Zero-range process with long-range interactions at a T-junction
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Zero-range process with long-range interactions at aT-junction
A G Angel, B Schmittmann and R K P ZiaCenter for Stochastic Processes in Science and Engineering,Department of Physics, Virginia Tech, Blacksburg, VA 24061-0435, USANovember 2, 2018
Abstract
A generalized zero-range process with a limited number of long-range interactionsis studied as an example of a transport process in which particles at a T-junction makea choice of which branch to take based on traffic levels on each branch. The system isanalysed with a self-consistent mean-field approximation which allows phase diagramsto be constructed. Agreement between the analysis and simulations is found to be verygood.
The zero-range process (ZRP) [1] is a simple model, in which particles hop from site tosite on a lattice, that has a soluble nonequilibrium steady state for a number of cases.As such it has been employed extensively as a model for the analysis of nonequilibriumphenomena. Studies have ranged from fundamental investigations of nonequilibriumsteady states and phase transitions [2, 3, 4, 5, 6] to simple models of real systemssuch as gel electrophoresis, sandpile dynamics, traffic and compartmentalized granulargases [7, 8, 9, 10]. Some interesting properties of the ZRP are the following. It hasa soluble nonequilibrium steady state, as stated before, and the statistical weight ofconfigurations in the steady state is described by a product measure. The relativesimplicity of this product measure form makes the steady state highly amenable toanalysis. The ZRP also displays condensation transitions where a finite fraction of theparticles in the system condense onto a single site. Of particular interest is the fact thatthese transitions can take place on a one-dimensional lattice, something that would notbe expected for an equilibrium system without long-range interactions. In homogeneoussystems these phase transitions can be of a spontaneous symmetry breaking nature,something that has been exploited in traffic models as a possible mechanism for the‘jam from nowhere’ phenomenon [11, 12]. The ZRP has also been proposed as a genericmodel for domain dynamics of one-dimensional driven diffusive systems with conserved ensity, through which it provides a criterion for the existence of phase separation insuch models [13].Recently, several generalizations of the basic ZRP have attracted much attentionin the literature, from both fundamental and application standpoints. One such gener-alization is a system with open boundary conditions, which can display condensationin cases where none exists for the periodic ZRP [14]. Systems with multiple species ofparticles have been proposed, to study inter-species mechanisms which can lead to newcondensation types [15, 16, 17, 18] and to model weighted directed networks [19]. TheZRP has also been generalized to continuous masses, and a general criterion for phasetransitions could be stated [20, 21, 22, 23]. One aspect that has long been known isthe fact that the ZRP can be solved on an arbitrary lattice [1], i.e., with a prescribedset of probabilities for a particle to hop from one particular site to another. This hasbeen generalized recently to continuous masses [24]. ZRP’s on complex networks havealso attracted much attention recently [25, 26, 27, 28, 29]. For an overview of the ZRPand many of its generalizations see [2].The aim of this work is to investigate the ZRP with limited long-range interactions.Specifically, we study a ZRP on a ring lattice, a section of which consists of twobranches, with T-junctions at both ends. The rates for a particle at the junctionto take one branch or the other are based on the total numbers of particles on thebranches. Another way of looking at this is a ZRP on an evolving lattice, with theevolution of the lattice dependent on the state of the system. While the connectivity ofthe sites is fixed, the probability of particles taking a particular branch at the junctionchanges dynamically with the state of the system. Thus, the structure of the latticeis static, but the properties of some of the links are allowed to change. Many realsystems involve processes which take place on evolving lattices or networks. Food websconstitute one example – see, e.g., [30] – with the network reflecting predator-preyor ”who eats whom” relations. Clearly these interactions will change with time. Forexample if one species is near extinction, its predators will preferentially prey on otherspecies, or even begin preying on new species entirely. Another example is a trafficnetwork; here the structure of the network is largely fixed, but the route which a driveris inclined to take may change depending on the levels of traffic in various parts of thesystem and any traffic calming measures that are employed (e.g., lanes reserved forbuses only during peak times).Often problems posed on evolving networks, possibly with long-range interactions,are difficult to solve. Thus, it will be helpful to identify simple exactly or approximately soluble models on such evolving networks or lattices. They can contribute importantsteps towards the understanding of such systems.The paper is organized as follows: In section 2, the model studied is introduced;in section 3 the model is analysed using the solution of the ZRP on a fixed arbitrarylattice as the starting point for a mean-field treatment; extensive numerical results arepresented in section 4; finally, in section 5, the implications of this work are discussedand conclusions are drawn. The simple model studied in this paper is defined as follows. Particles hop from siteto site on a lattice which consists of a “main stretch” of L m sites, branching at a T-junction into two other lanes: labeled “left” and “right” branches, with L ℓ and L r sitesrespectively. At the end, these branches converge to rejoin the main stretch (Figure 1).Since the system is periodic, the total number of particles on the lattice, N tot , is fixed.There is no exclusion, so that each site can hold any number of particles. The state ofthe system is completely described by the set of site occupancies { n i } . Particles hopfrom a site i with a rate u i ( n i ), i.e., a rate dependent on the number of particles at thatsite, n i . This corresponds to a particle on site i hopping with a probability u i ( n i )d t ina suitably small time interval, d t . Particles hop to the rightmost adjacent site, exceptat the T-junction where a particle hops to the left branch with a probability x and tothe right branch with a probability 1 − x . If