Maintenance of order in a moving strong condensate
Justin Whitehouse, André Costa, Richard A Blythe, Martin R Evans
MMaintenance of order in a moving strong condensate
Justin Whitehouse, Andr´e Costa, Richard A Blythe, Martin REvans
SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road,Edinburgh EH9 3JZ, UK
Abstract.
We investigate the conditions under which a moving condensate may existin a driven mass transport system. Our paradigm is a minimal mass transport modelin which n − n > u ( n ) of this move depends only on the occupation of the departure site. Westudy a hopping rate u ( n ) = 1+ b/n α numerically and find a moving strong condensatephase for b > b c ( α ) for all α >
0. This phase is characterised by a condensate thatmoves through the system and comprises a fraction of the system’s mass that tendsto unity. The mass lost by the condensate as it moves is constantly replenished fromthe trailing tail of low occupancy sites that collectively comprise a vanishing fractionof the mass. We formulate an approximate analytical treatment of the model thatallows a reasonable estimate of b c ( α ) to be obtained. We show numerically (for α = 1)that the transition is of mixed order, exhibiting exhibiting a discontinuity in the orderparameter as well as a diverging length scale as b (cid:38) b c .PACS numbers: 02.50.Ey, 05.70.Fh, 64.60.De a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t rder in a moving condensate
1. Introduction
In nonequilibrium statistical physics, condensation is used as a general term to describethe localisation of a finite fraction of some quantity—typically mass—in a wide varietyof fundamental models of dynamical processes. These include the flow of wealth [1],traffic flow [2–5], and the formation of hubs in complex networks [6, 7]. The archetypalmodel of this class is the Zero-Range Process (ZRP) [8,9]. In this minimal model, singleunits of mass hop between sites at a rate which is a function only of the total masson the site they are leaving (hence the name ‘zero-range’). Furthermore this modelsatisfies the conditions required for the steady state to factorise, which simplifies theanalysis of its condensate phase [10–12]. Given an appropriate choice of the hoppingrate u ( n ), which decreases suitably slowly with n , this process alone is enough tocreate a static condensate phase in which a finite fraction of the total mass of thesystem occupies a single site. For the case u ( n ) = 1 + b/n α the transition has beenextensively studied. When α = 1 there is a critical value of the parameter b , b c = 2,above which a condensation transition occurs when particle density ρ exceeds a criticalvalue ρ c = 1 / ( b − ρ < ρ c , the system is in a fluid phase where the mass isevenly distributed across sites, whereas for ρ > ρ c a condensate emerges. For α < b [8].There are certain cases in which the nature of the condensate phase is different fromthe ‘standard’ condensation described above. For example: the fraction of the totalsystem mass in the condensate can be equal to 1, creating a strong condensate [13, 14];the condensed phase can exhibit a subextensive number of smaller mesocondensates [15]or an extensive number of finite-sized quasi-condensates [16]. Also, it should be notedthat the existence of a condensate phase is not unique to models based on the ZRP orwith factorised steady states. For example, a non-Markovian simple exclusion processhas been shown to exhibit an immobile condensate phase [17].In all these examples, the condensates are static: they reside at the same pointin space for a long period until dissolving through a large fluctuation and reformingelsewhere [18]. However, in physical settings moving condensates or aggregates are oftenobserved, for example in traffic jams [19], gravitational clustering [20], sedimentation [21]and droplet formation [22]. In general, moving condensates are less well understood thanthe static variety and it is unclear what the physical mechanisms are that will allow themaintenance of the condensate.In this work we investigate conditions under which a condensate may maintain itsorder as it moves through the system. To understand why this is a pertinent questionwe first review how condensates move in a variety of simple model systems related tothe ZRP.In an important early contribution, Majumdar, Krishnamurthy and Barma [23]introduced a chipping model in which all the mass from a site can move, or ‘diffuse’, toan adjacent site. Additionally, a single unit of mass can ‘chip’ off from the departuresite and hop to an adjacent site. For symmetric diffusion of the mass, a condensed phase rder in a moving condensate u ( n ) = 1 + b/n , suggests that a condensate may be possible for largeenough b with the critical value b c somewhere close to two [27].Meanwhile, Hirschberg et al. have investigated what kinds of dynamical processeswill permit a moving condensate phase using variants of the ZRP with non-Markovianhopping rates [28, 29] and with hopping rates affected by spatial