Driven k-mers: Correlations in space and time
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Driven k -mers: Correlations in space and time Shamik Gupta, , , Mustansir Barma, Urna Basu, and P. K. Mohanty Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel Laboratoire de Physique de l’ ´Ecole Normale Sup´erieure de Lyon,Universit´e de Lyon, CNRS, 46 All´ee d’Italie, 69364 Lyon c´edex 07, France Theoretical Condensed Matter Physics Division,Saha Institute of Nuclear Physics, Kolkata 700064, India (Dated: November 12, 2018)Steady-state properties of hard objects with exclusion interaction and a driven motion along aone-dimensional periodic lattice are investigated. The process is a generalization of the asymmetricsimple exclusion process (ASEP) to particles of length k , and is called the k -ASEP. Here, we ana-lyze both static and dynamic properties of the k -ASEP. Density correlations are found to displayinteresting features, such as pronounced oscillations in both space and time, as a consequence of theextended length of the particles. At long times, the density autocorrelation decays exponentially intime, except at a special k -dependent density when it decays as a power law. In the limit of large k at a finite density of occupied sites, the appropriately scaled system reduces to a nonequilibriumgeneralization of the Tonks gas describing the motion of hard rods along a continuous line. Thisallows us to obtain in a simple way the known two-particle distribution for the Tonks gas. For largebut finite k , we also obtain the leading-order correction to the Tonks result. PACS numbers: 05.70.Ln, 05.60.Cd, 87.10.Hk
I. INTRODUCTION
The asymmetric simple exclusion process (ASEP), aparadigmatic model of nonequilibrium statistical me-chanics, involves hard-core particles undergoing biaseddiffusion on a lattice in the presence of an external drive[1–5]. The generalization of the ASEP to an exclusionprocess of hard-core extended objects ( k -mers, each ofwhich occupies k consecutive sites) is referred to as the k -ASEP. It was first introduced to model protein synthe-sis inside living cells [6, 7]. During the synthesis, ribo-somes move from codon to codon along messenger RNA,read off genetic information, and generate the proteinstepwise. Modelling the codons by lattice sites and theribosomes by k -mers, we recover the k -ASEP. The spatialextent of the k -mers takes care of the blocking of severalcodons by a single ribosome; steric hindrance, which pre-vents overlap of ribosomes, is modelled by the exclusionconstraint.Earlier studies of the k -ASEP in one dimension in-volved analyzing the steady-state density profile in anopen system [6, 7], the time-dependent conditional prob-abilities of finding the k -mers on specific sites at a giventime [8], the dynamical exponent [9], the phase diagramof the system with open boundaries [10–13], the hydro-dynamic limit governing the evolution of the density [14],and the effects of defect locations on the lattice on steady-state properties [15, 16]. Some aspects of the k = 2 caseof the k -ASEP were studied earlier in the context of amodel of driven, reconstituting dimers [17].Here, we are concerned with the k -ASEP on a one-dimensional (1D) periodic lattice. At long times, the pro-cess settles into a nonequilibrium steady state in whichall configurations with a given number of k -mers have equal weights [11]. In this work, our focus is on correla-tion functions, both static and dynamic.We compute static correlations in two different ways:(i) by counting the number of relevant configurations,and (ii) by mapping the k -ASEP to a zero-range process(ZRP) [18] and then by employing a matrix product for-malism [19]. Dynamic correlations in the k -ASEP arederived by mapping the k -ASEP to an equivalent ASEPwith a smaller number of sites and by using the knowndynamic properties of the latter [17]. We show that den-sity correlations exhibit pronounced oscillations in bothspace and time as a consequence of the extended lengthof the k -mers.One may also consider the k -ASEP in the continuumlimit, i.e., in the joint limit of large k and vanishing latticespacing, δ →
