Emergent motion of condensates in mass-transport models
EEmergent motion of condensates in mass-transport models
Ori Hirschberg, David Mukamel, and Gunter M. Sch¨utz Department of Physics of Complex Systems, Weizmann Institute of Science, 76100 Rehovot, Israel Institut f¨ur Festk¨orperforschung, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany (Dated: November 1, 2018)We examine the effect of spatial correlations on the phenomenon of real-space condensation indriven mass-transport systems. We suggest that in a broad class of models with a spatially correlatedsteady state, the condensate drifts with a non-vanishing velocity. We present a robust mechanismleading to this condensate drift. This is done within the framework of a generalized zero-rangeprocess (ZRP) in which, unlike the usual ZRP, the steady state is not a product measure. Thevalidity of the mechanism in other mass-transport models is discussed.
PACS numbers: 05.70.Ln, 02.50.Ey, 05.40.-a, 64.60.-i
Nonequilibrium condensation, whereby a macroscopicfraction of microscopic constituents of a system accumu-lates in a local region, is a common feature of many mass-transport systems. Examples include shaken granulargasses [1], vehicular traffic [2–4], the macroeconomics ofwealth distribution [5, 6], and others [7, 8]. Mechanismswhich can lead to the formation of condensates have beenstudied extensively in recent years, mainly by analyzingprototypical toy models. A primary role in these studieswas played by the zero-range process (ZRP), an exactly-solvable model in which particles hop between sites withrates which depend only on the number of particles inthe departure site [9–11]. Extensions and variations ofthe ZRP have been used to study the emergence of multi-ple condensates [12], first order condensation transitions[13, 14] and the effect of interactions [15] and disorder [16]on condensation. Moreover, one-dimensional phase sepa-ration transitions in exclusion processes and other drivendiffusive systems can quite generally be understood by amapping on ZRPs [17].The dynamics of condensates is less well explored. Inthe ZRP, where condensation takes place when a macro-scopic fraction of particles occupies a single site, the re-sulting condensate does not drift in the thermodynamiclimit [14, 18–20]. It is shown below that this is relatedto the fact that the steady state of the ZRP is a productmeasure. In some real-world systems, however, conden-sates are in continual motion. For example, traffic jams,which can be viewed as condensates, are known to prop-agate along congested roads [11, 21, 22]. Recently, twovariants of the ZRP were also found to relax to a timedependent state in which the condensate performs a driftmotion: one is a ZRP with non-Markovian hopping rates[23, 24], and the other is a model with “explosive conden-sation” [25]. To date, there is no systematic understand-ing of the mechanism by which a macroscopic condensatemotion emerges from the underlying nonequilibrium mi-croscopic dynamics.In this Letter, we study how spatial correlations inthe steady states may lead the condensate to drift witha non-vanishing velocity. We do so by introducing a generalization of the ZRP whose steady state does notfactorize. Within its framework, we identify the mech-anism which generates the drift. The analysis is basedon numeric simulations and on a mean-field approxima-tion which captures the essential effect of correlations inthe condensed phase, and thus elucidates the differentobserved modes of condensate motion. The drift mech-anism which we identify is robust and therefore it is ex-pected to be valid in a broad class of spatially-correlatedmass-transport systems.We focus on a class of stochastic one-dimensional mod-els defined on a ring with L sites. At any given time, eachsite i is occupied by n i particles with (cid:80) i n i = N , n i ≥ i to i + 1 with a ratewhich depends on the occupation numbers n i and n i − .This is a generalization of the usual ZRP in which therate depends only on n i . More specifically, we choose thehopping rates to be of the form n i − , n i , n i +1 w ( n i − ) u ( n i ) −−−−−−−−−→ n i − , n i − , n i +1 + 1 , (1)with rates u ( n i ) = 1 + bn i , w ( n i − ) = (cid:26) n i − (cid:54) = 0 α n i − = 0 . (2)The particular form of u ( n ) is motivated by the fact thatin the usual ZRP, which corresponds to α = 1, this choicewith b > w with α (cid:54) = 1 represents an interaction betweennearest-neighbor sites. According the dynamical rules(1)–(2), at every short time interval dt , each site i whoseoccupation n i ≥ u ( n i ) dt , as long as the preceding site ( i −
1) is occupied.If the preceding site is empty, this probability changes to αu ( n i ) dt . The model has three parameters: b , α , and thedensity ρ ≡ N/L which is conserved by the dynamics.In the usual ZRP (the case of α = 1), the stationarydistribution is known to factorize into a product of singlesite terms, and so can be exactly calculated [9, 27, 28].This factorization property renders the ZRP quite spe-cial, as slight variations of the ZRP dynamics result in a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec S i t e O cc upa t i on Time (10 time units) 0 0.5 1 1.5 2400405410 S i t e O cc upa t i on Time (10 time units) (a) (b) S i t e Most occupied2nd most occupied2 4 6 8 100500010000 O cc upa t i on Time (10 time units) (c) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5)(cid:5) (cid:1) (cid:6)(cid:7)(cid:8)(cid:8)(cid:4)(cid:9)(cid:8) S i t e O cc upa t i on Time (10 time units) (d) FIG. 1. The location (top) and occupation (bottom) of the most occupied ( • ) and 2 nd most occupied ( ◦ ) sites for several valuesof α . (a) α = 1, i.e., the usual ZRP. The condensate is stable for long times, and relocates to random distant sites. (b) α = 1 . α = 1 .
