Smith, DA

Wilson, R E

English

The random shuffle update method for the asymmetric exclusion process (ASEP) is introduced and the cluster dynamics technique is extended in order to analyse its dynamics. A sequence of approximate models is introduced, the first element of which corresponds to the classical parallel update rule whose two-cluster dynamics is reviewed. It is then shown how the argument may be extended inductively to solve for the two-cluster probabilities for each element of the sequence of approximate models. A formal limit is then taken, and macroscopic velocities and flow rates are derived

Explore Bristol Research

                          Smith, D. A., & Wilson, R. E. (2005). Extension of cluster dynamics tocellular automata with shuffle update.Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Extension of Cluster Dynamics to CellularAutomata with Shuﬄe UpdateDavid A. Smith1 and R. Eddie Wilson1Bristol Centre for Applied Nonlinear Mathematics, Department of EngineeringMathematics, University of Bristol, Queen’s Building, University Walk, Bristol BS81TR, United KingdomAbstract. The random shuﬄe update method for the asymmetric exclusion process(ASEP) is introduced and the cluster dynamics technique is extended in order toanalyse its dynamics. A sequence of approximate models is introduced, the first elementof which corresponds to the classical parallel update rule whose two-cluster dynamics isreviewed. It is then shown how the argument may be extended inductively to solve forthe two-cluster probabilities for each element of the sequence of approximate models.A formal limit is then taken, and macroscopic velocities and flow rates are derived.1 IntroductionThis paper is concerned with cellular automata of Nagel-Schreckenberg type [1],with the maximum velocity parameter vmax set equal to one. This type of modelis sometimes referred to as the Asymmetric Exclusion Process (ASEP) [2,3]. Inthis well-known set-up, space is discretised into a one-dimensional array of cellseach of which is either empty or occupied by exactly one agent, and each agentmoves according to a pair of very simple microscopic rules:1. If the cell immediately downstream is occupied, remain stationary. (R1)2. If the cell downstream is unoccupied, move forward into it with probabilityp, 0 < p ≤ 1. (R2)The only remaining subtlety (and the subject of this paper) concerns the preciseorder in which rules (R1,2) are applied.We consider the dynamics of rules (R1,2) under the shuﬄe update scheme,which has received very little attention in the literature to date [4–6]. At eachtime step in this scheme, a random order is generated which contains each agentexactly once. Rules (R1,2) are then applied to each individual agent in turn,according to this order, and the system is updated incrementally as each agenttakes its turn. After all agents have had their turns, a new random order isgenerated and the next time step begins.The shuﬄe update is similar to the random sequential scheme [7] in that theoccupancy of cells is updated incrementally as each agent applies its rules andconsequently, the shuﬄe update does not require conflict resolution to preservesingle-occupancy (even in multi-dimensional extensions). However, the shuﬄeupdate enjoys the modelling advantage that individual agents never receive large2 David A. Smith and R. Eddie Wilson?? ????? ?PSfrag replacementst∗t∗ − 1 Wpp1− pp2F(1)1F(1)2G(1),11G(1),12Probability Py(y/(1− c))(1− c− y)y(1− y/c)y(y/c)(y/(1− c))τ0 τ1 τ2 τ3 σ0 σ1 σ2 σ3Fig. 1. The list of all possible transitions to a (σ1, σ2) = (1, 0) two-cluster (highlightedin bold) at time step t∗, for the truncated process of order n = 1. The list is identicalto that for the parallel update rule. Probabilities for the window states at time stept∗ − 1 are denoted by P ; transition probabilities are denoted by W . Cells marked by ?can be either occupied or empty, with no effect on the P or W calculation: The stateprobability contribution is just a factor of one, and the movement or lack of movementof an agent in this cell cannot affect the monitored (σ1, σ2) two-cluster. The families ofleft hand column states labelled by F(n)i , G(n),mi are the building blocks of the inductiveprocess that follows laternumbers of consecutive turns, hence the possibility of unphysical velocities iseliminated.In this paper we give an outline of how the two-cluster analysis of Schreck-enberg et al [8], which analyses (R1,2) under the parallel update scheme, canbe extended to the more complicated case of the shuﬄe update. The argumenthere is more involved than [8] because under the shuﬄe update, it is possible forlarge blocks of contiguous agents to move forward in a single time step, if theirturns are served in upstream order.Due to this increase in