0, while keeping the product a = kδ andthe density of occupied sites fixed and finite. Such a limitwas considered previously to obtain the hydrodynamicbehavior of the continuum system [20]. In this limit, themodel describes hard rods of finite length a , undergoingbiased diffusion along a continuous line in the presenceof an external drive. In the case of unbiased motion, thiscontinuum model was studied earlier by Tonks [21]. Thisso-called Tonks gas has an equilibrium steady state inwhich quantities of physical interest, e.g., the equationof state and the two-particle distribution function, havebeen worked out exactly [21, 22]. The continuum limit ofthe k -ASEP is a nonequilibrium generalization, and maybe called the driven Tonks gas.The known two-particle distribution function of theTonks gas is recovered straightforwardly by taking thecontinuum limit of the k -ASEP, on noting that thesteady-state measure of configurations is the same,whether or not the system is driven. We also obtainthe leading-order correction to the Tonks result when k is large but finite. The Tonks result, after including theleading-order correction in 1 /k , turns out to be a goodapproximation to the k -ASEP equal-time spatial corre-lation even for not too large values of k (for example, k = 13 when the density of occupied sites is 0 . k -ASEP and discuss its steady-state measure. In thefollowing section, we derive closed-form expressions forthe steady-state equal-time engine-engine and density-density correlations. The former function describes corre-lation between the right end (the “engine”) of one k -merwith that of another at the same time instant. Consid-ering the continuum limit of the k -ASEP, we derive theknown two-particle distribution function for the Tonksgas. We also derive the leading finite- k correction to theTonks result. In Sec. IV, we address the steady-statedynamics by computing the k -mer current and kinematicwave velocity associated with transport of density fluc-tuations. We then discuss a mapping of the k -ASEPto an equivalent ASEP, and the behavior of the k -ASEPtemporal density-density correlation, whose scaling prop-erties are derived by utilizing the mapping. In AppendixA, we discuss a different method to obtain static correla-tions in the k -ASEP through a mapping to an equivalentZRP. II. THE k -ASEPA. Definition We consider a number, N , of k -mers that are subject tohard-core exclusion and are distributed on a 1D periodiclattice of L sites. The sites are labeled by the index i = 1 , , . . . , L . Each k -mer occupies k consecutive latticesites. The k -mers are hard objects that cannot break intosmaller fragments. The density of occupied sites in thesystem is given by ρ ≡ N kL . (1)We specify the location of a k -mer on the lattice by thesite index of its rightmost end, and we call this end the“engine” of the k -mer. We denote the occupation of the i th site by n i = 1 or 0, according to whether the site isoccupied or vacant, respectively. A k -mer is then repre-sented by a string of k consecutive 1’s and a configurationof the model by an L -bit binary string composed of 0’sand 1’s.The system of k -mers evolves according to a stochas-tic Markovian dynamics: in a small time dt , a k -meradvances forward (respectively, backward) by one latticesite with probability pdt (respectively, qdt ), provided thesite is unoccupied. The dynamics conserves the totalnumber of k -mers in the system. The elementary dy-namical moves may be represented as(111 . . . pdt ⇄ qdt . . . , (2) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) qp FIG. 1: (Color online) The k -ASEP on a ring, showing theallowed and disallowed dynamical moves of trimers ( k = 3).Here, k -mers are represented as k connected circles. Therightmost end of a k -mer (the “engine”) is indicated by afilled red circle. where the k -mer has been represented by a string of k consecutive 1’s enclosed by brackets. Figure 1 illustratesthe allowed and disallowed moves of trimers ( k = 3).For p = q , the k -mers move preferentially in one di-rection along the lattice, and the system at long timesreaches a nonequilibrium steady state with a steady cur-rent of k -mers.When p = q , the k -mers diffuse symmetrically to theleft and to the right. In this case, the model is a general-ization of the symmetric simple exclusion process (SEP)of hard-core particles to k -mers, and may be referred toas the k -SEP. At long times, the k -SEP settles into anequilibrium steady state. B. Relation to a model of diffusing, reconstitutingdimers
Earlier studies of a variant of the k -ASEP dealt withdimers ( k = 2) that, in general, do not retain their identi-ties and are allowed to reconstitute [17, 23]. Specifically,on a lattice of L sites, single particles or monomers (de-noted by 1) and paired particles or dimers (denoted by11) are distributed with at most one particle per site.The dynamics in a small time dt involves a dimer mov-ing by one lattice site, either forward with probability pdt or backward with probability qdt , without violating thehard-core constraint on site occupancies. The pairing ofthe dimers is impermanent, thereby allowing for recon-stitution. For example, in the sequence of transitions,11010 → → p = q ) [23] and in the asym-metric ( p = q ) [17] case, the phase space of the sys-tem breaks up into an infinite number of dynamicallydisjoint sectors. A non-local construct, called the ir-reducible string (IS), uniquely labels the different sec-tors. The IS for a given configuration is constructed fromthe corresponding L -bit binary string by deleting recur-sively any pair of adjacent 1’s until no further deletionis possible. The model exhibits dynamical diversity withquantities like the density autocorrelation showing strongsector-dependent behaviors that range from power lawsto stretched exponentials.The problem of hard non-reconstituting dimers, i.e.the case k = 2 of the k -mer system considered in thiswork, corresponds to a particular sector of the reconsti-tuting dimer problem, namely, the one with the null IS0000 . . . C. The steady state