05. The condensate remains on each sitefor a long time before “spilling” to the next. The definitions of T barrier and T spill are indicated. (d) α = 0 .
5. The condensateskips every other site. In all cases b = 3, ρ = 10 and L = 1000, except (a) where L = 400. Note the different time scales. non-factorizable models. To probe the effect of spatialcorrelations on the condensate, we choose for simplicity w to be of the form (2), which leads to a spatially corre-lated steady-state when α (cid:54) = 1. The drift motion whichwe describe below is also found for other forms of w ( n ),such as cases where w ( n ) (cid:54) = 1 for finitely many values of n , and for other choices of u ( n ) which give rise to con-densation [29].We have carried out Monte Carlo simulations of thedynamics (1)–(2) for several values of α in a system ofsize L = 1000 and density ρ = 10. After the system hasrelaxed to its steady state, the dynamics of the conden-sate was examined by tracking the position of the mostoccupied sites over time. The results are presented inFig. 1 and in videos in the supplemental material [30].In the usual asymmetric ZRP (Fig. 1a), it is known thatthe condensate is static up to timescales of order L b andthen it relocates to a random site due to fluctuations[14, 18–20]. There is a striking qualitative difference inthe dynamics of model (1)–(2) when α (cid:54) = 1 (Fig. 1b–1d),where the condensate is clearly seen to drift along thelattice. The condensate is seen to move from one site tothe next when α > α < α is not tooclose to 1 the motion is “slinky”-like, with the conden-sate “spilling” from an old site to a new one immediatelyafter the previous spilling has completed. The drift be-comes somewhat less regular when α is close to 1. Inthis regime, the slinky motion is interrupted by periodsof time when the condensate occupies a single site, beforethe spilling process is initiated (Fig. 1c). This, however,is argued below to be a crossover mode, and the intervalin α in which it is observed shrinks in the large L limit.To understand these results we propose a mean-fieldanalysis of the model in which the occupations of allsites are considered independent, but might not be identi-cally distributed. Within this approximation, the currentwhich arrives into site i from site i − J i . The probability P i ( n i ) to find n i particles in site i thus evolves according to dP i ( n i ) dt = P i ( n i − J i + P i ( n i + 1) (cid:104) w i (cid:105) u ( n i + 1) − P i ( n i ) (cid:0) J i + (cid:104) w i (cid:105) u ( n i ) (cid:1) , (3)where (cid:104) w i (cid:105) ≡ ∞ (cid:88) n =0 P i − ( n ) w ( n ) = 1 + ( α − P i − (0) (4)encodes the mean effect of site i − i . Equation (3) is valid also when n i = 0 withthe definition P i ( − ≡
0. Equations (3) and (4) areto be solved with the self consistency condition J i +1 = (cid:80) n P i ( n ) (cid:104) w i (cid:105) u ( n ).At low density, the system is in a subcritical, disor-dered phase (this will be shown below). In this homo-geneous phase, P i ( n ) = P ( n ) and J i = J for all sites i .At higher densities, however, condensation takes place,where the translational symmetry is spontaneously bro-ken and both P i ( n ) and J i depend on the distance of site i from the condensate. This dependence of P and J on i isa result of the correlations which exist in the steady stateof the model, and it provides the mechanism for the con-densate drift: a nonhomogeneous J i implies that in somesites the outflowing current is smaller than the incomingcurrent, leading these sites to accumulate particles whileother sites are similarly being depleted of particles. Weshall now demonstrate that this occurs in our model.In the homogeneous (subcritical and critical) phases,the model eventually reaches a steady state. In the non-homogeneous supercritical phase, however, the conden-sate location keeps moving with time. The analysis ofthis time-dependent phase is based on one key observa-tion: the timescale of the microscopic dynamics, whichfor the rates (2) is of order 1, is much faster than that ofthe condensate motion. As