complexity, we use a sequence of approximations tothe full model. We define the truncated process of order n to mean that rules(R1,2) are applied under the shuﬄe update scheme, with the proviso that theopportunity to move is offered only to agents who are in the first n positionsof a contiguous block at the beginning of the time step. Our procedure is thusto explain briefly how two-cluster dynamics works for n = 1 and then explainhow to extend it inductively to any truncated process of order n. Finally we letn→∞.2 Two-Cluster Analysis For n = 1This method is described fully in [8]. We suppose that the occupancy of neigh-bouring cell pairs is independent and we seek to compute the so-called two-clusterprobabilities P2 for all possible combinations of occupancy of two adjacent cells,i.e., the probabilities of two adjacent cells having states (1, 0), (0, 1), (1, 1) and(0, 0), where 0 and 1 denote empty and occupied respectively. It can be shownthat all such two-cluster probabilities can be calculated from y := P2(1, 0) andhence the goal is to seek this quantity.Shuﬄe Update Cellular Automata 3? ??? ? ?? ?? ? ? ?? ? ?PSfrag replacementsn = 1 n = 2 n = 3F(1)1F(1)2F(2)1F(2)2F(3)1F(3)2τ0τ0τ0 τ1τ1τ1 τ2τ2τ2 τ3τ3τ3 τ4τ4 τ5Fig. 2. The propagation of F(n)i states as n increases. These states correspond to theleft-hand column in Fig. 1 for n = 1. The added right hand cell in each window takesthe value ? meaning that it can be either occupied or empty but we need not considerwhich, since it has no effect on the ability or probability to produce (σ1, σ2) = (1, 0)The strategy is to list all possible configurations S at time t∗ − 1 which cangive rise to (1, 0) in a monitored two-cluster at time t∗. If we can calculate thetransition probability W (S) for each configuration, in addition to the probabilityP (S) of the configuration itself (which is usually expressed in terms of y), and ifwe assume the process has reached statistical stationarity, then we may employconditional probability to write y =∑S P (S)W (S), which for the truncation oforder n we re-write and express in the form fn(y; c, p) = 0 where c is the meandensity and p is the parameter of rule (R2).The truncated model with n = 1 is identical to the parallel update rule ofNagel and Schreckenberg [1], and for this standard case the two-cluster calcula-tions are derived in detail in [8]. Note further that the two-cluster method hasbeen shown to be exact in this case, in the sense that the spatial independenceassumption for neighbouring two-clusters is exact.For the case n = 1, Fig. 1 gives a listing of the relevant configurationsand their probabilities P and transition probabilities W . Rather than constructf1(y; c, p) and analyse its zeroes, we instead show now how inductive argumentsmay be used to extend Fig. 1 to truncated processes of arbitrarily large order n.3 Inductive Construction for Truncated ProcessesWe now generalise to look at the truncated process for arbitrary order n. As n isincreased, we need to consider bigger families of states, because there are moreways of obtaining (σ1, σ2) = (1, 0). The states also have wider windows, becauseas n increases, (σ1, σ2) can be affected by more sites further downstream. Thekey is to build the families of cell windows inductively from those with lower n.Our choice of labels for the states was chosen with this process in mind, andwe treat the F(n)i and G(n),mi states separately as they extend in quite differentways, summarised in Figs. 2–4.By considering Fig. 2, we observe that the F(n)i states and their probabilitiesdo not change in any substantive way, so that P (F(n)i ) = P (F(1)i ), W (F(n)i ) =W (F(1)i ), for all n.Now we look at Fig. 3 and consider the G(n),mi states. Let us look at the n = 2case and see how the P and W values relate to those for n = 1. Note P (G(2),1i ) =4 David A. Smith and R. Eddie Wilson? ? ??? ??? ?? ????PSfrag replacementsn = 1 n = 2 n = 3G(1),11G(1),12G(2),11G(2),12G(2),21G(2),22G(3),11G(3),12G(3),21G(3),22G(3),31G(3),32τ0τ0τ0τ1τ1τ1τ2τ2τ2τ3τ3τ3τ4τ4τ5New statesNew statesFig. 3. Propagation of the G(n),mi states as n increases. Existing states breed new onesas well as propagating in the same manner as the F(n)i states (shown in Fig. 2). Thecharacteristic feature of states which breed is that all cells with the dashed outlineshould be filled. For any given n, F(n)i and G(n),mi encompass all states capable ofproducing (σ1, σ2) = (1, 0) at the next time step. The breeding of G(n),mi means thatthe number of states increases by two each time n is increased by onePSfrag replacementsn = 1n = 2 n = 3G(1),11G(1),12G(2),11G(2),12G(2),21G(2),22G(3),11G(3),12G(3),21G(3),22G(3),31G(3),32pp2p2p22p36p32p32p46p− p22p− p22p22 −p36y(1− yc)y(1−yc)y(1− yc)2y(1− yc)2y(1− yc)3y3c(1−c)y3c(1−c)y3c(1−c)(1−yc) y3c(1−c)(1−yc)y3c(1−c) (1−yc)2WW PPFig. 