In the steady state of the k -ASEP, every microscopicconfiguration C with a given number of k -mers is equallylikely, and hence, occurs with probability 1 / Ω, where Ω isthe total number of configurations [11]. In such a state,for every transition away from C to another configuration C ′ , it is simple to construct a distinct and unique config-uration C ′′ that evolves to C at the same rate, therebyensuring stationarity. This argument holds for periodicboundary conditions and fails in the case of open bound-aries where the exact steady state is hard to obtain andis as yet unknown.Now, Ω is determined by counting the different possibleways of distributing a number, N , of k -mers over a latticeof L sites with periodic boundaries. First consider free boundaries, in which case we have an open chain of L sites on which the k -mers are placed. Then, the numberof possible ways of distributing the k -mers isΩ free N,L = (cid:18) L − N k + NN (cid:19) . (3)For a periodic lattice, the number of ways is determinedby first considering all configurations in which an arbi-trary but fixed site remains occupied by an arbitrarilychosen but fixed engine. The number of such configura-tions is simply Ω free N − ,L − k . However, the engine could bechosen to be one of the N identical engines available andmay be placed over any of the available L sites. Thus,the total number of distinct configurations is [11]Ω = LN Ω free N − ,L − k = LN (cid:18) L − N k + N − N − (cid:19) . (4)The steady-state probability that a randomly chosensite is occupied by an engine is obtained by consideringall possible ways of distributing a number, N −
1, of k -mers over a lattice of L − k sites with free boundaries.Since all configurations of the k -ASEP are equally likelyin the steady state, the desired probability is given by theratio Ω free N − ,L − k / Ω = ρ/k . Similarly, the probability thata randomly chosen site is vacant in the steady state maybe shown to be equal to 1 − ρ . Next, we consider the jointprobability that in the steady state, a randomly chosen site is occupied by an engine while the following site isvacant; this is given by the ratio Ω free N − ,L − ( k +1) / Ω. In thethermodynamic limit, i.e. in the limit N → ∞ , L → ∞ ,while keeping k and ρ fixed and finite, this joint proba-bility equals ρ (1 − ρ ) / [ ρ + k (1 − ρ )].We note that for the k -SEP, the condition of detailedbalance implies that all configurations with a given num-ber of k -mers have equal weights in its equilibrium steadystate. It thus follows that the exclusion process with k -mers has the same steady state, irrespective of whetherthe k -mers have a biased or an unbiased motion. III. STEADY STATE STATICSA. Engine-engine correlation
Let E i denote the engine occupation variable for site i , taking values 1 or 0 according to whether the site isoccupied by an engine or not, respectively. The (un-subtracted) equal-time engine-engine correlation func-tion E k ( r ) is defined as E k ( r ) ≡ h E i E i + r i , (5)where the angular brackets denote averaging with respectto the steady state. Note that E k ( r ) is identically zerofor r < k .Now, E k ( r ) for r ≥ k has non-zero contributions fromall configurations in which there are two engines at the i -th and ( i + r )-th sites, with the gap between the twoengines containing r − k sites that are occupied by anynumber, m , of k -mers between 0 and the maximum num-ber that may be placed in the gap, while the left over N − ( m + 2) number of k -mers is distributed over theremaining L − ( r − k + 2 k ) sites. The maximum numberof k -mers that may be placed over the gap of r − k sites isgiven by ⌊ ( r − k ) /k ⌋ , where ⌊ x ⌋ denotes the floor functionthat gives the largest integer not greater than x . Notingthat in the steady state of the k -ASEP, all configurationshave equal weights, we get E k ( r ) = ⌊ ( r − k ) /k ⌋ X m =0 Ω free m,r − k Ω free N − m − ,L − r − k Ω . (6)In the thermodynamic limit, one can show by usingEq. (3) that, for arbitrary integers m and m ,Ω free N − m ,L − m Ω free N,L = ρ m (1 − ρ ) m − m k , (7)where ρ ≡ ρk (1 − ρ ) + ρ . (8)Later, in Sec. IV, by using a mapping of the k -ASEP tothe usual ASEP, we show that the quantity ρ is in fact C k ( r ) E k ( r ) E k ( r ) C k ( r ) r V k ( r ) k (1 − ρ ) ρ E k ( r + k − FIG. 2: (Color online) The k -ASEP density-density correla-tion C k ( r ) and engine-engine correlation E k ( r ), evaluated nu-merically by using Eqs. (12) and (9) for k = 64 and ρ = 0 . V k ( r ) and [ k (1 − ρ ) /ρ ] E k ( r + k − the particle density in the latter. Using Eq. (7), we findthat in the thermodynamic limit, Eq (6) reduces to E k ( r ) = ρ ρ ( k − × ⌊ ( r − k ) /k ⌋ X m =0 (cid:18) r − k − km + mm (cid:19) ρ m (1 − ρ ) r − k − km ; r ≥ k. (9)For k = 2, the sum in Eq. (9) can be evaluated exactlyto obtain E k ( r ) = ρ (1+ ρ ) [1 − ( − ρ ) r − ] [17]. For k ≥ E k ( r ) may be numerically evaluated by us-ing Eq. (9). Figure 2 shows the result for k = 64. We seethat the engine-engine correlation exhibits damped oscil-lations in space, a hallmark of systems with hard-coreinteractions between the constituents [24], where purelyentropic considerations apply. For example, the fact that E k ( r ) has its first minimum at r = 2 k − r = 2 k may be understood to be due to the possibil-ity that when r = 2 k , there may be configurations withan additional engine between the two engines that areoccupying sites i and i + 2 k . B. Density-density correlation
The (unsubtracted) equal-time density-density corre-lation function C k ( r ) is defined as C k ( r ) ≡ h n i n i + r i . (10)Here, n i is the occupation variable for site i , taking values1 or 0, according to whether the site is occupied or isvacant, respectively. Evidently, n i can be expressed in terms of the engine occupation variable E i as n i = k − X m =0 E i + m . (11)Using the above equation, C k ( r ) may be expressed interms of the equal-time engine-engine correlation as C k ( r ) = k E k ( r ) + k − X m =1 m h E k ( r + k − m ) + E k ( r − k + m ) i , (12)where it is understood that E k ( r ) is zero for r < k .Alternatively, C k ( r ) can be computed in a straightfor-ward way from the equal-time vacancy-vacancy correla-tion, defined as V k ( r ) ≡ h n i n i + r i , (13)where n i = 1 − n i , so that C k ( r ) = V k ( r ) + (2 ρ − . (14)Now, V k ( r ) for r ≥ i -th and ( i + r )-thsites are vacant, with the gap between the two containing r − m , of k -mersbetween 0 and the maximum number that may be placedin the gap, while the left over N − m number of k -mers isdistributed over the remaining L − ( r − V k ( r ) = ⌊ ( r − /k ⌋ X m =0 Ω free m,r − Ω free N − m,L − r − Ω . (15)In the thermodynamic limit, we find that V k ( r ) = (1 − ρ ) ρ ( k − × ⌊ ( r − /k ⌋ X m =0 (cid:18) r − − km + mm (cid:19) ρ m (1 − ρ ) r − − km ; r ≥ . (16)It follows from the definition that V k (0) = 1 − ρ . Oncomparing Eq. (16) with Eq. (9), and noting that (1 − ρ ) /ρ = k (1 − ρ ) /ρ , we get V k ( r ) = k (1 − ρ ) ρ E k ( r + k − r ≥ . (17)Using Eq. (14), we get C k ( r ) = k (1 − ρ ) ρ E k ( r + k −
1) + (2 ρ − r ≥ . (18)For r = 0, we have C (0) = ρ .Figure 2 shows C k ( r ), computed using Eqs. (12) and(9), for k = 64 and ρ = 0 .
75. The oscillations in C k ( r ) f(r/k) = 16= 64= 256 k = 4 r/k a g ( r/k ) k E k ( r ) FIG. 3: (Color online) Scaling approach of the engine-enginecorrelation E k ( r ) to the Tonks limit: k E k ( r ) vs. r/k at fixed ρ = 0 .
75 and a = 400 shows data collapse for large k accordingto Eq. (23). may be related to those in E k ( r ) by using Eq. (18). Forinstance, since E k ( r ) has its first minimum at r = 2 k − C k ( r ) occurs at r = k . The inset of Fig. 2 shows that V k ( r ) and [ k (1 − ρ ) /ρ ] E k ( r + k −
1) for r ≥ k -ASEPthrough a mapping to a zero-range process and employ-ing a matrix product formalism [19]. C. Continuum limit: Driven Tonks gas
In a suitable continuum limit of the k -ASEP, discussedbelow, the model reduces to one of hard rods, which haveexclusion interaction, and which are undergoing driven,diffusive motion along a continuous line. In the absenceof drive, this continuum model was studied by Tonks asa 1D interacting system with an equilibrium steady statein which thermodynamic properties like the equation ofstate can be worked out exactly [21]. When the motionis driven, the continuum limit of the k -ASEP becomesthe driven Tonks gas.A quantity of physical interest for the equilibriumTonks gas is the two-particle distribution P ( R , R ), de-fined such that P ( R , R ) dR dR is the joint probabil-ity of finding the rightmost end of one hard rod be-tween R and R + dR and that of another between R and R + dR . For a translationally invariant system, P ( R , R ) is a function of the separation R ≡ | R − R | . k = 13 a g ( r/k ) + ( ρ/k ) h ( r/k ) k E k ( r ) r/k a g ( r/k ) FIG. 4: (Color online) Comparison of k E k ( r ) with the Tonksresult, a g ( r/k ), for k = 13, a = 400, and ρ = 0 .