shown below, the timescale ofthe spilling process is of order L , validating this observa-tion in the thermodynamic limit. Due to this timescaleseparation, while the condensate (i.e. the most occupiedsite), is static all other sites reach a quasi-stationary dis-tribution.In both phases, by equating the LHS of Eq. (3) to zerothe (quasi-)stationary distribution is found to be P i ( n ) = P i (0) z ni (cid:89) k ≤ n u ( k ) , with z i ≡ J i / (cid:104) w i (cid:105) . (5)Here, z i plays the role of a “fugacity” of site i . For ratesof the form (2), the normalization of P i ( n ) yields P i (0) = [ F (1 , b + 1; z i )] − , (6)where F is a hypergeometric function (note that P i (0)depends on the exact form of u ( n ) and not just on itslarge n asymptotics). The occupation probability isasymptotically given by P i ( n ) ∼ n − b z ni .We first examine the solution (5) in the subcritical andcritical phases, and show that the model undergos a con-densation transition. Since the system is homogenous inthese phases, the subscript i may be dropped from equa-tions (4)–(6). The fugacity can now be determined interms of the density by inverting the relation ρ ( z ) = (cid:88) n nP ( n ) = F (2 , b + 2; z )(1 + b ) F (1 , b + 1; z ) z, (7)where the RHS is obtained by substituting Eq. (5) in thesum. Similarly, Eq. (4) for (cid:104) w (cid:105) reads in the homogeneousphases (cid:104) w (cid:105) = 1 + ( α − F (1 , b + 1; z )] − .The density (7) is an increasing function of z that at-tains its maximum at z = 1, which is its radius of con-vergence about the origin. A finite density at z = 1indicates a condensation phase transition, which is math-ematically similar to Bose-Einstein condensation [9]. Bysubstituting z = 1 in (7) it is seen that condensationtakes place when b >
2, in which case the critical den-sity is ρ c = 1 / ( b − J c = (cid:104) w (cid:105) z → = 1 + ( α − b/ ( b − ρ < ρ c ,the system remains in a homogeneous subcritical phase.When ρ is increased, the current J increases until ρ and J reach their critical values and all sites of the systemare in a homogenous critical phase. When the densityis further increased, condensation sets in, breaking thetranslational invariance of the system.Let us now discuss the nonhomogeneous supercriticalphase and the mechanism of the condensate motion. Wefocus on the case of α >
1. In this case, the condensedphase is composed of a condensate, which at any giventime consists of two macroscopically occupied consecu-tive sites (say 1 and 2), while the rest of the sites are mi-croscopically occupied. We show that J > J i = 1for i (cid:54) = 2. This results in an increase of the occupa-tion of site 2 at the expense of site 1 over a macroscopic O ( L ) time scale, while the rest of the sites are in a quasi-stationary state. Therefore, the condensate drifts with avelocity of order L − . The analysis begins at site 1, whose occupation we as-sume is n = O ( L ) (cid:29)
1, and thus it emits a mean current J = (cid:104) w (cid:105) (1 + (cid:104) b/n (cid:105) ) (cid:39) (cid:104) w (cid:105) . At the moment, (cid:104) w (cid:105) isunknown. It is determined self-consistently at the end ofthe calculation. Since P L (0) (cid:54) = 0, as is established below,it is seen that J > α > P (0) = 0.It follows from (4) that (cid:104) w (cid:105) = 1. The fugacity of thesecond site is then z ≡ J / (cid:104) w (cid:105) (cid:39) J >
1, and there-fore its occupation distribution (5) cannot be normalized.This means that as long as site 1 is highly occupied, site2 tends to accumulate particles, implying that its occu-pation too becomes macroscopic (of order L ) for a longperiod of time [31, 32]. We call such a site with fugacity z i > supercritical .The analysis now continues site by site in a simi-lar fashion. For each site i , (cid:104) w i (cid:105) is calculated using(4) from the known value of P i − (0). The fugacityof site i (5) is then calculated from (cid:104) w i (cid:105) and the in-coming current into the site J i . Once the fugacity isknown, P i (0) and J i +1 are determined from (6) and from J i +1 = (cid:80) n P i ( n ) (cid:104) w i (cid:105) u ( n ), and the process is repeatedin the next site. Performing this analysis reveals thatsite 3 is critical (i.e., z = 1) and sites i = 4 , . . . , L are subcritical ( z i < J i = 1 and (cid:104) w i +1 (cid:105) =1 + ( α − / F (1 , b + 1; 1 / (cid:104) w i (cid:105) ) for all i ≥
