Disordered exclusion process revisited: some exact results in the low-current regime
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Disordered exclusion process revisited: some exactresults in the low-current regime
J Szavits Nossan SUPA, School of Physics and Astronomy, University of Edinburgh, MayfieldRoad, Edinburgh EH9 3JZ, United Kingdom Institute of Physics, Bijeniˇcka cesta 46, HR-10001 Zagreb, CroatiaE-mail: [email protected]
Abstract.
We study steady state of the totally asymmetric simple exclu-sion process with inhomogeneous hopping rates associated with sites (site-wisedisorder). Using the fact that the non-normalized steady-state weights which solvethe master equation are polynomials in all the hopping rates, we propose a generalmethod for calculating their first few lowest coefficients exactly. In case of binarydisorder where all slow sites share the same hopping rate r <
1, we apply thismethod to calculate steady-state current up to the quadratic term in r for someparticular disorder configurations. For the most general (non-binary) disorder,we show that in the low-current regime the current is determined solely by thecurrent-minimizing subset of equal hopping rates, regardless of other hoppingrates. Our approach can be readily applied to any other driven diffusive systemwith unidirectional hopping if one can identify a hopping rate such that the currentvanishes when this rate is set to zero.PACS numbers: 05.50.+q, 05.60.-kAMS classification scheme numbers: 82C22, 82C44, 82C70 Submitted to:
J. Phys. A: Math. Gen. isordered exclusion process in the low-current regime
1. Introduction
Despite numerous efforts conducted in the past, the understanding of macroscopicsystems out of equilibrium is still far from being put in a systematic theory. Unlikethe relaxation around the equilibrium which has become a standard textbook material,less can be said for systems that are maintained far from equilibrium. One possibleroute to fill this gap that proved useful in the past (e.g. in developing general theoryof critical phenomena) is to study particular microscopic models. Usually, one startswith a minimal model that admits analytical treatment (exact or approximate), lateradding more details to make it more realistic. Unfortunately, the lack of detailedbalance - a defining property of systems maintained far from equilibrium - means thatwe are generally deprived even of the knowledge of its steady state, not to mentionthe relaxation mechanism towards it. Exactly solved models far from equilibrium arethus rare, but have an important role in nonequilibrium statistical mechanics.One such model is the asymmetric simple exclusion process (ASEP), a minimalmodel of transport of (classical) particles driven by an external field and interactingonly through the exclusion principle that prevents them from coming too close to eachother. Although simplified, this interaction describes several real situations on variouslength scales, ranging from mobile ions in superionic conductors [1] to self-propelledparticles in mesoscopic (ribosomes on a mRNA [2, 3]) and macroscopic (cars [4])systems. From a theoretical viewpoint, the ASEP has become a paradigmatic modelof boundary-induced phase transitions that are, unlike their equilibrium counterpart(with short-range interactions), present even in one dimension. Originally proposedto model translation of mRNA more than five decades ago [2, 3], the exact solution in1993 [5,6] sparked a great interest that led to several important results relevant to thegeneral theory of nonequilibrium steady states. Using the exact solution of the ASEP,Derrida, Lebowitz and Speer [7] derived a non-equilibrium analogue of the free energythat describes coarse-grained fluctuations around the steady state, thus generalizingolder Onsager-Machlup theory [8] valid around equilibrium. This later helped Bertini et al [9] to build the macroscopic theory of fluctuations for driven diffusive systems,of which the ASEP is just one example.A lot of work has been devoted to improve the ASEP to better fit particularphenomena, most of which have their origins in vehicular traffic or biology (for acomprehensive and recent review see [10]). The exact solution has