current is uniform across bonds. As discussed in
Ref.[26],method(ii)ispreferable,asvaluesofJaboveathresholdvaluearenotallowed,andmethod(i)does
notconvergewellnearthethreshold.Theresultingmeanfielddensityprofile canbetestedagainsttheresults
of direct Monte Carlo simulation. The agreement is remarkably good, in that the mean field approximation
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tracks every one of a macroscopic number of ‘minishocks’ and correctly predicts their positions, though it
does not accurately reproduce their shapes [26].
The principal result is the existence of a density range D in which the current has a constant, density-
independent value (regime (b)), and where the system exhibits phase separation of densities [26]. A simple
qualitativeunderstandingofboththesefeaturescanbeobtainedbyreferringtoFig.3.Theargumentrelieson
aconjecturedmaximumcurrentprinciple,whichstatesthatforagivenvalueoftheoveralldensity,thesystem
settles into a state which maximizes the mean current. In our system, let us suppose that the density in each
stretchoflikebondsisuniform,andlabelledby1and2thestretchesofstrongandweakbonds,respectively.
The two parabolasinFig. 3arethecorrespondingJ versus rcurves forthepure 1and pure2 systems. Since
thecurrentisuniforminthedisorderedsystem,avalueofcurrentsuchasshownbytheupperdashedlineefin
the figure is ruled out, as such a value cannot be sustained in a 2-stretch. By contrast, a value of J
corresponding to say ad is quite possible, as the (1,2) densities can assume the values ðr ;r Þ if the density is
a b
relatively low, or alternatively ðr ;r Þ if the density is relatively high. Such choices correspond to the regime
c d
jr(cid:1)1j4D, and result in a density profile that is homogeneous on a macroscopic scale as strong and weak
2
bondsareinterspersedatthemicroscopiclevel.Ifthedensityistooclosetohalf-fillingjr(cid:1)1joD,thesystem
2
adjustsbykeepingthecurrentconstantatitsmaximumallowedvalue,andreplacingðr ;r Þin(1,2)stretches
a b
byðr ;r Þ,inafinitefractionofthefulllattice.Theresultingstatethenshowsphaseseparationofthedensity.
c d
The location of the high-density region is decided by the longest stretch of weak bonds in the system, as this
stretch (whose length is of order lnL) mimics the effect of a single defect weak bond in an otherwise pure
system, which is known to induce phase separation in a finite range of density [27,28].
Interestingly, it is possible to obtain upper and lower bounds [5] on the value of the threshold density for
phase separation rc ¼12(cid:1)D in terms of the ratio of bond strengths r¼u2=u1o1 and the fraction f of weak
bonds. An upper bound is obtained on noting that the maximum current that can flow through a stretch of
slowbondsdecreasesasthestretch lengthincreases. Itisthenplausiblethatthecurrentwouldbetheleastin
the fully segregated limit where all slow bonds form a single large stretch [26]. In such a system consisting of
two connected homogeneous parts, it is easy to figure out the way a given overall density would distribute
itself over the two portions. A lower bound on r was obtained [5] by arguing that the current in the
c
disorderedASEPisboundedabovebythecurrentinacorrespondingZRPwiththesamedisorderdistribution
and an equal number of particles.
Theresultsdiscussedaboveholdforthecaseofbinarydisorderinthetransitionrates.Similarphenomena,
e.g. the plateau in the J–r curve over a finite range D of density, have been found also for more general,
continuous distributions of the transition rates [29,30]. The problem has also been studied using open
boundaryconditions,specifyinginjectionandejectionrates.Anargumentbasedonmovingshockfronts[31]
suggests that in the presence of disorder, the high-current phase extends over a larger region of the phase
diagram[30]thaninthepuresystem[22].Further,anumericalstudyofthefirstordertransitionbetweenthe
low- and high-density phases shows that the transition remains first order, but the location of the transition
showsstrongsampletosamplefluctuationswhichdonotseemtodampdowninthethermodynamiclimit[32].
1
e f
J 2
a d
b c
(cid:1)
Fig.3. Current–densityplots.
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Going beyond the ASEP, the effect of slow sites on a lattice model of traffic was studied by numerical
simulationinRef.[33].Interestingly,aregimewithtwocurrentplateauregionswasobserved,andthesystem
was observed to switch between the two branches.
Toconclude thissection,we mentionanintriguingsymmetry thatholdsforthedisorderedASEP when no
backward hopping is allowed. A typical configuration of quenched disorder is not symmetric under space
inversion,andconsequentlynoristhesteadystatedensityprofile.Inspiteofthis,themagnitudeofthesteady
state current was observed to be invariant under reflection [26]. Subsequently, this invariance property has
been proved for both periodic [34,35] and open [34] systems.
3.2. Bidirectional disorder: with tilt
Whentheeasydirectiononabondisarandomvariable,oneneedstodistinguishbetweenthecasesfo1and
2
f ¼1, corresponding to a tilted and untilted potential, respectively (Figs. 2(c) and (d)). We may imagine
2
assigning arrows to denote the easy direction on the bonds of a periodic ring; a fraction ð1(cid:1)fÞ point
rightward,whileafractionfpointleftward.Aparticle–holeexchangeoccursinthedirectionofthearrowwith
rate u, and in the opposite direction with rate ul with lo1.
In the tilted case, there is an overall tendency for the particles to move rightward, but the question is
whetherthere isanonzerocurrent inthe thermodynamic limit. Localbarrierstotherightward motion occur
intheformofbackbendswhichconsistofachanceagglomerationofsuccessiveleft-pointingarrows.Anupper
boundJ onthecurrentthatcanbecarriedbyabackbendoflengthlisobtainedbyconsideringtheboundary
l
conditions,r¼1attheleftmostendofthebackbendandr¼0attherightmostend,whichforcethelargest
possiblecurrentthroughit.Withinameanfieldapproximation,thedensityprofile,andhencethecurrent,can
be calculated [36] with the result J (cid:3)lð1=2Þl. The factor 1 is a consequence of particle–hole symmetry which
l 2
impliesthattheparticledensityextendsuptohalf-waythroughthestretch,sothatthetopmostparticleneeds
tobe activated across 1l sites. An exact calculation [37] confirms the exponential decay ofcurrent withl.The
2
currentinthefullsystemofsizeLislimitedbythelengthl(cid:5)ðLÞofthelongestbackbend.Sincetheprobability
oftheoccurrence oflsuccessivebackwardbonds isfl,wemayestimatel(cid:5) from Lfl(cid:5) ’Oð1Þ.Theestimateof
the current ll(cid:5)=2 can be written as
JðLÞ(cid:3)L(cid:1)y=2, (6)
where the exponent
y¼ln l=ln f (7)
depends explicitly on the bias and fraction of reverse bonds. This dependence has been confirmed by Monte
Carlo simulations [26].
3.3. Bidirectional disorder: untilted
When there is no overall tilt, the mean current is identically zero. The problem then reduces to the
equilibrium problem of hard core particles at temperature T in a static random potential corresponding to a
height profile fhg. The Hamiltonian is
i
X
H¼ hn. (8)
i i
i
Height differences are related to the hopping rates of the model through the condition of detailed balance
u
exp½(cid:1)ðh (cid:1)hÞ=T(cid:2)¼ i;iþ1. (9)
iþ1 i u
iþ1;i
Since each local slope of the height is equal in magnitude but random in sign, the full height profile is
isomorphic to the trail of a random walker, with the height being the displacement of the walker and i being
time. Thus the problem reduces to that of hard-core particles in a Sinai potential. In the limit of large bias
(T !0),thesystemapproachesthegroundstate,whichcanbefoundasfollows.Makeaconstant-heightcut
of the height profile, fill particles up to that point, and then raise the level of the cut step by step and fill in
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M.Barma/PhysicaA](]]]])]]]–]]] 9
particles till they are exhausted. The lengths ‘ of particle clusters (which are present below the topmost filled
level)arethendeterminedbyreturnstotheoriginofthewalker,andtheirdistributionfollowsanasymptotic
power law Probð‘Þ(cid:3)‘(cid:1)3=2. This distribution has an infinite mean, and in a system of size L an estimate based
onextremalstatisticsshowsthatthelargestclustersizeisgL.Theprefactorgshowsstrongfluctuationsfrom
onesampletoanother,butisalwaysoforderunity.ThusatypicalconfigurationconsistsofatleastoneOðLÞ-
p p
ffiffiffiffi ffiffiffiffi
sized cluster in addition to (cid:3) L clusters of size (cid:3) L.