x is a constant, then this system can besolved using results for the ZRP on an arbitrary network [1, 3]. In this work, we extendthe model to cases where x depends on the number of particles in the left branch N ℓ and/or the number of particles in the right branch N r . Thus, x changes dynamicallywith the configuration of the system.To keep our model as simple as possible without being completely trivial, we takerates that are uniform within each section and proportional to the standard ZRP form: u ( n ) = 1 + b/n , with b > m ( n ) = u ( n ) (1a) u ℓ ( n ) = p u ( n ) (1b) u r ( n ) = q u ( n ) , (1c)where p and q are constants. The most general scenario involves arbitrary (positive) p, q . In this work, we restrict our simulations and analysis to the unit square in the p - q plane. Our main aim is to investigate what happens when a particle at the T-junctionhas advance knowledge of the states of the branches and stochastically selects whichbranch to take on the basis of this knowledge. An obvious case to consider is somerepulsion from highly occupied branches. In particular, one can consider the case inwhich where a particle only has knowledge of the state of one branch. It turns outthat this situation is one of the simplest to treat analytically. An example of such asystem is a car faced with the choice between a direct major route with frequent trafficreporting or a longer minor route without any traffic reporting. Heavy traffic mayoccur on either route and the question is: would it be faster to continue into a knownjam or take a longer route with an unknown state? A possible form of x to representthis idea is a linear dependence on the occupation of the left branch x = (cid:20) − QN ℓ N tot (cid:21) Θ (cid:18) − QN ℓ N tot (cid:19) , (2)where Q is a constant and the Heaviside function, Θ( n ), ensures that x , a probability,does not become negative. Another natural case to consider is the particle making achoice based on knowledge of the occupations of both branches. A form of x that fallswithin this category is x = N r N ℓ + N r . (3)This form implies that a particle is more likely to take the branch with the loweroccupation, a realistic choice for a driver trying to avoid a jam. Obviously, this choiceembodies an extra symmetry, allowing certain simplification in the analysis. The simple model defined above is amenable to an approximate analysis based onthe solution of the system with a fixed x , the steady state of which is known exactly[1, 3, 31]. As a precursor to the analysis for the variable- x system, results for thefixed- x system are reviewed briefly. x system Focusing only on the transitions between “fluid” (homogeneous) and condensed states,we may exploit the grand-canonical formalism. As in textbook treatments, the numberof particles in such an approach is allowed to vary and only the mean number of particles s controlled by the fugacity parameter: z . Within this framework, the steady-statedistribution of the occupations of the sites of the system is found to beP( { n i } ) = 1 Z " Y i ∈ ms f m ( n i ) z n i j ∈ lb f ℓ ( n j )( zx ) n j " Y k ∈ rb f r ( n k )( z (1 − x )) n k , (4)where the products go over the sites which are in the main stretch (ms), the left branch(lb) and the right branch (rb), respectively. Here, f µ ( n ) = n Y k =1 u µ ( k ) − for n >
01 for n = 0 , (5)and obviously, the subscript µ = m , ℓ , r for the ms, lb, and rb, respectively. Finally, Z is a normalization constant, akin to the grand-canonical partition function, Z = ∞ X n =0 ∞ X n =0 · · · ∞ X n L =0 " Y i ∈ ms f m ( n i ) z n i j ∈ lb f ℓ ( n j )( zx ) n j " Y k ∈ rb f r ( n k )( z (1 − x )) n k . (6)With the chosen hopping rates (1), we have f m ( n ) = f ( n ), f ℓ ( n ) = p − n f ( n ) and f r ( n ) = q − n f ( n ), so that this expression simplifies considerably: Z = X { n } " Y i ∈ ms f ( n i ) z n i j ∈ lb f ( n j )( zx/p ) n j " Y k ∈ rb f ( n k )( z (1 − x ) /q ) n k . (7) x system Let us first remind the reader that the grand-canonical treatment gives the correctdistribution of the system only below a certain critical density ρ c . Of course, due tothe presence of the branches, this quantity will depend on the parameters ( x, p, q ). Tosee how this arises, we will need the expression which relates the overall density of thesystem ( ρ ≡ N tot /L ) to the fugacity z . But we have three sections and the averagedensity of each must be considered separately. To facilitate, let us define g ( ζ ) ≡ P ∞ n =0 nζ n f ( n ) P ∞ n =0 ζ n f ( n ) , (8)which can be used to relate the density in each section to the effective fugacities: z m ≡ z ; z ℓ ≡ zx/p ; z r ≡ z (1 − x ) /q . (9) ummarizing, we write ρ µ ≡ h N µ i /L µ = g ( z µ ) , (10)so that ρ = (cid:28) N m + N ℓ + N r L (cid:29) = X µ ρ µ L µ L = L m L g ( z ) + L ℓ L g ( zx/p ) + L r L g ( z (1 − x ) /q ) . (11)From our experience with standard ZRP’s, it is clear that this approach is valid aslong as each effective fugacity remains less than unity (so that Z remains finite). Thus, z cannot exceed the minimum of 1, p/x , q/ (1 − x ). Meanwhile, g is a monotonicallyincreasing function of its argument and, if g (1) is finite, then there will be a density, ρ c ,beyond which (11) has no solution. This happens when the hop-rates in each sectiondecay to some constant value β more slowly than β (1 + 2 /n ) [3, 4]. In the system,the lack of a solution to the density equation manifests itself as a symmetry-breakingcondensation transition whereby the excess density will be taken up by a single site.In our case, this site will be located in the section where the effective fugacity firstreaches unity. For times much longer than the characteristic time scale associatedwith individual particles hopping, the condensate will be found on just one site in thissection. In simulations starting from random initial conditions, we often observe thecondensate to form on the first site of a stretch. However, with other initial conditionsthe condensate can form anywhere and we believe that, for finite systems and longtimes, the condensate will move slowly between sites, eventually exploring all of thepossible locations.To summarize, if u ( n ) decays more slowly than 1 + 2 /n , then we can expect con-densation for high densities. Further, the condensate will appear in the main stretch if p > x and q > − x . Otherwise, it will appear in the left or the right branch, dependingon whether p (1 − x ) is less or greater than qx . Though the critical density needed foreach of the sections is the same (i.e., g (1)), the overall critical density will depend onthe details of the parameter set. As an illustration, suppose x = 0 . p = 0 .