been successfullyextended to particle-dependent hopping rates [11–14] and multispecies systems [15,16],both of importance for traffic phenomena. In biology, some of the generalizationsinclude particles occupying more than one site [17, 18], position-dependent hoppingrates (termed site-wise disorder) [19], desorption and adsorption of particles in thebulk (i.e. Langmuir kinetics) [20], particles with internal states [21], extension to morethan one lane [22], dynamically extending lattice [23], etc. Unfortunately, common tomost of these generalizations is non-applicability of the exact methods that were usedto solve the original ASEP, namely the matrix-product ansatz (see e.g. [24]). Instead,most approaches utilize various mean-field approximations that neglect correlationsin a fashion similar to truncation in the Bogoliubov-Born-Green-Kirkwood-Yvon(BBGKY) hierarchy. While they give a satisfactory account for most of theaforementioned phenomena, it is less clear how to improve them in some controlledfashion. In some cases such as site-wise disorder, the mean-field approximation itselfbecomes hard to treat analytically as the number of inhomogeneities increases.In this article we study totally asymmetric simple exclusion process (TASEP) isordered exclusion process in the low-current regime
3- a version of the ASEP with unidirectional hopping - in the presence of site-wisedisorder and show that some of the information about its exact steady state can stillbe retained. Our starting point is the known fact, reviewed in Appendix A, thatthe non-normalized steady-state weights which solve the master equation can alwaysbe written as multivariate polynomials in all the hopping rates or alternatively, asunivariate polynomials in one of the hopping rates with polynomial coefficients thatdepend on all other hopping rates [25]. As our main result, we show how to computethe first few lowest polynomial coefficients exactly, which gives a good approximationof the exact steady-state weights if the hopping rate that we expand in is much smallerthan the other rates. Since in that case the particle current is small, we term thisregime the low-current regime. Our method of computing non-normalized steady-state weights in the low-current regime is not restricted to site-wise disorder but canbe readily applied to many other generalizations of the TASEP and other drivendiffusive system with unidirectional hopping.The paper is organized as follows. In section 2 we define the TASEP with site-dependent hopping rates and review some of the related exact results. A generalapproach for calculating steady state in the low-current regime is devised in section3 for arbitrary disorder distributions. Section 4 is devoted to few particular cases ofbinary disorder where all the slow sites share the same hopping rate r <
1. Our maininterest there is to calculate steady-state average of the particle current, i.e. its Taylorexpansion in one of the hopping rates which is considered to be small. Some of theexact results that we obtain have already been known, but here they are derived for thefirst time rather then being guessed from the exact solution of small systems [17, 26].Further applications, mainly devoted to the most general case of non-binary disorder,are discussed in 5 with some interesting new implications for the protein synthesis.
2. Model
We consider the totally asymmetric simple exclusion process (TASEP) on an one-dimensional lattice of L sites, where each site i = 1 , . . . , L is either occupied bya particle ( τ i = 1) or empty ( τ i = 0). Particles each move forward stochasticallyat rate p i (which is modelled by the random-sequential update), provided the site i + 1 in front is empty (exclusion principle). In this paper we consider mainly binarydisorder [27, 28], which means that p i is either r < p i ∈ R to which our approach applies as well is left forsection 5.) Boundary conditions can be either periodic ( τ L +1 = τ ) or open, the lattermeaning that new particles are injected at site site i = 1 at rate α provided it is empty,and are removed from the site i = L at rate β . An illustration of the process (notdrawing boundary rules explicitly) is presented in figure 1. Figure 1.