In Ref. [38], a similar model (the coarse-grained depth or CD model) was analysed, and the disorder-
averaged two-point correlation function C ðxÞ(cid:4)hnn i (cid:1)r2 was computed, using the cluster size
L i iþx L
distribution Probð‘Þ as input. The result is that C ðxÞ is a scaling function of argument x=L:
L
C ðxÞ¼Yðx=LÞ(cid:6)m2½1(cid:1)aðx=LÞa(cid:2); x=L51 (10)
L 0
with intercept m ¼1 and a¼1.
0 2
The argument of the scaling function indicates that it describes a phase-separated state [39]. The finite
interceptm2 isameasureoflong-rangeorder.Thecuspsingularityisamanifestationofill-definedinterfaces,
0
as it is known that a phase-separated system with well defined interfaces has a¼1, in concordance with the
Porod law [39]. We conclude that the state exhibits an unusual sort of phase separation, with unusual
interfacial characteristics and strong sample to sample fluctuations.
3.4. Time-dependent properties
In this subsection we discuss both the steady state dynamics and the coarsening dynamics of systems with
unidirectional and bidirectional disorder.
Withunidirectionaldisorder,thebehaviourofdensityfluctuationsinsteadystateresemblesthatintheZRP
discussed in the previous section. For jr(cid:1)r j4D, there is a kinematic wave which circulates through the
c
system,andinduceslong-periodoscillationsinthedensity[16].Thedissipationwasmonitoredbyperforming
aGalilean shifttokeepup withthewave.As expectedon thebasisofthekinematic wave criteriondiscussed
earlier,disorderwasfoundtobeirrelevant,andz¼3stillholds[16].Forr¼1,ontheotherhandthesystemis
2 2
phase separated, and the system supports two kinematic waves which move in opposite directions in the two
coexistingphases.Simulationssuggestthatthekinematicwavesemanatefromoneinterface,andterminateat
another, leading to a different universality class [16].
Inthephase-separatedregime,thekineticsofapproachtothesteadystateinvolvesacoarseningprocess[5].
Large-densitystretchesaccumulatebehindbottleneckregionswhicharestretchesofconsecutiveweakbonds.
If DJ is the difference of currents exiting from two such bottlenecks separated by a distance x, the region in
between would fill up in time t(cid:3)x=DJ. Estimating the probability of occurrence of stretches of length l and
recalling that they carry a current J ¼u =4ð1þa=lÞ [40] leads to the estimate:
l 2
t
xðtÞ(cid:3) . (11)
lnt
Turning to bidirectional disorder with tilt (fo1Þ, recall that in steady state, particles are held back behind
2
the longest backbend, while the region ahead of it is empty. The coarsening properties of this model were
obtainedbyKrug[5]whopointedoutthatbottleneckstretchlengthsaremuchsmallerthantheirseparations
implying that we may mentally replace stretches by single bonds that allow a maximum current J (cid:3)lð1=2Þl to
l
flow through. Following through, we find that the coarsening length is given by [5]
xðtÞ(cid:3)t1=ð1þy=2Þ. (12)
In the untilted case, the steady state is the equilibrium state of hard core particles in a random (Sinai)
potential. Not much is known about the dynamics in this case, though the single particle problem is well
p
ffiffiffiffi
studied [41]. The existence of large ð(cid:3)Oð LÞÞ potential barriers would make the approach to equilibrium
logarithmically slow, and it would be interesting to see how much slower the coarsening process is than the
familiar diffusion of a single particle in a Sinai landscape.