2, and q = 0 .
8, so that z should not exceed 2 /
3. A condensate will then appear on the leftbranch when the overall density exceeds [ L m g (2 /
3) + L ℓ g (1) + L r g (7 / /L . x system The aim is now to use the exact solution of the fixed- x system as a tool to understandthe system with the dynamically changing branch choice. The basic assumptions isthat if the system relaxes to a steady state, it will do so with some well-defined averagevalue for x . In particular, this assumption is valid if the fluctuations of the particlenumbers entering the definition of x , (2) or (3), are not too large. Within this self-consistent mean-field approximation the average numbers of particles in the left andright branches (with the exception of any condensate) can be calculated as a functionof x : h N ℓ i = L ℓ g ( zx/p ) , h N r i = L r g ( z (1 − x ) /q ) . (12) or our specific hopping rates, g is explicitly g ( ζ ) = ζ F (2 ,
2; 2 + b ; ζ )(1 + b ) F (1 ,
1; 1 + b ; ζ ) (13)where F is the hypergeometric function [32]. Note that g (1) = 1 / ( b − b > h N ℓ i and h N r i can then be fed into a self-consistent equation for x . Whether or not this equation admits a solution reveals much about the system.Although the grand-canonical treatment can describe only the sub-critical behaviourof the system in detail, it does lead us to some information about the phase diagram. Aswe will see, it can predict some simple aspects of the condensed phases. For example,for the fixed- x system, a single condensate appears to soak up all the excess mass inthe system. However, for the varying- x system, the feedback mechanism seems to beable to prevent a single condensate from absorbing all of the excess mass. Instead, theexcess is shared between two or more condensates. The first case chosen for study has the branch choice probability, x , dependent on thestate of the left branch only. For this system, which is defined by the branch choiceprobability (2), the self-consistent equation for x is x = 1 − QL ℓ N tot g ( zx/p ) (14)This equation depends on both x and z , which are also partially dependent on eachother. Within the self-consistent scheme we proposed, both (14) and (11) are to besolved simultaneously in x and z . If a simple solution exists, this implies that thereis no condensation. Much like the case for Bose-Einstein condensation presented intextbooks, if the naive approach fails to produce a solution, the specifics of this failurewill provide enough information for us to predict where condensation occurs.Recall that in the fixed- x system the maximum allowed value of the fugacity, z ,was z max = min { , p/x, q/ (1 − x ) } . Thus if (14) can be solved along with (11) underthis constraint on z , then there will be no condensation and the number of particlesin each section will be known simply by inserting the value of x from the solution intothe relevant expression.If, under the constraints on z , (14) can be solved and (11) cannot, then this impliesthat a condensate exists but not on the left branch. Recall that in the fixed- x system,inserting z at its maximum value into any of the expressions above provides a correctdescription of the system except for the condensed site. Thus, (14) is solved for themaximum value of z . The location of the condensate can then be determined byinspecting z and z (1 − x ) /q . If z < z (1 − x ) /q , then the condensate appears on theright branch, otherwise it appears on the main stretch.If neither (14) nor (11) can be solved, this suggests that x takes a value such thatthere is a condensation on the left branch. However, this condensation may not be thesame as in the fixed- x system, since the form (2) tends to suppress large numbers of articles on the left branch. In the fixed- x system the condensate grows indefinitelywith increasing total particle density. However, in this varying- x system, an increaseof the number of particles on the left branch results in a decreasing value of x . Thisimplies that a condensate on the left branch will not be able to take up the excessdensity for all densities. In this case, the maximum value of z will be such that either zx/p = z > z (1 − x ) /q or zx/p = z (1 − x ) /q > z . That is, the system organizes itselfinto such a state that two coexisting condensates are supported: In the grand-canonicalensemble two sections are critical and both hold a condensate. The locations of thecoexisting condensates can be determined from which of z , zx/p and z (1 − x ) /q areequal, e.g., if z = zx/p then the condensates appear on the left branch and the mainstretch.Unfortunately, the self-consistent and density equations (14), (11) are often difficultto solve analytically, due to the presence of hypergeometric functions. However, theycan be solved numerically and this can be used to map out the phase diagram of thesystem in the parameters p and q .The phase diagram becomes particularly easy to calculate if the following two as-sumptions are made. The number of particles in the system is large enough that thedensity equation (11) cannot be solved for any allowed values of z and x . Thus theremust always be a condensate somewhere in the system. This allows one to assumethat z must be at its maximum value of min [1 , p/x, q/ (1 − x )], instead of numericallysolving the density equation. Also, Q is sufficiently large that a condensate cannotappear on the left branch only. The phase diagram of this system can then be mappedout by first assuming a phase, and then working out the values of p and q necessaryfor its presence.The phases are identified by the location(s) of the condensate(s) in the system:L for the left branch, R for the right branch and M for the main stretch, while Nindicates that no condensates are present in the system Thus, LR denotes a phasewhere condensates appear on the left and right branches, for example. M Phase—