Schematic picture of the TASEP with binary disorder. Bonds whereparticles jump at rate r < r = 0 means no particle is allowed to move through the wall. The steady state is described completely by the probability P ( C ) to find isordered exclusion process in the low-current regime C = { τ , . . . , τ L } , which solves the followingcontinuous-time master equation, written here in a compact form,(periodic) 0 = L X i =1 p i ( τ i − τ i +1 ) P ( . . . , τ i − , , , τ i +2 , . . . ) (1 a )(open) 0 = − α (1 − τ ) P (0 , . . . ) + β (1 − τ L ) P ( . . . ,
1) ++ L − X i =1 p i ( τ i − τ i +1 ) P ( . . . , τ i − , , , τ i +2 , . . . ) . (1 b )In Appendix A we show that the steady-state probability P ( C ) is proportional tothe determinant of a matrix whose matrix elements are linear combinations of hoppingrates. That allows us to write P ( C ) in the following form, P ( C ) = f ( C ) P C f ( C ) , (2)where f ( C ) is a multivariate polynomial in all hopping rates,(periodic) f ( C ) = X k f k ( C ) r k (3 a )(open) f ( C ) = X i,j,k f i,j,k ( C ) α i β j r k . (3 b )The particular form of (2) with f ( C ) given by (3 a ) or (3 b ) means that the ensembleaverage of any physical observable g ( C ) (e.g. local density τ i at site i , or current p i τ i (1 − τ i +1 )) is a rational function with respect to any of the hopping rates (say r ) h g ( C ) i = X C g ( C ) P ( C ) = P C g ( C ) f ( C ) P C f ( C ) = P k a k r k P k b k r k , (4)where coefficients a k and b k are given by a k = X C g ( C ) f k ( C ) , b k = X C f k ( C ) . By expanding (4) in Taylor series around r = 0 we arrive at h g ( C ) i = ∞ X k =0 c k r k , (5)where c k = a k b − k X n =1 c k − n b n b , k = 0 , , . . . , M− M X n =1 c k − n b n b , k ≥ M + 1.Here M is the maximal degree of all f ( C )’s as polynomials in r , and can be calculatedby invoking Schnakenberg’s network theory [29]. The value of M is not important isordered exclusion process in the low-current regime h g ( C ) i in small r originates from the work of Janowskyand Lebowitz [26]. They obtained the first few coefficients in the expansion of current j L ( r ) around r = 0 in the TASEP with a single slow site both in the periodic and inthe open boundaries case with r ≪ α = βj L ( r ) = r − r + 1916 r − r + 77729356627146767085568 r − O( r ) . (6)The expansion (6) was obtained by solving the full master equation for small systemswith L ≤ L increases, low-order terms become independent of L . A similar approach, coined finite segment mean-field theory (FSMFT), has beendevised by Chou and Lakatos in [19] for the open boundaries case with few slow sites.When slow sites are confined to a small segment, the idea is to solve the masterequation exactly for that small segment and then calculate the current j ( σ − , σ + ) asa function of the average densities σ − and σ + at its ends. Assuming that the densityprofiles are flat outside the segment, σ − and σ + can be then calculated numericallyby solving σ − (1 − σ − ) = j ( σ − , σ + ) = σ + (1 − σ + ). For r ≪ α, β > /
2, Chou andLakatos were able to deduce the analytical expression for the current up to O( r ) fortwo particular configurations of disorder, namely j L ( r ) = dd + 1 r + O( r ) , (7)for two slow sites placed d sites apart, and j L ( r ) = l + 14 l − r + O( r ) , (8)for a bottleneck of l slow sites. As d → ∞ in (7), the current for r ≪ α, β > / j L ( r ) ≈ r +O( r ) as in the TASEP with one slow site. In (8), the current isclearly dominated by the capacity of the bottleneck and approaches r/ l → ∞ .The downside of FSMFT is that the present computing power restricts the size of thesegment that is treated exactly to ≈
20 sites, thus excluding more complex disorderconfigurations. Also, FSMFT is basically a brute force attack on the master equationand tells us little about where the coefficients e.g. in (6)-(8) come from. In thenext section we will present a general approach for computing low-order terms in theexpansion f ( C ) = f ( C ) + γf ( C ) + O( γ ), where γ stands for any of the model’shopping rates such that the current j L ( γ ) → γ → γ = r in the periodicboundaries case or γ ∈ { r, α, β } in the open boundaries case).