3. This re-cursion relation defines a sequence (cid:104) w i (cid:105) which converges(exponentially rapidly) to a unique fixed point w ∗ ( α, b )which is the solution of the equation w ∗ = 1 + α − F (1 , b + 1; 1 /w ∗ ) , (8)and thus satisfies w ∗ ( α, b ) > α >
1. When L (cid:29)
1, the periodic boundary conditions imply that (cid:104) w (cid:105) (cid:39) w ∗ >
1, and thus Eq. (5) confirms that P L (0) > ρ BG ,which can be defined as the mean density of all butthe two most occupied sites (since the condensate istypically carried by two sites). Below the transition, ρ BG = ρ , which approaches ρ c = 1 / ( b −
2) as the tran-sition is approached from below. Above the transition,all sites outside of a finite boundary layer around thecondensate are subcritical with a mean occupation of ρ BG (cid:39) ρ ( z = 1 /w ∗ ) < ρ (1) = ρ c since the function ρ ( z ),Eq. (7), increases monotonically with z . Therefore, thecondensation transition is found to be a discontinuous(first order) one. This is in contrast to the usual ZRPwith rates (2) and α = 1 where the transition is continu-ous. A similar discontinuity exists in the current, whichjumps from J c > ρ c to J = 1 just above it.We now discuss the emergent dynamics of the conden-sate and identify two distinct modes of motion: a reg-ular slinky motion, and an irregular motion through abarrier. The motion of the condensate from one site tothe next consists of two stages: a “spilling” stage duringwhich it is supported on two sites, and a period beforethis spilling is initiated, when the condensate is carriedby a single site. We first consider the spilling process.According to the calculation above, the number of par-ticles that accumulate in the second condensate site perunit time is on average J − J = w ∗ −
1. As there are N cond (cid:39) ( ρ − ρ BG ) L particles in the condensate, the to-tal spilling time T spill scales, to leading order, linearlywith the system size: T spill = ( ρ − ρ BG ) L/ ( w ∗ − α →
1, the spilling time diverges.Once a spilling is complete, there is a moment that thecondensate is located solely on a single site. We now re-label this site as site 1. At this moment, the occupationof the following site is n ≈ ρ BG = O (1). The rate atwhich particles leave the second site is, at this stage, ap-proximately J ≈ b/ρ BG , which should be comparedwith the rate of incoming particles, J (cid:39) w ∗ . Accordingto Eq. (8) and the definition of ρ BG , the two rates areequal when α = α ∗ which is the solution of the equation α ∗ = 1 + b F (cid:0) , b + 1; 1 /w ∗ ( b, α ∗ ) (cid:1) /ρ BG ( b, α ∗ ). Themode of condensate motion now depends on whether α is larger or smaller than α ∗ . (i) When α > α ∗ , the initialcurrent into site 2 is larger than the mean current outof this site, and a spilling of the condensate is initiatedimmediately. In this case, the condensate drifts continu-ously in a slinky motion as in Fig. 1b. (ii) On the otherhand, J < J when 1 < α < α ∗ , and thus particlesdo not immediately accumulate on site 2. The incomingcurrent into the site surpasses the outgoing current andspilling sets in only after fluctuations bring the occupa-tion of the second site to a value n ∗ ( α, b ) which is definedby w ∗ = 1 + b/n ∗ . The ensuing motion of the condensateis more irregular, with a stable condensate which occa-sionally spills to the next site as in Fig. 1c. Note that themean time T barrier it takes before a fluctuation brings n over the barrier n ∗ does not scale with the system size.Thus, in a large enough system the condensate regularlydrifts and is typically supported by two neighboring sitesfor any value of α > α is increased con-form with numerical findings (Fig. 1). In particular, thespilling mechanism between the two condensate sites inwhich