For a condensate to be present only on the main stretch, z must beequal to 1, both p/x and q/ (1 − x ) must be greater than 1 and the self-consistentequation (14) for x must have a solution. Thus, this phase is bordered by the line p = min( x ), where min( x ) refers to the smallest possible value of x that can be foundfor (14) in the grand-canonical treatment. It is given bymin( x ) = 1 − QL ℓ N tot ( b − . (15)It is also bordered by a line which comes from the solution of the self-consistent equation(14) where q/ (1 − x ) becomes less than one indicating a shift to an R phase. This lineis given by q = QL ℓ g ( r ) N tot , (16)where r = (1 − q ) /p . LM Phase—
For condensates situated on the left branch and the main stretch tocoexist, z must be equal to 1, p/x must be equal to 1 (so x = p ), q/ (1 − x ) must begreater than 1 and the self-consistent equation (14) must have no solution. Thus, this hase is bordered by a line already identified for the M phase ( p = min( x )) and alsoby the line q = 1 − p . LR Phase—
For condensates situated on the left and right branches to coexist, z must be less than 1, p/x must be equal to q/ (1 − x ), both of these must be equalto z and the self-consistent equation for x (14) must not have a solution. Thus, thisphase is bordered by a line already identified for the LM phase ( q = 1 − p ) and the line q = (1 / min( x ) − p which is the border for (14) to have a solution. R Phase—
For a condensate to be present on the right branch only, z must be lessthan 1, p/x must be greater than z and q/ (1 − x ) must be equal to z . The boundariesfor this phase have already been identified as the boundaries of two other phases; thisphase is bounded by the line q = (1 / min( x ) − p and the line which comes from thesolution of the self-consistency equation (14) where x is such that q/ (1 − x ) = 1 whichis given by (16).The other possible phases (L, RM, LRM and N) are not observed when both thedensity of particles and the repulsion parameter, Q , are sufficiently high, as has beenassumed above. For an L phase the density must be sufficiently low, or the repulsionsufficiently weak, to hold a condensate without x becoming too small to sustain it.For an RM phase to be present, there must be some repulsion from the right branch,otherwise the condensate will prefer to form on only one of the sections. For an LRMphase one must have z = 1, zx/p = 1 and z (1 − x ) /q = 1, which can only happen onthe line q = 1 − p , and even so, it is likely that at least one of the sections will merelybe at criticality and not hold a condensate. Finally for the N phase the density mustbe so low that there does not need to be a condensate anywhere in the system.Assembling all this information together, phase diagrams for given parameters canbe produced. As an example, the phase diagram for Q = 20, N tot = 8000, L ℓ = 300, L r = 600 and L m = 1000 calculated via this method is shown in Figure 2 (a). Alsoshown are some of the values of p and q for which simulations were run to verify thephase behaviour.It is straightforward to calculate how the phase diagram will be affected by changingthe repulsion parameter Q and/or the lengths of the branches. The fact that all thephase boundaries meet at a single point does not change when these parameters arevaried; this point is simply displaced. Assuming a sufficiently high density, the line q = 1 − p always forms the boundary between the LM and LR phases; only its lengthdepends on the parameters noted above. The vertical boundary between the LM and Mphases is displaced horizontally and its length changes also. For the boundary betweenthe LR and R phases, the slope changes. Finally the boundary between the R and Mphases retains its curved nature, but is displaced and shortened/lengthened.The theory can also be applied to systems at lower densities and with weakerrepulsion, but it is less straightforward. The simplification associated with high densitysystems - z being pinned to a maximum value - no longer holds. As a result, we mustgenerally rely on numerical methods to find a solution for both equations (14) and(11). With this approach, the phase boundaries can be mapped out, as in the specificcase of Q = 2, N tot = 800, L ℓ = 300, L r = 600 and L m = 1000 (Figure 2 b). For theseparameters, the N and L phases are realised. We also see that the phase diagram hasa much richer structure. q p LR LM R M q p LR L LM MNR(a) (b)
Figure 2: Phase diagrams for the left-branch feedback system with L ℓ = 300, L r = 600, L = 1900 and (a) Q = 20, N tot = 8000 and (b) Q = 2, N tot = 800. Also shown arepoints where simulations have been run and the expected behaviour verified. The phases arelabelled by the position(s) of the condensate(s): L indicates a condensate on the left-branch,R indicates a condensate on the right-branch, M indicates a condensate on the main stretchand N indicates that no condensates are present.10 simpler task than computing the full phase diagram for any particle number iscalculating how and when the phases emerge. Clearly at low enough particle num-ber, no condensates will be present. The M phase will emerge at p = q = 1 when N tot = L ℓ g ( x ) + L r g (1 − x ) + L m g (1), with x coming from the solution of x =1 − QL ℓ g ( x ) / ( L ℓ g ( x ) + L r g (1 − x ) + L m g (1)). For Q = 2 this takes place at N tot = 610.The L (R) phase first emerges at sufficiently small p ( q ) when there are enough particlesto support such a phase, i.e., N tot = L ℓ g (1) ( L r g (1)) and the LR phase when there areenough particles to support both condensates, N tot = ( L ℓ + L r ) g (1) and both p and q are sufficiently small. The LM phase will emerge from the q = 1 boundary when boththe L and M phase can first support condensates at N tot = L ℓ g (1)+ L r g (1 − x )+ L m g (1)where x is given by x = 1 − ( Q/N tot ) L ℓ g (1); for Q = 2 this is N tot = 714. The R andM phases first meet at the p = 1 boundary when N tot = L ℓ g (1 − q ) + ( L r + L m ) g (1),with q coming from the solution of q = QL ℓ g (1 − q ) / ( L ℓ g (1 − q ) + ( L m + L r ) g (1)),which in the case Q = 2 corresponds to N tot = 869. The N phase disappears whenthe LR and LM phases meet up at N tot = ( L ℓ + L r + L m ) g (1), the first point whereall three sections can have critical occupancies. The LR and LM phases then begin toborder