75, show-ing the discrepancy between the two for large but finite k .The discrepancy is resolved on including the correction to theTonks result, to leading order in 1 /k , as is shown by a com-parison of k E k ( r ) with the function a g ( r/k ) + ( ρ/k ) h ( r/k )(Eq. (25)). Then, if a is the rod length, it is known that [22] P ( R ) = g ( x ) ,g ( x ) ≡ la ∞ X m =1 A ( x − m ) ( x − m ) m − ( m − l − m × exp (cid:16) − x − ml − (cid:17) , (19)where x = R/a is a reduced distance. Here, l = 1 /aρ T ,where ρ T is the density of rods, and A ( x ) is the unit stepfunction: A ( x ) = (cid:26) x < , x ≥ . (20)We now show that the continuum limit of the k -ASEPengine-engine correlation easily yields Eq. (19). Sucha derivation is justified by the fact that, as discussed inSec. II C, the k -ASEP has the same steady-state measureof configurations for both unbiased and biased motion ofthe k-mers. This fact further implies that Eq. (19) alsoholds for the driven Tonks gas.The continuum limit of the k -ASEP is obtained byconsidering the joint limit k → ∞ , r → ∞ , and thelattice spacing δ →
0, while keeping R = rδ , a = kδ , and ρ fixed and finite [20]. The k -ASEP then describes biasedmotion of hard rods of length a along a continuous line.The density of hard rods is ρ T = ρ/a . In this limit, when ρ = ρδ/ [ a (1 − ρ )] and (1 − ρ ) = exp h − ρδ/ [ a (1 − ρ )] i ,Eq. (9) reduces to E k ( r ) = δ la ⌊ R/a − ⌋ X m =0 [ R/a − ( m + 1)] m m !( l − m +1 × exp (cid:16) − R/a − ( m + 1) l − (cid:17) . (21)Comparing the right-hand side of the last equation withEq. (19), and noting that R/a = r/k , we find that E k ( r ) = δ g (cid:16) rk (cid:17) . (22)Now, since δ = a/k , we find that in the continuum limit,i.e., in the limit k → ∞ , r → ∞ , δ →
0, while keeping a , R , and ρ fixed and finite, E k ( r ) for different k has thescaling form E k ( r ) = a k g (cid:16) rk (cid:17) . (23)Moreover, in the continuum limit, defining P ( R ) = E k ( r ) /δ , we find from Eq. (22) that P ( R ) is preciselyin the form of Eq. (19). We have thus derived the two-particle distribution, valid for both the equilibrium andthe driven Tonks gas, by considering the continuum limitof the k -ASEP.Figure 3 shows plots of k E k ( r ) for different k at fixed ρ and a , evaluated using Eq. (9). As k increases, thecurves show a good scaling collapse, in accordance withEq. (23). For large but finite k , the right-hand sideof Eq. (23) has finite- k corrections, as is suggested bythe discrepancy between k E k ( r ) and a g ( r/k ), shown inFig. 4. The leading-order correction to Eq. (23) will bediscussed in the following subsection.The behavior of the density correlation C k ( r ) in thecontinuum limit may be easily obtained by using Eqs.(18) and (23). We find that in this limit, for different k at fixed ρ and a , we have C k ( r ) = a (1 − ρ ) ρ g (cid:16) rk + 1 (cid:17) + (2 ρ − . (24)Figure 5 shows plots of C k ( r ) for different k at fixed ρ and a , evaluated using Eqs. (9) and (12). As k increases,the curves show a good scaling collapse, in accordancewith Eq. (24). D. Finite- k corrections to Tonks two-particledistribution In order to compute finite- k corrections to Eq. (23),we evaluate the engine-engine correlation for finite r , k , and δ , with r ≫ k ≫
1, and δ ≪
1, keep-ing R = rδ, a = kδ , and ρ fixed and finite. Then,on substituting ρ ≈ ρk (1 − ρ ) − ρ k (1 − ρ ) and (1 − ρ ) ≈ r/k + 1 k = 4= 64= 16= 256 a (1 − ρ ) ρ g ( r/k + 1) + (2 ρ − C k ( r ) FIG. 5: (Color online) Scaling approach of the density-densitycorrelation C k ( r ) to the Tonks limit: C k ( r ) vs. r/k + 1 atfixed ρ = 0 .
75 and a = 400 exhibits data collapse for large k in accordance with Eq. (24). The data are obtained bynumerically evaluating Eqs. (9) and (12). exp (cid:16) − ρk (1 − ρ ) (cid:17) exp (cid:16) ρ k (1 − ρ ) (cid:17) into Eq. (9), and keep-ing terms to leading order in 1 /k , we get k E k ( r ) ≈ a g (cid:16) rk (cid:17) + ρk h (cid:16) rk (cid:17) , (25)where h ( x ) = ∞ X m =1 A ( x − m ) ( x − m ) m +1 ( m − l − m exp (cid:16) − x − ml − (cid:17) × h m ( m − x − m ) − m ( l −
1) + x − m l − i ; x = 1 . (26)Here, A ( x ) is the unit step function defined in Eq. (20).Figure 4 shows that inclusion of the leading-order correc-tion to the Tonks result, as in Eq. (25), indeed resolvesthe discrepancy between k E k ( r ) and a g ( r/k ). IV. STEADY STATE DYNAMICSA. Current and kinematic wave velocity
In discussing the current in the system, we need todistinguish between that associated with the motion ofengines, and that with the k -mers. Contributions to theengine current across a bond ( i, i +1) arise when either (i)the i -th site is occupied by an engine, while the ( i + 1)-thsite is vacant, or, (ii) the ( i + 1)-th site is occupied by anengine, while the ( i − k +1)-th site is vacant. On using theresults of Sec. II C, we find that in the thermodynamiclimit, the average engine current in the steady state isgiven by [11] J e = ( p − q ) ρ (1 − ρ ) ρ + k (1 − ρ ) . (27)To compute the k -mer current, J , note that associatedwith the motion of the engine to an adjacent site is thesliding of the corresponding k -mer across ( k −