correlations [30]. Bothmodels exhibit a condensate phase which drifts with a finite, nonvanishing velocity. Inthe former, temporal correlations between departure and arrival sites allow the formationof a condensate over two adjacent sites, which then moves with a ‘slinky’-like motionthrough the system. In the latter, the effect of spatial correlations is that the condensatealso moves with a slinky-like motion, but with certain differences in the details dependingon the values of certain hopping parameters. Condensate motion of a similar slinkynature has also been observed in a totally asymmetric model [31,32] in which the hoppingrate is a monotonically increasing function of the mass at both departure and arrivalsites. The condensation found in this model is found to be “explosive” in as much asthe condensate moves with a superextensive velocity and forms instantaneously in aninfinite system.Taken together these studies pose the intriguing question that we pursue here:What are the key dynamical processes that permit a moving condensate phase, andwhich processes will destabilise the phase? This question is of broader relevance tophenomena such as flock formation and schooling of fish [33].In this work, we introduce a minimal mass-transport model in the spirit of theZRP which allows large-mass hopping events. The new feature is the incorporationof a ‘backchip’ move described below and illustrated in figure 1b. We find that themodel exhibits a moving condensate phase with a distinctive mechanism of formationand maintenance. Most notably, the moving condensate is a strong condensate in thesense that a fraction tending to one of the particles occupy a single site. This condensatetravels through the system followed by a tail of low occupancy sites that collectivelycomprise a vanishing fraction of the mass. As we show below, the dynamics of the masswithin this tail is responsible for maintaining the structure of the condensate. Usingnumerical simulations, we find that above a critical value for a rate parameter b , andat all densities, a strong condensate forms. Numerically we are able to classify thetransition as being of mixed order, exhibiting features of both first and second orderphase transitions. We further provide an approximate theory of the mechanism whichgives a reasonable prediction for the critical value, and also discuss the behaviour of thesystem below this critical rate. rder in a moving condensate u(1) (a) u(3) (b) Figure 1: The elementary dynamical processes. (a) Hop: When a site contains only oneparticle, this particle hops onto the next site with rate u (1) and leaves behind an emptysite. (b) Backchip: When a site contains n (here, 3) particles, n − u ( n ) onto the next site and leave behind a single particle.
2. Model and motivation
To motivate the specific features of our model let us first consider the limit of zerochipping rate in the models of [24, 27]. In the absence of any chipping, the dynamicsis simply diffusion combined with irreversible aggregation. The stationary state of thisprocess on a finite system comprises a single condensate containing all the system’smass. The work of [25, 26] has shown that the condensate is unstable to the effectof single particles chipping away from an aggregate with rate u = 1 + b/n , where n is the number of particles contained within the aggregate, unless b is greater than acritical value b c >
0. Therefore it is of interest to consider what other perturbationsmay destabilise the condensate exhibited in diffusion with irreversible aggregation.Here we consider a different perturbation of the diffusion with aggregationdynamics: when the aggregate moves forward it leaves one unit of mass behind—seefigure 1. We retain the zero-range feature that the rate of movement u ( n ) only dependson the number of particles within the aggregate.To be precise, we consider a system of N particles upon a one-dimensional latticeof L discrete sites. We are interested in the large N , L behaviour where the density ρ = N/L is fixed. In the case where the site has occupancy n >
1, a ‘backchip’ takesplace: n − u ( n ) given by u ( n ) = 1 + bn α , (1)where n is the total number of particles on the departure site. With this form, largervalues of the rate parameter b bias the dynamics towards faster hopping from lessoccupied sites, causing larger groups of mass to move slowly in comparison.The dynamical rule has of course to be modified for a single unit of mass at a site,i.e. when the site’s occupancy is n = 1. This single particle moves to the next site andleaves behind an empty one (figure 1a). We take the rate of this ‘hopping’ process to rder in a moving condensate (a) m k k data: ρ = 1.0, b = 2.0, α = 1.0data: ρ = 2.0, b = 2.0, α = 1.0data: ρ = 5.0, b = 2.0, α = 1.0data: ρ = 10.0, b = 2.0, α = 1.0 (b) Figure 2: Illustration of the condensate phase. (a) A typical configuration of the systemin the strong condensate regime, sketched using data taken directly from a simulationwith L = 1000, N = 1000, b = 2 . α = 1 .