each other along the line q = 1 − p and this line extends as N tot is increaseduntil the L phase is extinguished at N tot = Q ( L r + L m ) g (1) / ( Q − Next, we study the case where the branch choice probability depends on the occupationof both the left and right branches in a repulsive fashion, i.e., the value of x favourssending particles to the least occupied branch. The form used, given in (3), is clearlymore symmetric than the previous case. We can write the self-consistent equation for x in a way that makes this symmetry transparent: L r (1 − x ) g ( z (1 − x ) /q ) = L ℓ xg ( zx/p ) . (17)Again, the solution of this equation (or lack thereof) in conjunction with the densityequation (11) reveals a great deal about the system.The phase diagram for this system can be mapped out in much the same manneras above, though the analysis is slightly more involved.As before, simplifications occur if the total number of particles in the system ishigh enough so that a condensate must appear somewhere in the system. Then, thephase diagram can be constructed by assuming the location(s) of the condensate(s)and examining the values of q and p for which a solution can persist. As an example,in order for the system to display a condensate on only the main stretch, we must have z = 1 and a solution to the self-consistency equation (17) such that both x < p and(1 − x ) < q .For a sufficiently large N tot , there are three possible ways in which equation (17)cannot be solved. For the system to be in the L, R, or LR phase, we must have x = p ,(1 − x ) = q , or x = p and (1 − x ) = q respectively. By calculating when these equalitieshold in conjunction with z being pinned to its maximum value of min { , p/x, q/ (1 − x ) } ,the rest of the phase diagram is mapped out. M Phase—
As noted above, for a condensate to be present only on the mainstretch, z must be equal to 1, both p/x and q/ (1 − x ) must be greater than 1 and the elf-consistent equation (17) for x must have a solution. Thus, this phase is borderedby a line which comes from the numerical solution of (17) which in this case takes theform L r (1 − x ) g ((1 − x ) /q ) = L ℓ xg ( x/p ) . (18)When no solution exists, the system must shift to a phase with a condensate presenton one or more of the branches. L Phase—
For a condensate to exist only on the left branch, we require z < z = p/x , z < q/ (1 − x ) and no solution to the self-consistent equation (17) for x such that the occupation of the left branch is greater than the critical value, but theoccupation of the right branch is not. Thus, in the region q < − p this phase isbordered by the line coming from the solution of N tot = L r g (1)( p + q ) /p + L m g ( p + q ) . (19)Along this line the system shifts into a phase with condensates present on the left andright branches. In the region q > − p this phase is bordered by the line from thesolution of N tot = L r g ((1 − p ) /q ) /p + L m g (1) . (20)Along this line the system shifts into a phase with condensates present on the leftbranch and the main stretch. R Phase—
For a condensate to exist only on the right branch, we require z < z < p/x , z = q/ (1 − x ) and the right branch occupation to be greater than critical sothat there is no solution to the self-consistent equation for x (17). Thus in the region q < − p this phase is bordered by a line coming from the solution of N tot = L ℓ g (1) p/q + L m g ( p + q ) . (21)Along this line the system moves into the LR phase. In the region q > − p the phaseis bordered by a line given by N tot = L ℓ g ((1 − q ) /p ) p/ (1 − p ) + L m g (1) . (22)Along this line the system moves into the LM phase. LM Phase—
For condensates situated on the left branch and the main stretch tocoexist, we must have z = 1, x = p , 1 − x < q , and no solution to the self-consistentequation for x (17). Thus, this phase is bounded by the line already identified as aboundary for the L phase (20), the line q = (1 − p ) and a line coming from the numericalsolution of (17) which in this case takes the form L r (1 − p ) g ((1 − p ) /q ) = L ℓ pg (1) , (23)and gives the limit of the region in which no solution can be found. This coincides withpart of the boundary from the M phase; in fact, the only way in which the equationgiving the boundary for the M phase (18) cannot be solved under the high densityassumption is if either x = p or (1 − x ) = q . RM Phase—
Similarly, for condensates situated on the right branch and the mainstretch to coexist, we must have z = 1, x < p , 1 − x = q , and no solution to the self-consistent equation for x (17). Thus, this phase is also bounded by the line q = (1 − p ), line (different from the one above) coming from the limits of the numerical solutionof (17) and a line already identified as a boundary for the R phase (22). The first linecorresponds to the part of the M boundary not already matched by the LM boundary L r qg (1) = L ℓ (1 − q ) g ((1 − q ) /p ) . (24)Note that if the lengths of the left and right branches are the same, as chosen here,this phase is symmetric with the LM phase. LR Phase—
Finally, for condensates situated on the left and right branches tocoexist, the appropriate conditions are z < x = p , 1 − x = q , and no solution to theself-consistent equation for x (17). Thus, this phase is bounded by the line q = 1 − p ,a line from solving (17) which happens to touch, but not cross, q = 1 − p for the casestudied here and lines that have already been identified as boundaries for the L and Rphases (19), (21).As before, the other possible phases (LRM and N) are not observed when the densityis sufficiently high, as has been assumed. The LRM phase can again only possibly existon the line q = 1 − p and it is likely that the condensate will only form on at most twoof the sections. Finally the N phase will not be seen when the density is sufficientlyhigh as the system must take on a condensate to accommodate the number of particlesin a steady state.The phase diagram for a system with N tot = 8000, L = 2000 and L ℓ = L r = 500is shown in Figure 3 (a). Also shown are the p and q values for simulations that