1) bonds, sothat J = kJ e . It can be checked that J has a maximumat the density ρ c = √ k/ ( √ k + 1).The kinematic wave velocity v K ≡ ∂J/∂ρ accounts forthe transport of density fluctuations through the systemin the steady state [25]. We find v K = k ( p − q ) (cid:20) ( k − ρ + k (1 − ρ )[ ρ + k (1 − ρ )] (cid:21) . (28)Evidently, v K vanishes if the density is ρ c . As we dis-cuss below, v K plays an important role in determiningthe form of the temporal decay of the density autocorre-lation. B. Mapping to the ASEP: Wheeling velocity (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
49 25 2549 2549 49 25235678 10 11 235678 10 1 235678 10 1 235678 10 134 634 6 134 6 134 6 1 k -ASEP configuration ASEP configuration FIG. 6: (Color online) Mapping of a typical k -mer configura-tion to an ASEP configuration and their subsequent evolutionin time. The first site of the ASEP lattice may be defined inmore than one way; the figure illustrates one possibility. We now discuss a mapping of the k -ASEP with a num-ber, N , of k -mers on a 1D periodic lattice of L sites to anASEP of N hard-core particles on a 1D periodic lattice of L ′ = L − N ( k −
1) sites. We show that as a result of themapping, a fixed site in the k -ASEP corresponds to an ASEP site that moves around the ASEP ring with a finitemean velocity. This phenomenon is known as wheeling,and the mean velocity is called the wheeling velocity W [17]. This velocity plays an important role in the scalingproperties of the temporal density-density correlation ofthe k -ASEP, as we discuss in the next subsection.The mapping involves representing each k -mer by ahard-core particle that corresponds to the engine of the k -mer, as illustrated in Fig. 6. In this way, every k -ASEPconfiguration is mapped to a unique configuration in theASEP. Associated with the motion of a k -mer is that ofthe corresponding ASEP particle according to the ASEPdynamics. We now see that the quantity ρ in Eq. (8) isthe particle density in the equivalent ASEP.It can be seen from Fig. 6 that in the process of map-ping, the ASEP image of a fixed k -mer site moves aroundthe ASEP lattice as a result of the k -mer motion. For ex-ample, consider the transition (111 . . . → . . . k -ASEP sites containing the 0 and the leftmost 1 inthe string (111 . . . . . . → (111 . . . k -ASEPsites containing the 0 and the rightmost 1 in the string0(111 . . .
1) do not change, while the images of those con-taining the remaining 1’s in the string decrease by oneunit. This motion of an ASEP site corresponding to afixed site in the k -ASEP is the phenomenon of wheeling.As a result, the displacement of the ASEP image of afixed k -ASEP site in time t is given by∆ r ( t ) = W t + φ ( t ) , (29)where φ ( t ) is a random variable with zero mean, arisingfrom the stochasticity in the dynamics. Referring to theresults on joint occupation probabilities in Sec. II C, wefind that the wheeling velocity is given by W = ( p − q )( k − ρ (1 − ρ ) ρ + k (1 − ρ ) . (30) C. The temporal density-density correlation
The temporal density-density correlation function inthe steady state of the k -ASEP is defined as C k ( r = | i − j | , t ) ≡ h n i (0) n j ( t ) i − ρ , (31)where n i ( t ) denotes the occupation index of site i at time t . In particular, the density autocorrelation is given by C k ( t ) ≡ C k (0 , t ).We study C k ( t ) by performing Monte Carlo simulationsof the k -ASEP in the steady state. Figure 7 shows that C k ( t ) oscillates in time. The inset shows that for large k ,the autocorrelation for different k at a fixed density ρ isinitially a function of t/k . This behavior holds up to a k -dependent time. To understand the dependence of C k ( t ) -0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 200 400 600 800 1000k=60 -0.04 0 0.04 0.08 0.12 0.16 0 3 6k=28=60=124=252=508 t/kk = 60 ρ = 0 . Time t C k ( t ) C k ( t ) = 508= 252 k = 28= 60= 124 FIG. 7: (Color online) Short-time behavior of the k -ASEPdensity autocorrelation C k ( t ) for k = 60 and ρ = 0 .
8. Thedata are obtained from Monte Carlo simulations. The insetshows C k ( t ) vs. t/k at fixed ρ = 0 .
8, illustrating scaling forlarge k . on the ratio t/k , we note that at short times, the relevanttime scale is set by the time τ ( k ) that an occupied sitetakes to fall vacant. This is consistent with C k ( t ) beinga function of t/τ ( k ). An upper bound on τ ( k ) may beestimated as the time τ e ( k ) ≈ k/v e , the time that anengine takes to move by its own length. Here v e is thevelocity of an engine, which may be obtained from Eq.(27) as v e = k ( p − q )(1 − ρ ) ρ + k (1 − ρ ) . In the limit of large k , onefinds that τ e ( k ) ≈ k , so that C k ( t ) is a function of t/k , asobserved.We now discuss the behavior of C k ( t ) at long times.To proceed, we examine the function C k ( r, t ) for whichan earlier study in the case k = 2 has illustrated thatin the limit of long times and large distances, it assumesa particular scaling form [17]. To obtain the scaling forgeneral k , we utilize the mapping to the ASEP discussedin Sec. IV B and invoke known scaling properties of thedensity correlation in the latter.In the ASEP ( k = 1), the temporal density-densitycorrelation function, C ( r = | i − j | , t ) ≡ h n i (0) n j ( t ) i − ρ ,in the scaling limit follows the form [26] C ( r, t ) ∝ t − / F ( u ); u = 12 ( J t ) − / ( r − v K t ) . (32)Here, J and v K are, respectively, the steady-state cur-rent and the kinematic wave velocity in the ASEP, givenby J = ( p − q ) ρ (1 − ρ ), and v K = ( p − q )(1 − ρ ) [2]. Inthe limit of large u , it is known that F ( u ) ∼ exp( − µ | u | ),with µ ≃ − .