0. The columns represent the massoccupying the site, with the exact size shown above, and the direction of motion isindicated by the arrow. (b) The average mass m k at a site k sites behind the condensatefor ρ = 1 , , ,
10. In the strong condensate regime all of the system’s mass is typicallyin the condensate, with any remaining units of mass trailing closely behind.be u (1).Our choice of hop rates allows us to compare our model to the standard formulationof the ZRP where only a single particle can hop at a time. This type of hop is oftenreferred to as a ‘chip’, as in [23–26], and is in some sense symmetric to our definition ofa backchip. The name ‘chip’ can be conceptually understood in the context of a singlemass unit chipping off from a site with large occupancy. We first study the case where α = 1, as it can be shown exactly that with the hop rate given in (1) and α = 1 thestandard ZRP undergoes a condensation transition at ρ c = 1 / ( b −
2) for b >
3. Monte Carlo simulations
We implemented Monte Carlo simulations of the system on a one-dimensional periodiclattice. From these we see that, above a critical value of b , the system exhibits a strong condensate, where almost all the mass occupies a single site, which moves through thesystem. This is immediately followed by a short tail of sites with very low occupancy,leaving all other sites empty (figure 2). This behaviour can also be seen from the plotsof the site occupancy distribution (figure 3), which is strongly peaked at n/N = 1,indicating that we have a strong condensate. It tells us that typically at an instant intime nearly all of the mass occupies a single site.The maintenance of order in the strong condensate phase can be attributed to thedynamics of this tail of mass which trails directly behind the condensate itself. Since alarger hopping rate b biases the rate function u ( n ) towards hops from sites with a lowoccupancy n , hops from sites with low n will occur much more frequently than from rder in a moving condensate p ( n / N ) n/ N ρ = 0.25 ρ = 0.50 ρ = 0.75 ρ = 1.00 ρ = 1.25 ρ = 1.50 ρ = 1.75 ρ = 2.00 Figure 3: Plot of the distribution p ( n ) of site occupancies n in the strong condensateregime at b = 2 . L = 1000. N is the total number of particles in thesystem. The points at and near n/N = 1 indicate the presence of s strong condensatewhich contains effectively all of the system’s mass.those sites with large n . When the condensate hops, it leaves behind a site of occupancy1. A single unit of mass has the largest possible hop rate, and thus it seems plausiblethat it is much more likely for the single mass unit immediately behind the condensateto recombine with it than it is for the condensate to hop again and away from the massit left behind. In this way, it appears that the strong condensation is a consequence ofthe condensate being unable to escape from the tail of mass trailing behind. As such,the structure of a very strongly occupied condensate and its very short tail of a fewmasses is maintained as they move through the system. We provide further evidencefor this intuitive picture of strong condensation within a theoretical treatment below.
4. Analysis of the moving strong condensate
The fact that the system exhibits a coherent moving structure, comprising a condensateand its tail, implies that the occupancies of the sites near the condensate are likely to becorrelated. An exact solution for the stationary distribution of this model is thereforeunlikely to be easily attainable. In order to construct an approximate theory thatallows us to estimate the critical value b c at which the transition to a strong condensateoccurs, we make two main assumptions. First, we work in the frame of reference of thecondensate, labelling sites k = 1 , , , . . . according to how far behind the condensatethey are, and assuming that almost all of the mass occupies site k = 0 (i.e., that a strongcondensate has formed). Second, we allow the probability distribution for the number rder in a moving condensate n on each site k to take a different form, p k ( n ), on each site but , to allowanalytical progress, we assume that the occupancies on different sites are uncorrelated.We distinguish between the dynamics of particles in the tail, and of the condensateitself. When mass is transferred in the tail (either by a single particle hop, or by abackchip), it moves in the negative k direction, towards the condensate , from k + 1 → k .This leads to a mass current J k due to hopping from site k to k − J k = u (1) p k (1) + ∞ (cid:88) n =1 ( n − u ( n ) p k ( n ) . (2)Meanwhile, the condensate hops with rate (cid:39) N and weconsider the limit of large N . In the frame of reference of the condensate, this causesthe whole tail to shift its position which is accounted for by relabelling the indices k → k −
1. Consequently, the total average current arriving at site k in the positive k direction i.e. from site k − k is K k − = n k − − J k . (3)By continuity, the mean occupancy of site k changes with time asdd t n k = K k − − K k . (4)In the steady state dd t n k = 0, so we find K k = K for all sites k . Thus, (3) becomes K = n k − − J k . (5)Inserting the explicit form (1) for u ( n ) (with α = 1) into (2) we obtain J k = (1 + b ) p k (1) + n k − (1 − b )(1 − p k (0)) − b (cid:100) n − k (6)where (cid:100) n − k = ∞ (cid:88) n =1 p k ( n ) n , (7)that is, an average of 1 /n over the part of the distribution where n >