havebeen run to verify the phase behaviour.As before the changes in the phase diagram due to changing the lengths of thebranches (which do not have to be the same) are straightforward to compute. Theboundary of the LR phase remains unchanged as the line q = 1 − p , but the pointwhere the other boundary lines meet with this one does move along this line and theboundaries between the LM and L and LR and L change length and slope accordingly.The boundary lines between the M and the LM and RM phases respectively retaintheir curved shape, but they move and the points at which they touch q = 1 − p and q = 1 or p = 1 also change.As in the previous, case, the analysis can also be applied to systems with densitiesthat do not a priori guarantee the presence of a condensate. Again, however, theapplication is less straightforward and relies on simultaneous numerical solutions of thedensity equation (11) and the self-consistent equation for x (17). The phase diagramcalculated in this way for the system with L ℓ = L r = 500, L m = 1000 and N tot = 800(i.e., the same system as above but at a lower density) is shown in Figure 3 (b). Herethe N phase is realised, the LM and RM phases are lost and the phase diagram hasbecome richer in structure.Again, how and when the phases emerge as the total number of particles is increasedis perhaps more straightforward to calculate than the full phase diagram. Clearly,at low enough particle numbers there will be no condensation in the system. Forsufficiently small q ( p ) the R (L) phase will emerge simply when there are sufficientparticles to support such a phase i.e., N tot = L r g (1) ( L ℓ g (1)). Likewise, the LR phasewill emerge for sufficiently small p and q when N tot = ( L l + L r ) g (1). The M phase mustfirst emerge at p = q = 1 when the number of particles on the main stretch first becomes q p LRLM M RML R q p LRL N MR(a) (b)
Figure 3: Phase diagrams for the system with repulsive feedback from both branches, with L ℓ = L r = 500, and L m = 1000 and (a) N tot = 8000, (b) N tot = 800. Also shown are pointswhere simulations have been run that verify the predicted behaviour. Note that only halfof the diagrams have been explored in this way due to the symmetry of the system. Thephases are labelled by the position(s) of the condensate(s): L indicates a condensate on theleft branch, R indicates a condensate on the right branch, M indicates a condensate in themain stretch and N indicates that no condensates are present.14 ritical, and will do so at N tot = L m g (1)+ L r g (1 − x )+ L ℓ g ( x ) with x being the solutionof x ( L r g (1 − x ) + L ℓ g ( x )) = L r g (1 − x ). For the symmetric system with L m = 1000, L ℓ = L r = 500 this turns out to be N tot = 634. The RM phase emerges from the p = 1boundary at when both the right branch and main stretch occupancies first attaintheir critical values together. This is at the point N tot = L r g (1) / (1 − q ) + L m g (1)with q coming from the solution of (1 − q )( L ℓ g (1 − q ) + L r g (1)) = L r g (1); for thespecific system studied here this is N tot = 860. Also, for the specific system studiedhere the RM phase is symmetric with the LM phase. At N tot = 1000, the N phasedisappears completely as this is the point where the LR, LM and RM phases first meetup i.e., when N tot = L ℓ g (1) + L r g (1) + L m g (1). The L and R phases are present forall high particle densities, although they will eventually become infinitesimally small,the thickest part of the phases behaving as p, q ∼ /N tot respectively.Finally, we note that in the condensed phases the size of the condensate(s) canchange to allow a solution of both the x balance equation and the density equation.This will break down only when the x , z solution becomes such that the condition foranother phase will be met or when the condensate has to take its critical value to satisfythe equations. Due to the way that the occupations of each section depend on z it canbe shown that approaching a boundary from either side must result in an unambiguousanswer. Thus, it is expected that many different forms of the x -function will give thekind of multiple condensate behaviour seen for the two choices, (2) and (3), consideredin this study. The only constraints which should be placed on a suitable x -functionare the following: it has a range between 0 and 1; it gives a unique solution to theself-consistent equation for x and the density equation for any given ( p , q ) pair; it isrepulsive in some way so as to inhibit a sole condensate; and it is sufficiently smooththat the solution will not be discontinuous within any phase. To verify the predicted phase diagrams, we performed extensive Monte Carlo simula-tions on this system with both feedback mechanisms. Our simulation method is simple:A site is picked at random and a particle there is moved to the next site with the rele-vant probabilities, (1), (2), and (3). We have studied a range of branch lengths, N tot ’s, p ’s, q ’s and Q ’s. In this paper, we only present the results of L m = 1000, 0 < p < < q < N tot = { , } , with L ℓ = 300, L r = 600, Q = 20 for the system withfeedback from the left branch only and L ℓ = L r = 500 for the system with feedbackfrom both branches. In the phase diagrams, shown in Figures 2 and 3, we displaythe behaviour of the condensates. N indicates no condensates are present. If a singlecondensate appears in the system, its location is denoted by L, R, and M - indicatingthe left, right, and main sections, respectively. Similarly, the labeling of coexistence oftwo condensates is self-explanatory. The lines are predictions of the phase boundariesfrom our mean-field theory. The crosses (+) indicate points at which simulations havebeen performed. All numerical results showed the expected behaviour for the relevantphase. Occasionally, small discrepancies between prediction and simulation were seenvery close to the boundaries, but in all cases these discrepancies lessened when largersystem sizes were used. Note that, since systems with the second feedback mechanism (3) and Figure 3) are symmetric under p ↔ q , only