295 [26].It is evident from the k -ASEP to ASEP mapping dis-cussed above that at long times, neglecting the stochas-tic part φ ( t ) in the displacement of a mapped site in theASEP, the correlation C k ( r, t ) has a behavior similar to C ( r + W t, t ) [17]. Thus, for a fixed k , in the limit of long C k ( t ) = 1 / t = 0 . k = 4 ρ = 0 . FIG. 8: (Color online) Long-time decay of the k -ASEP densityautocorrelation C k ( t ), as a power law in time at the compen-sating density ρ c , and as an exponential at other densities.The data are obtained from Monte Carlo simulations. Here k = 4, so that ρ c = 1 /
3. The black line has the slope of − / times and large distances, C k ( r, t ) follows the scaling form C k ( r, t ) ∝ t − / F ( u ′ ); u ′ = 12 ( J t ) − / (cid:16) r +( W − v K ) t (cid:17) . (33)The autocorrelation C k ( t ) behaves asymptotically as C k ( t ) ∝ t − / e − κt , (34)where κ is a constant determined by the difference( W − v K ). Thus, at long times, C k ( t ) decays as an ex-ponential in time, unless the density ρ is such that thedifference vanishes. In this case, the autocorrelation atlate times decays in time as a power law: C k ( t ) ∼ t − / .The corresponding ASEP density is called the compen-sating density ρ c [17], and satisfies ρ c ( k −
1) + 2 ρ c − . (35)Solving for the positive root, we get ρ c = 1 √ k + 1 , (36)which matches with the result for k = 2 derived in [17].Note that corresponding to ρ is the k -ASEP density ofoccupied sites ρ c , mentioned in Sec. IV A, at which the k -mer current J is maximized and the kinematic wavevelocity v K is zero.Figure 8 shows C k ( t ) as a function of time for threevalues of the ASEP density, namely, the compensatingdensity ρ c , and two other values on either side. We seean asymptotic t − / decay of the autocorrelation at thecompensating density and an exponential decay at otherdensities, in accordance with our analysis above. (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) ............ .... k -ASEPZRP FIG. 9: (Color online) Mapping of the k -ASEP to the ZRP.The k -ASEP vacancies are considered as sites in the ZRP,while an uninterrupted sequence of k -mers in front of a va-cancy is regarded as a set of particles occupying the corre-sponding ZRP site, such that the number of particles equalsthe number of k -mers in the sequence. V. ACKNOWLEDGEMENT
Part of this work is based on the Ph. D. thesis of SGat the Tata Institute of Fundamental Research, Mumbai.He acknowledges support of the Israel Science Founda-tion (ISF), and the French contract ANR-10-CEXC-010-01. We thank G. M. Sch¨utz and R. K. P. Zia for dis-cussions and for pointing out Ref. [20] and Ref. [16],respectively, to us.
Appendix A: Mapping of the k -ASEP to the ZRP In this appendix, we discuss a mapping of the k -ASEPto a zero-range process (ZRP) and show how static corre-lations in the former are obtained by using the mapping.This method of obtaining the correlations is an alterna-tive to the one discussed in Sec. III.In the ZRP, an unrestricted number of particles resideson lattice sites and hops between sites with a rate thatdepends only on the number of particles on the departuresite [18]. Each k -ASEP configuration can be mapped toa unique ZRP configuration in the following way: oneconsiders vacancies (0’s) in the k -ASEP as sites in theZRP and an uninterrupted sequence of k -mers followinga vacancy as particles residing on the corresponding ZRPsite, with the number of particles equal to the number of k -mers in the sequence. The ZRP has N particles and M ≡ L − N k sites labeled by the index i . A generic ZRPconfiguration is the set { m i } ≡ ( m , m . . . m M ), where m i is the number of particles on the i -th site (see Fig.9). The motion of a k -mer in the k -ASEP translates tohopping of a particle from a ZRP site to its right or leftneighbor with rates p or q , respectively.For an arbitrary hop rate, the ZRP has a product mea-sure stationary state [18]. In our case, where the rates donot depend on the number of particles on the departuresite, the steady-state weight of any configuration { m i } is P ( { m i } ) = Y i f ( m i ) δ (cid:16) N − M X i =1 m i (cid:17) , f ( m ) = 1 . (A1) The delta function stands for overall particle conserva-tion.The steady state weight of any k -ASEP configuration isobtained by mapping it to a configuration in the equiva-lent ZRP and computing its weight by utilizing Eq. (A1).In our case, since f ( m ) = 1 for all m , all k -ASEP config-urations with a given number, N , of k -mers are equallylikely. Now, using the formalism