0. To proceed wemust also determine an appropriate form for p k ( n ). We have performed some numericalanalysis of p k ( n ) in the sites immediately preceding the condensate to allow us to makethe appropriate choice. As shown in figure 4, we find that it is not easy to fit a simplefunction to the distribution p k ( n ), but to make progress we assume p k ( n ) = (1 − a k ) a nk , (8)a geometric distribution, which describes the mass in different parts of the tail of thecondensate reasonably well (figure 4). This is much easier to work with analytically thanother possible assumed distributions as it has the useful property that the parameter a k can be expressed in terms of the mean occupancy n k at site k as a k = n k / (1 + n k ).This allows us to express the current (6) entirely in terms of n k and b , using p k (0) = 11 + n k (9) rder in a moving condensate p k ( n ) for having n particles at site k behind the condensate. Here we have plotted the data for the sites closest behind thecondensate, and to the measured distribution for k = 2 (large blue +) have fitted thefunction c (1 − a ) a n (red solid line) to the middle of the tail, and (1 − a ) a n (green dashedline) to the front of the tail. (Similar fits can be made for other values of k .) Above n ∼
15 the data is too noisy to be fitted to reliably. The data shown is for sites k = 2 , , ρ = 0 . L = 1000, b = 1 . α = 1 . p k (1) = n k (1 + n k ) (10) (cid:100) n − k = ln(1 + n k )1 + n k . (11)Inserting these expressions into (6), we find J k = n k + (1 + b ) n k ( n k + 1) − (1 − b ) n k ( n k + 1) − b ln( n k + 1)( n k + 1) . (12)Next, we make the continuum approximation k → x in (5) by Taylor expanding n k − about x = k to first order. This leads to the equation ∂n∂x = f ( n ) − K (13)where f ( n ) = − (1 + b ) n ( n + 1) + (1 − b ) n ( n + 1) + b ln( n + 1)( n + 1) . (14)The boundary condition is n = 1 which comes from the fact that every time thecondensate hops it leaves one particle behind. In the continuum limit, this boundarycondition becomes n (0) = 1. rder in a moving condensate (a) (b) Figure 5: Using the boundary condition ¯ n ( x = 0) = n = 1 we see graphically that (a)if n c < n then as x increases, as we move further away from the condensate, so too does¯ n . It does so indefinitely, resulting in an infinite mean occupancy infinitely far from thecondensate. (b) If n c > n , then we see that as x increases, ¯ n ( x ) decreases to 0.For the case of the strong condensate, there is no mass at x → ∞ which meansthat K = 0 and therefore that ∂n∂x = f ( n ). The form of f ( n ) is illustrated in figure 5.We note the limits f (0) = 0 and f ( n ) → − b for n → ∞ . We also observe that there isan unstable fixed point at n c , the non-zero root of f (¯ n ), at which d¯ n d x = 0. As illustratedin figure 5, the value of n c relative to the boundary condition n = 1 will iterativelydetermine the values of ¯ n ( x ) at successively larger x .If n c < n (figure 5a) then d¯ n d x > n > n c . This means that ¯ n willincrease indefinitely as x increases, resulting in an infinite mean occupancy far fromthe condensate. This contradicts our assumption that there is no mass as x → ∞ andtherefore we discard this solution as unphysical.If n c > n (figure 5b) then the gradient of ¯ n is negative, and it remains negativeup to the stable fixed point at ¯ n = 0. This means that successively further from thecondensate ¯ n decreases continuously to 0. This is the physical solution. The consistencycondition for this solution gives us a condition for the existence of the strong condensate: n c > n = 1. This can be translated into a condition on b by using the fact that f ( n c ) = 0. Substituting n = n = 1 in (14) and setting the resulting expression to zero,we find an equation for the critical value b = b c such that we have a strong condensatefor b > b c . The result is b c = 13 − (cid:39) . . (15)This prediction for b c agrees fairly well with our numerical results displayed infigure 6 and figure 7. In that figure the sample variance σ = (cid:80) i ( n i /N − ρ ) of theoccupancy per particle n/N is plotted against b and shown to increase sharply from σ (cid:39) σ (cid:39) b (cid:39) .
5. This corresponds to the transition from thefluid, in which the mass is evenly distributed and the sample variance is small, to thestrong condensate phase in which the sample variance approaches 1. This transition rder in a moving condensate ρ and sharpens as the system size increases.
5. Classification of the phase transition
Figure 6: Plots of the order parameter σ against the rate parameter b , for various systemsizes L with density ρ = 0 .
5. A transition is seen to occur for all system sizes, with aclear crossover point in the σ - b curves at b = 0 .
5, which is indicative of the critical value b c = 0 . b c and the nature of the transition we analysedata from different system sizes L at the same density ρ = 0 .
5. First, by studying a plotof the order parameter σ against b for this data in figure 6, we see the transition occursover a similar range of b for all system sizes L . Furthermore, the transition sharpens withincreasing system size and there is a stable intersection of the curves at b = 0 .
5. Thisstrongly suggests that in the thermodynamic limit the transition would be discontinuousin σ at b c = 0 .
5. Our confidence in our measurement of b c is reinforced by the resultof applying a finite size scaling procedure to the same data, as shown in figure 7. Weplot the order parameter σ against a rescaled hop rate parameter b (cid:48) = L X ( b − b c ) and,through our choice of b c and X , find the best data collapse when b c = 0 .
5, and thescaling exponent X = 0 . k behind the condensate(figure 8) we see that there is a decay length associated with the average shape of thetail of mass behind, which increases as b (cid:38) .
5. To better understand how this decaylength changes as the b approaches the (numerical) critical value b c from above, we fit anexponential distribution to the tail at different values of b , and then plot the dependenceof the fitted decay length λ on b . rder in a moving condensate σ against the rate parameter b , for varioussystem sizes L with density ρ = 0 .