4. The state and transition probabilities for the G(n),mi for increasing n. We seethe state probabilities unaltered along the propagating (horizontal) arrows, and gain-ing a factor of the conditional probability 1 − y/c on the breeding (diagonal) arrow.Transition probabilities gain a factor of p/n on a diagonal arrow, while W (G(n),n−11 ) =W (G(n−1),n−11 )(1− p/n). All others remain unchanged on the horizontal arrowsP (G(1),1i ), since a ? contributes a factor of one to the probability, and P (G(2),2i ) =(1 − y/c)P (G(1),1i ), gaining a factor of the conditional probability 1 − y/c fromthe extra agent appearing in the blocks at τ3. For the transition probabilitieswe find W (G(2),12 ) = W (G(1),12 ). Moreover W (G(2),2i ) = (p/2)W (G(1),1i ), becauseShuﬄe Update Cellular Automata 5one more agent is required to move, gaining a factor of p, whilst the 1/2 comesfrom the probability that the agents will update in an order which will allow thesecond agent to move. Finally W (G(2),11 ) = (1 − p/2)W (G(1),11 ) because it hasbecome necessary to specify that the second agent in the block does not move,although it now can. These n = 2 probabilities are shown in Fig. 4. By lookingat both Figs. 3 and 4, we see that the (n = 1) 7→ (n = 2) transition involvedthe breeding of states. This process generalises inductively to give the breedingbehaviour at higher n values. Non-breeding G(n),mi states propagate unalteredin the manner of the F(n)i states.4 General Solution for the Two-Cluster ProbabilityBy employing conditional probability we may writeyn =∑iP (F(1)i )W (F(1)i ) +∑i,mP (G(n),mi )W (G(n),mi ) , (1)where yn denotes P2(1, 0) for the truncated process of order n and F(n)i , G(n),miare the families of states described in the previous section. This formula, on sub-stitution of the relevant quantities, may be rearranged in the form fn(y; c, p) = 0,wherefn(yn; c, p) = p−py1− c+n∑i=1(−yc)i n∑j=ipjj!(j − 1i− 1) (1−py1− c). (2)By letting n → ∞ we solve f(y; c, p) = 0 for the two-cluster probability of thefull process, wheref(y; c, p) = −(1− p) +1c− y(1−py1− c)(c− yep(1−y/c)). (3)In general, this equation appears to have a unique solution for y, but requiresnumerical solution.5 Steady State Velocities and Flow RatesWe now find the mean velocity and flow rate in terms of the two-cluster proba-bility y. We proceed by using y to find the probability distribution of the lengthof the block to which an agent chosen at random belongs. This quantity may beconstructed from conditional two-cluster probabilities in the formPl =(yc)2(1−yc)l−1, (4)where l is the length of the block in question. We now use the fact that theagent is equally likely to be in any position within the block, and we sum the6 David A. Smith and R. Eddie Wilsonprobability that it moves from the kth position over all k = 1, 2, . . . , l positions.By using (4), we thus obtain the mean velocity (equivalent to the probabilitythat an agent chosen at random moves) in the formvˆ =∞∑l=1lPl1ll∑k=iP (block serves at least k agents)=( yc− y)(exp(pc(c− y))− 1). (5)We can therefore write down flow rateq =( cyc− y)(exp(pc(c− y))− 1), (6)as the product of the system density and mean velocity.6 ConclusionBy extending the two-cluster analysis of the well-known parallel update model[1,8], we have been able to derive expressions for steady state distributions andflow rates of the more complicated shuﬄe update case. The results found hereagree with the recent paper of Wo¨lki et al [5] who employed a car-orientedmean field (COMF) method, as opposed to the site-oriented (SOMF) methodthat we use. However, it remains to be shown whether exactness, i.e. the spatialindependence of neighbouring clusters, holds for the full model and the truncatedprocesses of order n > 1.References1. K. Nagel, M. Schreckenberg: Journal de Physique I 2, 2221 (1992)2. N. Rajewsky, L. Santen, A. Schadschneider, M. Schreckenberg: J. Stat. Phys. 92,151 (1998)3. D. Chowdhury, L. Santen, A. Schadschneider: Phys. Rep. 329, 199 (2000)4. G. Lunt: Cellular Automaton Models for Flows Without Momentum. Masters The-sis, University of Bristol (2001)5. M. Wo¨lki, A. Schadschneider, M. Schreckenberg: preprint, arXiv:cond-mat/0509546v1 21 Sep 20056. D. Smith, R. E. Wilson: preprint7. B. Derrida, E. Domany, D. Mukamel: J. Stat. Phys. 69, 667 (1992)8. M. Schreckenberg, A. Schadschneider, K. Nagel, N. Ito: Phys. Rev. E 51, 2939(1995)

Cellular Automaton Models for Flows Without Momentum. Masters Thesis,

Extension of cluster dynamics to cellular automata with shuffle update

https://research-information.bris.ac.uk/files/2990496/dasmithtgf.pdf

Extension of cluster dynamics to cellular automata with shuffle update

Abstract

Similar works

Full text

Available Versions

Explore Bristol Research