3. Main idea and general results
We start by writing f ( C ) as a polynomial in one of the model’s hopping rates γ , f ( C ) = X k f k ( C ) γ k , (9)where we assume f k ( C ) to be dependent on C and all other hopping rates = γ .Inserting (9) in the stationary master equation (1 a ) or (1 b ) and collecting all the isordered exclusion process in the low-current regime γ k ) gives recursion relations0 = X C ′ (1 − δ γ,W ( C ′ → C ) ) W ( C ′ → C ) f ( C ′ ) − X C ′ (1 − δ γ,W ( C → C ′ ) ) W ( C → C ′ ) f ( C ) (10)0 = X C ′ (1 − δ γ,W ( C ′ → C ) ) W ( C ′ → C ) f k ( C ′ ) + X C ′ δ γ,W ( C ′ → C ) ) f k − ( C ′ ) − X C ′ (1 − δ γ,W ( C → C ′ ) ) W ( C → C ′ ) f k ( C ) − X C ′ δ γ,W ( C → C ′ ) ) f k − ( C ) , k > . (11)It is useful to picture our system as a directed graph made of vertices (configurations)and directed edges (transitions between configurations) weighted by the hopping rates.Edges that are weighted by γ are called slow, and all the others are called regular. Aspecial role here is played by configurations that have all their outgoing edges slow. Wewill call such configurations blocked (with respect to γ ) because if the system gets intoone of these configurations and γ →
0, it will stay there forever. If B is a non-emptyset of all such configurations then terms f ( C ) are clearly absent from (10) for any C ∈ B . For example, the TASEP with a slow site and periodic boundary conditionshas only one blocked configuration with respect to r , the one in which all particles arebehind the slow site. In the open boundaries case there are two additional blockedconfigurations, one with respect to α (an empty lattice) and one with respect to β (full lattice).The fact that some terms are missing when collecting zeroth-order terms mayseem confusing, because we have to go to the next order in γ to calculate f ( C ), butwe need f ( C ) to calculate the next order terms. In the next section we show how toeliminate all first-order terms, thus ending up with a closed set equations for f ( C ), C ∈ B . We start by outlining the general procedure for calculating zeroth-order terms andthen show how it works on an explicit example. Since f ( C ) is absent from (10) forany C ∈ B , we can assume that f ( C ) = 0 for C ∈ B and then solve (10) by settingall the remaining zeroth-order terms to 0, f ( C ) = 0 , C / ∈ B. (12)Thus the existence of blocked configurations reduces the calculation of f ( C ) to asmaller set of configurations B .To get a closed set of equations for the remaining zeroth-order terms we have toinspect terms that are linear in γ ,0 = X C ′ (1 − δ γ,W ( C ′ → C ) ) W ( C ′ → C ) f ( C ′ ) + X C ′ δ γ,W ( C ′ → C ) ) f ( C ′ ) − X C ′ (1 − δ γ,W ( C → C ′ ) ) W ( C → C ′ ) f ( C ) − X C ′ δ γ,W ( C → C ′ ) ) f ( C ) . (13)It will prove useful to write (13) in the following form, f ( C ) = X C ′ ∈ B λ ( C, C ′ ) f ( C ′ ) , C / ∈ B, (14) isordered exclusion process in the low-current regime λ ( C, C ′ ) is some unknown matrix. Inserting (14) in (13), the equation for f ( C )now reads A γ ( C ) f ( C ) = X C ′′ ∈ B κ ( C, C ′′ ) f ( C ′′ ) , C ∈ B (15)where A γ ( C ) and κ ( C, C ′′ ) are given by A γ ( C ) = X C ′ δ γ,W ( C → C ′ ) , (16) κ ( C, C ′′ ) = X C ′ (1 − δ γ,W ( C ′ → C ) ) λ ( C ′ , C ′′ ) + δ γ,W ( C ′′ → C ) . (17)To find f ( C ), we have to find λ ( C, C ′ ), calculate κ ( C ′ , C ′′ ) and then solve (15).Luckily, κ ( C ′ , C ′′ ) can be found without using the expression (17). The algorithmthat we give below is essentially the same as the one we’ll use later for constructing λ ( C, C ′ ).To start with, let’s call a path P in configuration space any sequence ofconfigurations C , C , . . . , C n such that none of W ( C → C ), W ( C → C ), . . . , W ( C n − → C n ) are zero. A regular path is a path in which none of the edges is slow.With all these preliminaries, we rewrite the equation (13) for f ( C ), C ∈ B , A γ ( C ) f ( C ) = X C ′ W ( C ′ → C )(1 − δ γ,W ( C ′ → C ) ) f ( C ′ )+ X C ′ δ γ,W ( C ′ → C ) f ( C ′ ) , C ∈ B. (18)Configurations C ′ on the r.h.s. are obtained by moving one of the particles backwards .Any move to a configuration C ′ / ∈ B across a slow edge should be discarded since f ( C ′ ) = 0 for C ′ / ∈ B and the second term on the r.h.s. vanishes in that case. Nowlet’s focus on C ′ for which W ( C ′ → C ) = γ . Since we started from C ∈ B andmoved one particle backwards across a regular edge, C ′ has the following properties:(a) C ′ / ∈ B , (b) f ( C ′ ) = 0 and (c) A ( C ′ ) = 1. The last one is simply because C ′ is just one jump from C and therefore cannot have any other regular outgoing edgesexcept the one pointing towards C . Now, let’s write the equation for f ( C ′ ), A ( C ′ ) | {z } =1 · f ( C ′ ) + A γ ( C ′ ) f ( C ′ ) | {z } =0 = X C ′′ δ γ,W ( C ′′ → C ′ ) f ( C ′′ )+ X C ′′ W ( C ′′ → C ′ )(1 − δ γ,W ( C ′′ → C ′ ) ) f ( C ′′ ) . (19)The l.h.s. of (19) is exactly what we get on the r.h.s. of (18) by moving oneparticle across a regular edge from C ∈ B . We can then insert (19) in (18) and thuseliminate f ( C ′ ). The idea is to repeat this process of moving particles backwards andsubstituting f ( C ′′ ) from A ( C ′′ ) f ( C ′′ ) + A γ ( C ′′ ) f ( C ′ ) | {z } =0 = X C ′′′ δ γ,W ( C ′′′ → C ′′ ) f ( C ′′′ )+ X C ′′′ W ( C ′′′ → C ′′ )(1 − δ γ,W ( C ′′′ → C ′′ ) ) f ( C ′′′ ) , (20) isordered exclusion process in the low-current regime C ′′ / ∈ B that we reach along. Mathematically speaking, we must exhaust allbackward paths originating from C that have one slow edge at the end and all otheredges regular. Moreover, since f ( C ′ ) = 0 for all C ′ / ∈ B , we should consider onlypaths that end in configurations belonging to B .Now, consider any C ′′ / ∈ B that is reached by moving particles backwards from C without crossing a slow edge. Then if we start at C ′′ and move particles forwardwithout crossing a slow edge, we must end at C . This ensures that by moving particlesbackwards in order to eliminate any f ( C ′′ ) along the way, f ( C ′′ ) will appear exactly A ( C ′′ ) times. We can then complete the l.h.s. of (20) (since A γ ( C ′′ ) f ( C ′′ ) = 0)and substitute (20) in (18). In the end, when all paths have been exhausted and allfirst-order terms eliminated, the final result is A γ ( C ) f ( C ) = X C ′ ∈ S ( C ) f ( C ′ ) , C ∈ B (21)where S ( C ) is the set of all configurations C ′ such that (a) C ′ can be reached from C by moving particles backwards and crossing a slow edge in the last move only and(b) f ( C ′ ) = 0. (From this definition, S ( C ) ⊂ B for any C ∈ B .) Going back to (17)we have therefore proved that κ ( C, C ′ ) = ( , C ∈ B and C ′ ∈ S ( C ),0 , otherwise. • • ◦| ◦ ◦|◦ ◦ •| ◦ •| ◦ • ◦| ◦ •|◦ ◦ •| • ◦| • ◦ ◦| ◦ •|◦ • ◦| • ◦| • ◦ ◦| • ◦|◦ • •| ◦ ◦| ◦ ◦ ◦| • •|• ◦ •| ◦ ◦| rr r r
111 11 r rr
Figure 2.