5. (b) We perform a finite size scaling procedure on b , rescaling it to ( b − b c ) L X . The choice of parameters for the best collapse of the dataonto a single curve is b c = 0 . X = 0 . k behind the condensate site, measurednumerically. The decay length in the region 0 < k <
500 can be seen to slowly increaseas b (cid:38) . λ appears to diverge like a power law as b (cid:38) . b − . rder in a moving condensate λ ( b ) measured from the tails in figure 8 fits a power lawin the scaling regime but ‘flattens out’ as b (cid:38) .
5. Numerically, the system size onlyaffects the b dependence of λ as b approaches 0 .
5. As the system size is increased the λ ( b ) at b (cid:38) . b (cid:38) . < k < mixed order or hybrid phase transition. Transitions of thisnature are not unprecedented, for example, in [34, 35] mixed order transitions in longrange lattice models are considered. In another example a phase transition in the sizeof the giant viable cluster of a multiplex network, is shown to be discontinuous in theorder parameter but also to exhibit critical behaviour above the critical point [36].This asymmetry is attributed to the specifics of the dynamics, which only provide amechanism for critical behaviour above the critical point and not below.We speculate that the mixed order transition in the present work may also beattributed to the difference in mechanisms above and below the transition: from above,the transition is brought about by the divergence of a length scale in a coherent structure,namely the tail; from below we see no coherent structures until the condensate itself isformed.Another interesting observation is that the critical exponent here is measured to rder in a moving condensate .
7, which is similar to the value 1.7338 of the numerically measuredcritical exponent associated with the temporal correlation length in directed percolation(DP) [37]. Similarities can be seen between the dynamics of the mass in the tail inthe frame of reference of the condensate and the dynamics of the driven asymmetriccontact process (DACP) which exhibits a phase transition in the DP universality class,specifically with the temporal exponent measured at 1.7(2) [38]. Here, the backchipprocess generates new singly-occupied sites which, in the language of the DACP, become“inactive” by recombining with other occupied sites, over timescales which are shortcompared to the timescales at which mass moves from greater-than-singly occupiedsites. It may be of interest to investigate such potential similarities in greater depth inthe future.
To see whether our approximate theory also captures the existence of a mixed ordertransition we numerically integrated (14) in order to measure how a length scale inthe profile of the solution for n ( x ) changes as b (cid:38) b c . In contrast to the results fromsimulation (figure 9), we find (figure 10) that according to the theory the length scale λ ( b ) of the tail diverges as λ ( b ) ∼ log (cid:18) b − b c (cid:19) . (16)The divergence is still indicative of a mixed-order transition, but it points to one whichhas a weakly diverging length scale.This can also be seen in the approximate theory by studying the mass in the tailvery close to the condensate. By considering (cid:15) = b − b c (cid:28) η = n − ¯ n = 1 − ¯ n (cid:28) f (¯ n, b ) given in (14), to find f (1 − η, b c + (cid:15) ) = f (1 , b c ) − A(cid:15) + Bη + O ( η , (cid:15) , (cid:15)η ) , (17)where A = 14 b c , B = 1 − b c ln 24 , and f (1 , b c ) = 0 . (18)Using the relationship f (¯ n, b ) = d¯ n d x we can then writed η d x = A(cid:15) + Bη (19)and integrate to find η ( x ) = AB (cid:15) (e Bx − . (20)A characteristic length scale λ can then be defined by the value of x at wich η reachessome arbitrary finite value, yielding AB (cid:15) e Bλ = constant . Thus λ ( (cid:15) ) = | ln (cid:15) | B + constant (21)So we see that as (cid:15) (cid:38) BηA , the characteristic length scale λ diverges slowly as a logarithm. rder in a moving condensate b (cid:38) b c . The length scale is defined as the distance from x = 0to the position behind the condensate at which the absolute value of d n/ d x is largest,which was the most distinct and well defined feature of n ( x ).