one half of the phase diagram(above or below the line q = p ) was tested.One phenomenon appeared in the majority of the runs, namely, a tendency forcondensates to form on the first site of the section. The reason for this behaviour isneither transparent nor intuitive. The exact solution for the fixed- x system implies thatthe condensate is equally likely to be found anywhere in the stationary state. Clearly,the dynamics breaks overall translational invariance, so that the favouring of the firstsite may be a subtle manifestation of the underlying dynamics. Exploring this issue isbeyond the scope of this work, but would be interesting for future studies. To insurethat other sites are equally favoured, in the steady state, for condensation, we carriedout runs where one or more condensates were initially placed at “interior” sites. It isreassuring that such condensates did not move to the first site. We should emphasize,of course, that the probability for a condensate to appear on any site (of the allowedsections) is equal, so that the condensate will move between sites. Given sufficienttime, all sites will eventually be explored in a finite system. This wandering behaviourhas been observed, especially when a small condensate on one section coexists with alarge condensate in another: The large condensate remains stationary, but the smallerone moves between several sites over the duration of the run.We also considered a more sensitive test for our mean-field theory. Since the pre-dictions from this approach are based on the solution of a system with fixed x , weperformed simulations on such systems - with x fixed at the value expected by theory.In particular, we studied distributions for the occupation of a site on each of the threesections (M,L,R) for both these fixed- x systems and the original model. Comparisonsare shown in Figures 4 and 5. To be consistent with the theory, these distributionsshould match everywhere except in those parts which represent a condensate. Specifi-cally, for the varying-x system at high enough densities, the single condensate charac-teristic of the fixed- x system can be split into two residing on two different sections ofthe lattice. In general, the agreement was observed to be excellent. There was someapparent discrepancy when a “small” condensate was present, but such differences areconsistent with finite-size effects observed in the standard ZRP at densities above, butclose to, criticality.We also studied how the quality of the agreement between the theory and sim-ulations depends on the lengths of the branches, focusing only on the first feedbackmechanism - x given by (3). We chose L ℓ = L r in the range of 10 to 500 sites and ranthe system in the LM, LR and M phases. The global density of particles was fixed.We measured both the fractional deviation of x from its expected value and the fluc-tuations in x on the scale of the mean, see Figure 6. Generally, the agreement worsensas the branches are shortened. However, this general trend is not observed for the LRphase. One possible difference is the following. For the LM and M phases, at least oneof the branches will have a low density of occupation, leading to effects of discreteness.In very short branches and with low densities, there are typically only a few particleson the sites, so that changes in occupation happen through comparatively large jumps.Also, the fact that the LM phase deviates and fluctuates more than the M phase isprobably due to the fact that the condensate on the left branch is relatively smalland so, more susceptible to instability and collapse. For very small branch lengths, ome blurring of the phase boundaries was observed. Close to the boundaries expectedfrom theory, the states observed sometimes did not match with those predicted. It isthought that these states may be metastable, as systems artificially nucleated in theexpected state remained there for the duration of all runs.Finally, we also studied numerically the validity of the theory at low densities whereno condensation occurs. Through simulations, we can deduce the values of x and z .These values are then inserted into (14) and (11) to see how well the latter are satisfied.In all cases the agreement is very good. Apparently, the grand-canonical analysis isquite successful for a self-consistent, mean-field approach, despite suggestions that thistype of analysis should work well only for the condensed state. In this paper a generalization of a ZRP with limited long-range interactions was studied.The basic model was that of a ring lattice with a section which splits into two branchesat a T-junction before rejoining. Apart from the branch point (BP), we impose thestandard ZRP rules for totally asymmetric particle hopping. The rates are uniformwithin each branch, but may be different for the three sections. For the particle at theBP, we must assign probabilities to hop to each branch. If these are fixed parameters,there exists an exact solution for the stationary state distribution. Here, we focus ona simple generalization, i.e., that these probabilities depend on the total occupationwithin the branches. Thus, there is a long-range interaction between the particle atthe BP and those in the branches. We study this system both analytically and withextensive Monte Carlo simulations. Our self-consistent mean-field theory is based onthe exact solution with fixed probabilities.The long-range feedback mechanism has interesting consequences for the system.For systems with sufficiently high densities and fixed branching probabilities, a con-densate forms only on one of the three sections. In other words, there can be only fourregions in a phase diagram (involving the overall density, the relative hopping rates,and the branching probabilities). By contrast, our model displays three additional phases, associated with coexistence of condensates on two of the three branches. Asa side remark, we note that coexistence of condensates on all three branches can onlyhappen on a line rather than in an extended region in parameter space and is thereforedifficult to