A part of the graph representing the TASEP on a ring of L = 5sites with N = 2 particles and slow sites placed at i = 2 and i = 5. Blockedconfigurations are designated with ellipses and slow edges with double arrows. To illustrate this, let’s consider the TASEP on a ring with L = 5 sites, N = 2particles and two slow sites placed at i = 3 and i = 5. There are 5! / (2!3!) = 30configurations in total of which | B | = 3 are blocked, B = {◦ • •|◦ ◦| , ◦ ◦ •|◦ •| , ◦ ◦ ◦|• •|} (as in figure 1, sites in front of the walls are considered slow). A portion of the graphcontaining configuration C = ◦ ◦ •| ◦ •| is presented in figure 2. The equation for f ( ◦ ◦ •| ◦ •| ) reads2 f ( ◦ ◦ •| ◦ •| ) = f ( ◦ • ◦| ◦ •| ) + f ( ◦ ◦ •| • ◦| ) == [ f ( ◦ • ◦| ◦ •| ) + f ( ◦ • ◦| ◦ •| ) | {z } =0 ] + f ( ◦ ◦ •| • ◦| ) , (22) isordered exclusion process in the low-current regime f ( ◦ • ◦| ◦ •| ) = 0 to complete the master equationfor f ( ◦ • ◦| ◦ •| ), f ( ◦ • ◦| ◦ •| ) + f ( ◦ • ◦| ◦ •| ) = f ( • ◦ ◦| ◦ •| ) + f ( ◦ • ◦| • ◦| ) . (23)Master equation for the remaining term f ( ◦ ◦ •| • ◦| ) reads f ( ◦ ◦ •| • ◦| ) = f ( ◦ • ◦| • ◦| ) . (24)Substituting (23) and (24) in (22), the equation for f ( ◦ ◦ •| ◦ •| ) now reads2 f ( ◦ ◦ •| ◦ •| ) = f ( • ◦ ◦| ◦ •| ) + 2 f ( ◦ • ◦| • ◦| ) . (25)Equations for the terms on the r.h.s. of (25) are f ( • ◦ ◦| ◦ •| ) = f ( • ◦ ◦| • ◦| ) , f ( ◦ • ◦| • ◦| ) = f ( • ◦ ◦| • ◦| ) + f ( ◦ • •| ◦ ◦| ) , which upon substitution in (25) becomes2 f ( ◦ ◦ •| ◦ •| ) = 2 f ( • ◦ ◦| • ◦| ) + f ( ◦ • •| ◦ ◦| ) . (26)Finally, inserting2 f ( • ◦ ◦| • ◦| ) = f ( • ◦ •| ◦ ◦| ) | {z } =0 + f ( ◦ ◦ ◦| • •| ) , in (26) gives the final equation for f ( ◦ ◦ •| ◦ •| )2 f ( ◦ ◦ •| ◦ •| ) = f ( ◦ • •| ◦ ◦| ) + f ( ◦ ◦ ◦| • •| ) . (27)This daunting task that we have just performed in fact has a remarkably simpleinterpretation. To see it, let’s write equations for the remaining blocked configurations, f ( ◦ • •| ◦ ◦| ) = f ( ◦ ◦ •| ◦ •| ) , (28) f ( ◦ ◦ ◦| • •| ) = f ( ◦ ◦ •| ◦ •| ) . (29)The process described by (27)-(29) can be interpreted as follows: any particle thatjumps from a slow site immediately joins the queue in front, while at the same timethe whole queue that the particle has just left moves one step forward. Intuitively, thisis easy to understand: the limit r → r → partial exclusion process(not to be confused with partially asymmetric exclusion process). Interestingly, thepartial exclusion process has been introduced long time ago by Sch¨utz [30], but hasbeen rarely studied since [31, 32].For zeroth-order terms, our result can be summarized as follows: consider theTASEP with periodic boundary conditions on a lattice of L sites, N particles and D slow sites placed at sites k , k , . . . , k D = L (due to periodic boundary conditions, wehave a freedom to place the last slow site at the end). Then f ( C ) for C ∈ B solves isordered exclusion process in the low-current regime Figure 3.