6. Condensation transition for the generic hop rate
The transition to a strong condensate phase is found for the generic hop rate (1) for all α >
0. We can repeat the α = 1 calculation we performed previously but with α > s ( z ) = (cid:80) ∞ n =1 z n n s to find J k = n k − b ) n k ( n k + 1) + 1( n k + 1)+ b ( n k + 1) (cid:20) Li α − (cid:18) n k n k + 1 (cid:19) − Li α (cid:18) n k n k + 1 (cid:19)(cid:21) . (22)By making the same continuum approximation as before (13) we find f ( n ) = 1 − (2 + b )( n + 1) + (1 + b )( n + 1) − b ( n + 1) (cid:20) Li α − (cid:18) nn + 1 (cid:19) − Li α (cid:18) nn + 1 (cid:19)(cid:21) . (23)Finally, using the boundary condition n (0) = 1 and the constraint on the stable fixedpoint n c ( b, α ), we obtain b c ( α ) = (cid:20) (cid:18) Li α − (cid:18) (cid:19) − Li α (cid:18) (cid:19)(cid:19)(cid:21) − . (24)This function (figure 11) is monotonically increasing from b c (0) = and is boundedfrom above by 1. rder in a moving condensate Α b c (cid:72) Α (cid:76) Figure 11: A plot of b c ( α ) (red, solid). b c increases monotonically from b c (0) = 1 / b c = 1 (blue, dashed).It is important to note that (24) holds for α > α = 0 is singular,because for α > α = 0 all masses,including any condensate, move with rate 1 + b . Thus at α = 0 our mechanism for themaintenance of a moving condensate is no longer valid, as there is no reason small masseswould tend to catch up to large masses ahead. This is confirmed by our simulationresults, shown in figure 12a, for α = 0. Measuring the variance of the mass distributionwe see no evidence of a condensate forming above a certain value of b .On the other hand, we can probe the validity of (24) as α (cid:38) u log ( n ) = 1 + b ln( n + 1) . (25)As ln n increases more slowly than any power of n we can consider (25) as approximatingthe limit of an arbitrarily small, positive choice of α . The results from our simulationswith u log ( n ) presented in figure 12b show that there is a transition to the strongcondensate phase in the region b ∼ . − .
4, which gives us yet more confidence inour analytic result (24).We have also performed simulations at α values larger than 1 (figure 13). We findthat the transition becomes more gradual for larger values of α , and occurs over a regionof values of b which are larger than the value of b c predicted by (24). By studying varioussystem sizes (figure 13c) we see that evidence that the gradual nature of the transition isa finite size effect, as it becomes more sharp when we increase the system size. The hoprate u (1) = 1 + b for all α , but for large α the hop rate from sites with low occupancies(greater than 1) is reduced significantly. This has the effect of suppressing all singleoccupancy sites because single units of mass always catch up with a site ahead of massgreater than one, in the same way that the condensate is maintained. We note thatalthough our prediction for b c seems to agree well with simulations for α = 0 and α = 1, rder in a moving condensate (a) (b) Figure 12: Plots of the variance σ of the occupancy distribution against the rateparameter b , for systems of size L = 1000. (a) With α = 0 all masses move with thesame rate. Thus there is no mechanism for the formation of the strong condensate, andno transition is observed in b . (b) Using the modified hop rate u log ( n ) given in (25) wecan probe b c ( α ) as α (cid:38)
0. A transition occurs when b is in between 0 . .
4, in goodagreement with the prediction b c ( α = 0) = 1 / α , b c appears to overshoot the asymptote b c = 1 predicted by the approximatetheory. (a) (b) (c) Figure 13: Plots of the variance σ of the occupancy distribution against the rateparameter b . (a) α = 2, L = 500. (b) α = 10, L = 1000. The formula (24) predicts that b c ( α = 2) = 0 . b c ( α = 10) = 0 . α is increased, and takes place over a range of values of b which are greaterthan the b c predicted by (24). (c) α = 10. By increasing the system size the transitionbecomes sharper, which is evidence that its gradual nature is a finite size effect. We canestimate b c ∼ . ± . rder in a moving condensate L . This means that the critical density ρ c has L dependence of the form ρ c ( L ) ∼ ln( L ).
7. Subcritical region
Our numerical data and analytical work has shown the existence of a strong condensatephase when b > b c at all densities. When b < b c we find a transition from a homogeneousfluid phase when ρ (cid:46) ρ (cid:38)
1. Althoughthis phase quantitatively and qualitatively looks like a standard condensate, numericalanalysis shows that the transition density ρ c diverges as ln( L ), in a similar way to thatobserved for biased hopping rates in [26]. This can be seen from figure 14 where wehave analysed the size of the position of the peak, n peak , in the distribution of thestandard condensate phase. If the fluid contains on average N c = Lρ c ( L ) particles, then n peak ( L ) (cid:39) ( ρ − ρ c ( L )) L/N = (1 − ρ c ( L ) /ρ ). Our measurements of the value of n peak ( L )(figure 14) show that n peak ( L ) ∝ − ln( L ), and thus that ρ c ∝ ln( L ).Putting this finding together with the results of the previous sections, we are finallyable to sketch a phase diagram for the model in the ρ − b plane. This is shown in figure 15.