discussed in [19], onemay rewrite the steady-state weight of any k -ASEP con-figuration ( n , n , . . . , n L ) in a matrix product form byreplacing each occupation number n i by either a matrix D or a matrix E depending on whether n i is 1 or 0,respectively. From the correspondence between the k -ASEP and the ZRP, we have P ( { m i } ) = Tr[ ED km . . . ED km M ] δ (cid:16) N − M X i =1 m i (cid:17) , (A2)where Tr denotes the usual matrix trace operation.Without loss of generality, one may take E = | α ih β | ,where the vectors | α i and h β | are to be determined. Thischoice of E together with Eqs. (A2) and (A1) demandthat for any positive integer m , the matrix D satisfies h β | D mk | α i = f ( m ) = 1 . (A3)Also, h β | D j | α i = 0 for positive integers j which are notmultiples of k. A simple k -dimensional representation ofmatrices E and D is when they have non-zero elements E = 1 and D k = 1 = D i,i +1 , that is, E = | α ih β | , with | α i = | i , h β | = h | ,D = k − X i =1 | i ih i + 1 | + | k ih | . (A4)Here, the set {| i i} represents the standard basis vectors in k -dimensions. This choice ensures that the weight of anyconfiguration with one or more blocks of l particles is zeroif l is not an integral multiple of k ; all other configurationsare equally probable.Let us mention that the matrix formulation discussedabove is different from the Matrix Product Ansatz(MPA) of Derrida et al. [27], in which matrices satisfyspecific algebraic relations dictated by the system dy-namics. For models with ZRP correspondence, the ma-trices generically satisfy Eq. (A3), and therefore, dependonly on the weights f ( m ) . It is always possible to get onerepresentation of these matrices, whereas finding explicitrepresentation of the MPA matrices is non-trivial.
Partition function Z L ( z ) : The first task in comput-ing k -ASEP static correlations is to find the partitionfunction of the system, which is conveniently done inthe grand canonical ensemble by associating the fugac-ity z with any occurrence of the matrix D . This gives Z L ( z ) = Tr[ C L ], where C = zD + E . The configurationwith no vacant site is not dynamically accessible. Thus, Z L ( z ) is given by weights of all configurations with at0least one vacant site: Z L ( z ) = L X n =1 Tr (cid:2) ( zD ) n − EC L − n (cid:3) = L X n =1 h β | C L − n ( zD ) n − | α i . (A5)To proceed, we use the following generating function: Z ( z, γ ) = ∞ X L =1 γ L Z L ( z ) = h β | γ I − γC I − γzD | α i = γ [1 + ( k − γz ) k ][1 − γ − ( γz ) k ][1 − ( γz ) k ] , (A6)where I is the k -dimensional identity matrix. Note that Z ( z, γ ) may be interpreted as the partition function inthe variable length ensemble. The parameters z and γ together determine macroscopic observables like the den-sity of occupied sites and the average system size. Thedensity of occupied sites is ρ = 1 − h ¯ n i i , where h ¯ n i i = γ Z h β | I − γC | α i = 1 − ( γz ) k k − γz ) k . (A7)The average system size is given by h L i = γ Z ∂ Z ∂γ . One may check that h L i has the radius of convergence z ∗ = 1 γ (1 − γ ) /k . (A8)Thus, the thermodynamic limit h L i → ∞ is achieved at z = z ∗ , when every observable of the system becomes afunction of γ only . For example, ρ is obtained from Eq.(A7) as ρ = k (1 − γ ) γ + k (1 − γ ) , which may be inverted to obtain γ = k (1 − ρ ) ρ + k (1 − ρ ) = 1 − ρ . (A9) Equal-time correlations V k ( r ) , E k ( r ) : One may com-pute V k ( r ) by writing it in terms of matrices as follows: V k ( r ) = γ r +1 Z h β | C r − | α ih β | I − γC | α i = γ r (1 − ρ ) h β | C r − | α i . (A10)Now, for any integer j , one may check that h β | C j | α i = ⌊ j/k ⌋ X m =0 (cid:18) j − mk + mm (cid:19) z mk , (A11)which results in V k ( r ) = γ r (1 − ρ ) ⌊ ( r − /k ⌋ X m =0 Ω free m,r − z mk . The equal-time engine-engine correlation E k ( r ) may besimilarly calculated by using the matrix formulation: E k ( r ) = γ r + k Z h β | I − γC | α i ⌊ ( r − k ) /k ⌋ X m =0 Ω free m,r − k z ( m +2) k = (1 − ρ ) γ r + k − ⌊ ( r − k ) /k ⌋ X m =0 Ω free m,r − k z ( m +2) k . (A12)In the thermodynamic limit, using Eq. (A8), we get V k ( r ) = γ r +1 γ + k (1 − γ ) ⌊ ( r − /k ⌋ X m =0 Ω free m,r − (cid:18) − γγ k (cid:19) m , E k ( r ) = γ r + k (1 − γ ) γ + k (1 − γ ) ⌊ ( r − k ) /k ⌋ X m =0 Ω free m,r − k (cid:18) − γγ k (cid:19) m . Replacing γ by 1 − ρ in the expressions on the rightreduce them to those in Eqs. (16) and (9), respectively. [1] T. M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985).[2] G. M. Sch¨utz, in
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