Mapping the TASEP with binary disorder in the limit r → the master equation of the totally asymmetric partial exclusion process on a latticeof D sites, where each site i = 1 , . . . , D can hold at most k i − k i − particles, where k = 1 (see figure 3a).One striking thing about this result is that κ ( C, C ′ ) and therefore any f ( C )are completely independent of the other hopping rates (if they exist) and insteaddepend only on the “connectivity” of blocked configurations. We will go back to thisobservation more in section 5 where we discuss two or more different slow hoppingrates. For now, let’s just focus on what this means for the binary disorder in the openboundaries case. In the open boundaries case, all blocked configurations (with respectto r ) have the first compartment occupied by particles and the last one empty. If wehave a lattice of L sites and D slow sites placed at k , . . . , k D , then f ( C ) for C ∈ B solves master equation of a totally asymmetric partial exclusion process on a latticeof D − i = 1 , D − k i +1 − k i particles. Atthe boundaries, particles jump into the first compartment at rate 1 if it’s not full andleave the lattice from the last compartment at rate 1 (see figure 3b). Thus the ratesat which particles are exchanged with reservoirs are always maximal, i.e. equal to 1.For a small number of blocked configurations we may hope to solve (21)analytically (as in section 4) or numerically (even for L large). As the number ofblocked configurations increases, since the exact solution for the steady state of thetotally asymmetric partial exclusion process is not known, we may end up with aproblem no less harder that the one we started with. In the next section we give ageneral recipe for how to calculate first-order terms f ( C ) if we can somehow solve(21). To calculate f ( C ), C / ∈ B , we have to determine λ ( C, C ′ ) in (14). The equation for f ( C ) for any C / ∈ B reads f ( C ) = 1 A ( C ) X C ′′ ∈ B δ γ,W ( C ′′ → C ) f ( C ′′ )+ X C ′′ W ( C ′′ → C ) A ( C ) (1 − δ γ,W ( C ′ → C ) ) f ( C ′′ ) , (30) isordered exclusion process in the low-current regime f ( C ′′ ) = 0 for all C ′′ / ∈ B . Leaving allzeroth-order terms intact, we insert expressions like (30) recursively for the remainingfirst-order terms, until we are left with zeroth-order terms only. Again, we end up withas many zeroth-order terms as there are blocked configurations that can be reachedfrom C by moving particles backwards but crossing a slow edge in the last move only.Recalling that the set of all such configurations is S ( C ), let’s index a backward pathfrom C to any C ′ ∈ S ( C ) with P C,C ′ = C, C , C , . . . , C n , C ′ ( n can, of course, varyfrom path to path). For any given C , λ ( C, C ′ ) is then given by λ ( C, C ′ ) = X P C,C ′ W ( C → C ) A ( C ) W ( C → C ) A ( C ) · . . . · W ( C n − → C n − ) A ( C n − ) 1 A ( C n ) , (31)for C ′ ∈ S ( C ) and is 0 for C ′ / ∈ S ( C ).To calculate f ( C ) for C ∈ B , we use the same recipe as for f ( C ). Starting from C ∈ B , we look for paths P C,C ′ from C to any C ′ such that (a) C ′ can be