8. Conclusion
In conclusion, the hopping dynamics invoking ‘backchip’ processes that we have studiedin this work give rise to a strong condensate phase in which the condensate and its shorttail of trailing particles move together through the system. This phase is present at alldensities ρ when the parameter b in the hopping rate u ( n ) = 1 + b/n α is greater thana critical value b c , which appears to be independent of the system size. Numerically wehave measured b c = 0 . α = 1. This is in fairly good agreement with the rder in a moving condensate ρb ρ (L) c b c STRONG CONDENSATEFLUID STANDARD CONDENSATE
Figure 15: A sketch of the phase diagram for our system. Above the value b c , the systemexhibits a strong condensate phase. Below b c and for low densities the system existsin a fluid phase with a homogeneous distribution of mass throughout. Above a certainvalue ρ c , the system exhibits the characteristics of a standard condensate phase. Notehowever that this critical density diverges as L → ∞ .value b c (cid:39) .
62 found using the condensate frame analysis of section 4. We classify thetransition as being mixed order as it exhibits a discontinuity in the order parameter σ ,which is indicative of a 1st order phase transition, as well as a diverging length scale,in this case the decay length of the tail of the condensate, which is a characteristic of a2nd order transition.Our results also show a number of additional interesting features. First, thecondensate and its tail comprise a coherent object that moves throughout the system,and the stability of the condensate lies in the dynamics of the vanishingly small fractionof particles in the tail. Once a few particles are left behind through backchippingthey quickly rejoin the condensate. This picture is substantiated by the theory ofsection 4 which demonstrates that the tail of a moving strong condensate necessarilydecays quickly to zero for b greater than a critical value b c . Second, by extending ouranalysis to values of α (cid:54) = 1 in (1), we find that the strong condensation phenomenon isgenerically present for any α >
0. As illustrated in figure 11 the function b c ( α ) increasesmonotonically from b c (0) = 1 / α → ∞ . We recallthat in the standard ZRP, condensation is present only for α <
1. Simulation resultsconfirm the existence of the strong condensate for α >
1, although the approximatetheory appears to underestimate the transition point.Below the critical b c , we see behaviour more reminiscent of the standard ZRP, inwhich there is an apparent transition from a fluid phase at low density, to a standard rder in a moving condensate ρ . However we see numerically that thiscritical value ρ c ∼ ln( L ) as L → ∞ , in a similar way to that observed in [26]. Here, thecondensate is not a true feature of the system in the large L limit, but rather a finitesize effect. This suggests that systems in which aggregates diffuse and chip, the mostrelevant quantity in determining whether condensation occurs is the rate of decay of thechip rate with the aggregate size.To understand the interaction between these dynamical processes better, it wouldbe interesting to study a generalisation of this model where n − a particles hop in unisonfor n > a and a single particle hops for n ≤ a . We have shown here that in the case a = 1a strong condensate forms and travels through the system, with its structure maintainedby the effects of hops from its tail. On the other hand the case a = N yields the zerorange process, where only one particle may hop at a time and a condensation occurs forsufficiently large choice of b , but with a static condensate. It would be interesting thento ask how a should scale with N for one to observe a moving, as opposed to static,condensate and at what speed it would travel. Acknowledgments
JW acknowledges financial support from EPSRC (UK) via the SUPA CM DTC. Thiswork was partially supported by the EPSRC under grant EP/J007404/1.
References [1] Burda Z, Johnston D, Jurkiewicz J, Kami´nski M, Nowak M A, Papp G and Zahed I 2002
Phys.Rev. E (2) 026102 URL http://link.aps.org/doi/10.1103/PhysRevE.65.026102 [2] O’Loan O J, Evans M R and Cates M E 1998 Phys. Rev. E (2) 1404–1418 URL http://link.aps.org/doi/10.1103/PhysRevE.58.1404 [3] Chowdhury D, Santen L and Schadschneider A 2000 Physics Reports
199 – 329 ISSN 0370-1573URL [4] Levine E, Ziv G, Gray L and Mukamel D 2004