the accumulation of particles is linear in time isverified (Fig. 1b). Furthermore, the first order nature ofthe transition, as manifested by the ρ bg ( ρ ) curve, and the ρ ρ bg L = 250L = 500L = 1000L = 2000 −6 −4 −2 n P ( n ) α = 1 (ZRP) α = 1.5 FIG. 2. A first order phase transition is seen in the back-ground density ρ BG as a function of the density ρ . Numericalresults for several system sizes are plotted, along with an ex-trapolation to L = ∞ (thick black line). Here α = 1 . b = 3. The inset shows that the occupation of a site far awayfrom the condensate has a subcritical distribution (i.e., withan exponential tail) when α >
1. This differs from the usualZRP where the background fluid is known to be critical. Re-sults are for L = 1000, ρ = 10, and b = 3. subcritical nature of the background fluid are presentedin Fig. 2.The mechanism for the condensate drift found in thismodel can be summarized as follows: the spontaneousbreaking of translation invariance by the formation ofthe condensate may induce an accumulation of parti-cles in a nearby site. This accumulation results in acontinually drifting condensate, since whenever a con-densate is established on a new site, another one be-gins to form further ahead. This mechanism holds ina much more general setting, including when other forms w ( n ), partially asymmetric hopping and higher dimen-sional lattices are considered, and more widely in othernon-factorized mass-transport models [29]. Note that thedrift discussed here, in which the two most occupied sitesare typically nearest neighbors, cannot occur in modelswith a factorized steady state, the latter being symmet-ric under site permutations. In this respect, our mech-anism differs from that studied recently in [25], whereunbounded hopping rates generate a drift (with infinitevelocity) in a model whose steady state factorizes.An important point to note is that in general, the newcondensate site does not have to be a neighbor of the oldone. For instance, in our model (1)–(2) with α <
1, a sim-ilar analysis shows that the condensate skips every othersite, as observed in Fig. 1d [29]. In this case, the super-critical site is site 3, rather than 2 (when the condensateis located on site 1). In principle, it may happen thatthere is more than one supercritical site, possibly lead-ing to more complicated condensate drifts. It may alsohappen that no other site is supercritical, in which casethe condensate would not drift. A precise and generalclassification of the conditions under which a condensatedrift occurs remains an interesting open problem. How-ever, in many specific models, a study of condensationand the condensate motion can be carried out followingthe mean-field procedure outlined in this Letter. For in-stance, a recently proposed accelerated exclusion process(AEP) [33] can be analyzed in a similar fashion, yieldingthe phase diagram of the model and revealing that theAEP condensate drifts in the steady-state [34].We thank Ariel Amir, Amir Bar, Or Cohen and TridibSadhu for useful discussions. The support of the IsraelScience Foundation (ISF) is gratefully acknowledged. [1] K. van der Weele, D. van der Meer, M. Versluis, andD. Lohse, Europhys. Lett. , 328 (2001).[2] O. J. O’loan, M. R. Evans, and M. E. Cates, Phys. Rev.E , 1404 (1998).[3] D. Chowdhury, L. Santen, and A. Schadschneider, Phys.Rep. , 199 (2000).[4] J. Kaupuˇzs, R. Mahnke, and R. J. Harris, Phys. Rev. E , 056125 (2005).[5] J. P. Bouchaud and M. M´ezard, Physica A , 536(2000).[6] Z. Burda et al. , Phys. Rev. E , 026102 (2002).[7] S. N. Majumdar, S. Krishnamurthy, and M. Barma,Phys. Rev. Lett. , 3691 (1998).[8] S. N. Dorogovtsev and J. F. F. Mendes, Evolution ofNetworks (Oxford University Press, Oxford, 2003).[9] M. R. Evans and T. Hanney, J. Phys. A , R195 (2005).[10] S. N. Majumdar, Real-space Condensation in Stochas-tic Mass Transport Models, in Exact Methods in Low-Dimensional Statistical Physics and Quantum Comput-ing: Lecture Notes of the Les Houches Summer SchoolJuly 2008 , edited by J. Jacobsen et al.
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