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10 M.Barma/PhysicaA](]]]])]]]–]]]
4. Dynamical disorder
In this section, we study the properties of an ASEP subjected to bidirectional bond disorder, where the
disorder is itself time-dependent. Specifically, we consider an untilted potential (as in Fig. 2(d)) which is
evolvingstochasticallyintimethroughthesingle-stepmodel[42]whichpreservestheoverallno-tiltcondition
under time evolution. The basic dynamical move is the interchange of nearest neighbour links of the height
fieldatacertainrateiftheyrepresentoppositeslopes.IfLandRstandforleft-leadingandright-leadinglinks,
respectively,thenalinkconfiguration::::LR::::canevolveinto::::RL::::atacertainrate,andthereversemove
isallowedatanequalrate.Thelargespaceandtimepropertiesoftheheightfieldofthismodelaredescribed
by the continuum Edwards–Wilkinson model [43]. The ASEP particles are passive sliders on the surface, in
that they are influenced by, but do not back-influence, the dynamics of the fluctuating surface.
ThisproblemhasbeenstudiedinRef.[38].Particlestendtoformclusterswhentheyfallintovalleys,andto
decluster when valleys overturn and form hills. However, declustering involves activation and is a slow
process,whileclusteringisrelativelyrapid.Thisasymmetryleadstooverallclustering.Numericalsimulations
show that this tendency to cluster persists at all length scales, and leads finally to a steady state that exhibits
phaseseparation.However,thisphaseseparatedstateisunusualinthatischaracterizedbylargefluctuations
oftheorderparameter,whichdonotdampdowninthethermodynamiclimit.Despitethepresenceofstrong
fluctuations,thesystem neverlosesitsorderedcharacter[44].Fluctuationscarrythesystemrelativelyrapidly
fromoneorderedconfigurationtoanothermacroscopicallydistinctone.Buttheprobabilityforthesystemto
leave this attractor of ordered states vanishes exponentially in the system size.
A quantitative measure of the ordering is provided by the two point correlation function. Results of
simulations show that Eq. (10), derived for the disordered problem with an untilted, bidirectional potential
describes the form of C ðxÞ in this case as well. When the update rates for the landscape and particles are
L
equal, the value of m2 is close to 0.8, indicating long-range order, consistent with the existence of a phase
0
ordered state. Moreover, the value of the cusp exponent a is close to 1, its value for the problem discussed in
2
theprevious section.Thisquantitativesimilaritybetween thescalingproperties ofphaseorderinginducedby
quencheddisorder,andbydynamicaldisordercorrespondingtoanevolvingEdwards–Wilkinsonlandscape,is
not completely understood.
4.1. Time-dependent properties
The time dependence of the particles is dictated by fluctuations of the driving height field. Except at very
smalltimes,theautocorrelationfunctionA ðtÞ(cid:4)hnð0ÞnðtÞiisfoundnumerically[44]tobeascalingfunction,
L i i
A ðtÞ¼Xðt=LzÞ(cid:6)m2½1(cid:1)dðt=LzÞb(cid:2); t=Lz51 (13)
L 0
withz¼2beingthedynamicalexponentgoverningtheheight-fielddynamicsanddaconstantoforderunity.
Simulations show that m2 ’0:8;b’0:2. Like its counterpart for static correlations (Eq. (10)), the scaling
0
functionXshowsanapproachwithacuspsingularitytoafinitevalueatsmallargument,onceagainsignalling
longrangeorderofanunusualsort.Finally,tounderstandthecoarseningpropertiesofthismodel,recallthat
clusteringoccursbycollectingparticlesinvalleysofthedrivingheightfield.Sincethelargestvalleysthatform
in time t are typically of size t1=z, we expect the coarsening length in this case to follow:
xðtÞ(cid:3)t1=2. (14)
Thetime-evolvingtwo-pointcorrelationfunctionhnðtÞn ðtÞi(cid:1)r2isexpectedtobesimilartotheformofEq.
i iþx
(10), except that the system size L is replaced by the coarsening length xðtÞ. The results of numerical
simulations agree well with this [38].
5. Conclusion
We have reviewed the properties of the one-dimensional ASEP with various types of disorder—particle-
wise, space-wise (unidirectional and bidirectional), and dynamical. All forms of disorder bring about phase
separation, though under different conditions, and of different types.