Journal of Statistical Physics http://dx.doi.org/10.1007/s10955-004-5706-6 [5] Kaupuˇzs J, Mahnke R and Harris R J 2005
Phys. Rev. E (5) 056125 URL http://link.aps.org/doi/10.1103/PhysRevE.72.056125 [6] Krapivsky P L, Redner S and Leyvraz F 2000 Phys. Rev. Lett. (21) 4629–4632 URL http://link.aps.org/doi/10.1103/PhysRevLett.85.4629 [7] Angel A G, Evans M R, Levine E and Mukamel D 2005 Phys. Rev. E (4) 046132 URL http://link.aps.org/doi/10.1103/PhysRevE.72.046132 [8] Evans M R 2000 Brazilian Journal of Physics
42 – 57 ISSN 0103-9733[9] Evans M R and Hanney T 2005
Journal of Physics A: Mathematical and General R195 URL http://stacks.iop.org/0305-4470/38/i=19/a=R01 [10] Evans M R, Majumdar S N and Zia R K P 2004
Journal of Physics A: Mathematical and General L275 URL http://stacks.iop.org/0305-4470/37/i=25/a=L02 [11] Majumdar S N, Evans M R and Zia R K P 2005
Phys. Rev. Lett. (18) 180601 URL http://link.aps.org/doi/10.1103/PhysRevLett.94.180601 [12] Evans M R, Majumdar S N and Zia R K P 2006 Journal of Statistical Physics http://dx.doi.org/10.1007/s10955-006-9046-6 rder in a moving condensate [13] Jeon I, March P and Pittel B 2000 The Annals of Probability pp. 1162–1194 ISSN 00911798URL [14] Jeon I 2010 Journal of Physics A: Mathematical and Theoretical http://stacks.iop.org/1751-8121/43/i=23/a=235002 [15] Schwarzkopf Y, Evans M R and Mukamel D 2008 Journal of Physics A: Mathematical andTheoretical http://stacks.iop.org/1751-8121/41/i=20/a=205001 [16] Thompson A G, Tailleur J, Cates M E and Blythe R A 2010 Journal of Statistical Mechanics:Theory and Experiment
P02013 URL http://stacks.iop.org/1742-5468/2010/i=02/a=P02013 [17] Concannon R J and Blythe R A 2014
Phys. Rev. Lett. (5) 050603 URL http://link.aps.org/doi/10.1103/PhysRevLett.112.050603 [18] Godr`eche C 2003
Journal of Physics A: Mathematical and General http://stacks.iop.org/0305-4470/36/i=23/a=303 [19] Lighthill M J and Whitham G B 1955 Proceedings of the Royal Society of London.Series A. Mathematical and Physical Sciences
Preprint http://rspa.royalsocietypublishing.org/content/229/1178/317.full.pdf+html ) URL http://rspa.royalsocietypublishing.org/content/229/1178/317.abstract [20] Silk J and White S 1978
The Astrophysical Journal
L59–L62[21] Horvai P, Nazarenko S and Stein T 2008
Journal of Statistical Physics http://dx.doi.org/10.1007/s10955-007-9466-y [22] Family F and Meakin P 1989
Phys. Rev. A (7) 3836–3854 URL http://link.aps.org/doi/10.1103/PhysRevA.40.3836 [23] Majumdar S N, Krishnamurthy S and Barma M 1998 Phys. Rev. Lett. (17) 3691–3694 URL http://link.aps.org/doi/10.1103/PhysRevLett.81.3691 [24] Majumdar S N, Krishnamurthy S and Barma M 2000 Journal of Statistical Physics http://dx.doi.org/10.1023/A%3A1018632005018 [25] Rajesh R and Majumdar S N 2001 Phys. Rev. E (3) 036114 URL http://link.aps.org/doi/10.1103/PhysRevE.63.036114 [26] Rajesh R and Krishnamurthy S 2002 Phys. Rev. E (4) 046132 URL http://link.aps.org/doi/10.1103/PhysRevE.66.046132 [27] Levine E, Mukamel D and Ziv G 2004 Journal of Statistical Mechanics: Theory and Experiment
P05001 URL http://stacks.iop.org/1742-5468/2004/i=05/a=P05001 [28] Hirschberg O, Mukamel D and Sch¨utz G M 2009
Phys. Rev. Lett. (9) 090602 URL http://link.aps.org/doi/10.1103/PhysRevLett.103.090602 [29] Hirschberg O, Mukamel D and Sch¨utz G M 2012
Journal of Statistical Mechanics: Theory andExperiment
P08014 URL http://stacks.iop.org/1742-5468/2012/i=08/a=P08014 [30] Hirschberg O, Mukamel D and Sch¨utz G M 2013
Phys. Rev. E (5) 052116 URL http://link.aps.org/doi/10.1103/PhysRevE.87.052116 [31] Waclaw B and Evans M R 2012 Phys. Rev. Lett. (7) 070601 URL http://link.aps.org/doi/10.1103/PhysRevLett.108.070601 [32] Evans M R and Waclaw B 2014
Journal of Physics A: Mathematical and Theoretical http://stacks.iop.org/1751-8121/47/i=9/a=095001 [33] Toner J, Tu Y and Ramaswamy S 2005 Annals of Physics
170 – 244 ISSN 0003-4916 specialIssue URL [34] Bar A and Mukamel D 2014
Phys. Rev. Lett. (1) 015701 URL http://link.aps.org/doi/10.1103/PhysRevLett.112.015701 [35] Bar A and Mukamel D 2014 arxiv.org ( Preprint ) URL http://arxiv.org/abs/1406.6219 [36] Baxter G J, Dorogovtsev S N, Goltsev A V and Mendes J F F 2012
Phys. Rev. Lett. (24)248701 URL http://link.aps.org/doi/10.1103/PhysRevLett.109.248701 [37] Hinrichsen H 2000
Advances in Physics Preprint rder in a moving condensate com/doi/pdf/10.1080/00018730050198152 ) URL [38] Costa A, Blythe R A and Evans M R 2010 Journal of Statistical Mechanics: Theory and Experiment