observe in simulations. As a result, the global phase diagram is considerablyricher.In simulations, we often observe the condensate forming on the first site of a section.This behaviour parallels the one displayed in a ZRP with open boundaries [14], in whichparticles are inserted into the left boundary and extracted from the right boundary withconstant rates. If the insertion rate is low and the extraction rate is high, no condensateappears and much can be understood through a treatment similar to the one used here:exploiting a grand-canonical function with fugacities dependent on the boundary rates.For high insertion and/or low extraction rates, condensation was seen on the end sites,but these condensates tended to continue growing with time. In our branching ZRP(with periodic boundary conditions), the formation of a condensate will feed back intothe equivalent of the insertion rate and so, a condensate with a stable size can form. n, number of particles at a site -6 -5 -4 -3 -2 -1 n p ( n ) n, number of particles at a site -6 -5 -4 -3 -2 -1 n p ( n ) n, number of particles at a site -6 -5 -4 -3 -2 -1 n p ( n ) n, number of particles at a site -6 -5 -4 -3 -2 -1 n p ( n ) (a) (b)(c) (d) Figure 4: Residence plots of the number of particles on a site in the main stretch, leftbranch and right branch for the system with branch choice probability, x , dependent on thestate of the left branch only (given by the form (2)) compared with the same distributionsfrom a system with the branch choice probability fixed at the value predicted by theory.The system had L = 1900 sites with L ℓ = 300 and L r = 600, N tot = 8000 particles andrepulsion parameter Q = 20. The distributions are shown for p and q values which fit in thevarious phases: (a) LM phase ( p = 0 . q = 0 . p = 0 . q = 0 . p = 0 . q = 0 . p = 0 . q = 0 . x system in the main stretch ( ◦ ), the left branch ( (cid:3) ) and the rightbranch ( ⋄ ) and for the varying- x system in the main stretch (+), the left branch ( × ) and theright branch ( ∗ ). From the theory it is expected that the distributions from the system with x fixed and the system with x varying should match, with the exception of the condensateregion. This is seen in (a) and (b) where in the fixed- x system the condensates are on themain stretch and right branch respectively and in the varying- x system small amounts ofthese are shared with a condensate on the left branch. In general the agreement is very good.18 n, number of particles at a site -6 -5 -4 -3 -2 -1 n p ( n ) n, number of particles at a site -6 -5 -4 -3 -2 -1 n p ( n ) n, number of particles at a site -6 -5 -4 -3 -2 -1 n p ( n ) (a) (b) (c) Figure 5: Residence plots of the number of particles on a site in the main stretch, leftbranch and right branch for the system with branch choice probability, x , dependent on thestate of both branches (given by the form (3)) compared with the same distributions for asystem with x fixed at the value predicted by theory. The system had L = 2000 sites with L ℓ = L r = 500 and N tot = 8000 particles. The distributions are shown for p and q valueswhich fit in the phases: (a) LM phase ( p = 0 . q = 0 . p = 0 . q = 0 . p = 0 . q = 0 . x system in themain stretch ( ◦ ), the left branch ( (cid:3) ) and the right branch ( ⋄ ) and the varying- x system inthe main stretch (+), the left branch ( × ) and the right branch ( ∗ ). From the theory it isexpected that the distributions from the system with fixed x and the system with varying x should agree, except for the part of the distribution which describes the condensate. Forthis system the agreement is very good. 19
100 200 300 400 500 600 number of sites in each branch -0.6-0.4-0.200.20.40.60.81 fr ac ti on a l d e v i a ti on o f x LM phase (p=0.25, q=0.85)LR phase (p=0.2, q=0.4)M phase (p=0.75, q=0.85)
Number of sites in each branch F l u c t u a ti on s i n x LM phase (p=0.25, q=0.85)LR phase (p=0.2, q=0.4)M phase (p=0.75. q=0.85) (a) (b)
Figure 6: Behaviour of the system with branch choice probability, x , dependent on left andright branches (with the form given by (3) with varying, but equal, numbers of sites in thebranches. The system has L m = 1000 sites in the main stretch and a fixed global density of4 particles per site. (a) The fractional deviation of the branch choice probability, x , from thevalue predicted by the mean-field theory. (b) Fluctuations of the branch choice probability, x , on the scale of the mean. The success of the grand-canonical treatment in the open boundary model may berelated to the good agreement between the mean-field theory and simulation results inour model.Remarkably, this agreement continues to be quite good for very short ( ∼
10 sites)branches. This is somewhat surprising, since mean-field theories are expected to bereliable only in the thermodynamic limit. We suspect, however, that the effects due tothe finite size of the main stretch may be more serious, and we intend to explore theseeffects in a future publication. In any case, the good agreement we found provideshope that the mean-field approach may be suitable for further generalizations, such asmore branches/loops and increasingly complex long-range feedbacks. We believe that,with the promise of further surprises, such systems deserve further investigations. Inparticular, we believe that there is considerable potential for applying such models to abroad spectrum of physical systems. In particular, there are many transport processeson networks where individual “agents” make decisions on which route to take, basedon the existing state of the rest of the network. Examples include vehicular trafficsystems and data transport in computer networks. The model studied here is perhapsthe simplest of this class and we hope that it serves as a springboard for complexgeneralized models and advances the understanding of realistic systems.
This work was supported in part by a grant from the US National Science FoundationDMR-0414122. eferences [1] Spitzer F 1970 Adv. Math. J. Phys. A: Math. Gen. R195[3] Evans MR 2000
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