reached from C by moving particles backwards and crossing a slow edge in the last move only and(b) f ( C ′ ) = 0. Let’s call S ( C ) the set of all such configurations and V ( C ) the setof all configurations that are visited in going from C to all C ′ ∈ S ( C ) (not including C ′ and C ). Starting from the equation for f ( C ), C ∈ B , A γ ( C ) f ( C ) = X C ′′ W ( C ′′ → C )(1 − δ γ,W ( C ′ → C ) ) f ( C ′′ )+ X C ′′ δ γ,W ( C ′′ → C ) f ( C ′′ ) , C ∈ B, (32)the idea is to eliminate W ( C ′′ → C ) f ( C ′′ ) by noting that A ( C ′′ ) | {z } =1 f ( C ′′ ) + A γ ( C ′′ ) f ( C ′′ ) = X C ′′′ δ γ,W ( C ′′′ → C ′′ ) f ( C ′′′ )+ X C ′′′ W ( C ′′′ → C ′′ )(1 − δ γ,W ( C ′′′ → C ′′ ) ) f ( C ′′′ ) , (33)This time, however, f ( C ′′ ) on the left is not necessarily zero. To eliminate f ( C ′′ )in (32), we add A γ ( C ′′ ) f ( C ′′ ) to both sides of (32) and substitute A ( C ′′ ) f ( C ′′ ) + A γ ( C ′′ ) f ( C ′′ ) with (33). By repeating the process of moving particles backwards andeliminating any A ( C ′′ ) f ( C ) by adding and subtracting A γ ( C ′′ ) f ( C ′′ ), we finally geta closed system of equations A γ ( C ) f ( C ) = X C ′ ∈ S ( C ) f ( C ′ ) − X C ′′ ∈ V ( C ) A γ ( C ′′ ) f ( C ′′ ) , C ∈ B. (34)Since S ( C ) also contains blocked configurations, we can rewrite (34) as A γ ( C ) f ( C ) − X C ′ ∈ S ( C ) ∩ B f ( C ′ ) = h ( C ) , C ∈ B, (35)where h ( C ) is given by h ( C ) = X C ′∈ S C ) C ′ / ∈ B f ( C ′ ) − X C ′ ∈ V ( C ) A γ ( C ′ ) f ( C ′ ) , C ∈ B. (36)If it weren’t for the h ( C ), (35) would be just the same as in (21). Because of thenon-zero terms on the right, (35) no longer describes a stochastic process as in (21).It may also be much difficult to solve (34) than (21) if B is large. isordered exclusion process in the low-current regime In principle, the same procedure can be applied to higher order terms. Let’s denotewith S k ( C ) the set of all configurations C ′ such that (a) C ′ can be reached from C by moving particles backwards and crossing a slow edge in the last move only (b) f k ( C ′ ) = 0. Then for C / ∈ B we have f k ( C ) = X C ′ ∈ S k − ( C ) λ ( C, C ′ ) f k − ( C ′ ) − A γ ( C ) A ( C ) f k − ( C ) , C / ∈ B (37)where λ ( C, C ′ ) = 0 for C ′ / ∈ S k − ( C ), and is given by (31) for C ′ ∈ S k − ( C ). Theadditional term on r.h.s. of (37) was not present for k = 1 only due to the fact that f ( C ) = 0 for C / ∈ B .Now, for C ∈ B , let’s denote with V k ( C ) the set of all configurations that arevisited in going from C to all C ′ ∈ S k ( C ). To find f k ( C ), C ∈ B , we have to solve thefollowing system of equations, A γ ( C ) f k ( C ) = X C ′ ∈ S k ( C ) f k ( C ′ ) − X C ′′ ∈ V k ( C ) A γ ( C ′′ ) f k ( C ′′ ) , C ∈ B. (38)As for k = 1, we can rewrite (38) as A γ ( C ) f k ( C ) − X C ′ ∈ S k ( C ) ∩ B f k ( C ′ ) = h k ( C ) , C ∈ B, (39)where h k ( C ) is given by h k ( C ) = X C ′∈ Sk ( C ) C ′ / ∈ B f k ( C ′ ) − X C ′ ∈ V k ( C ) A γ f k ( C ′ ) , C ∈ B. (40)For the reasons evident in the following section, going beyond linear order becomeshighly non-trivial for larger systems. In the rest of this paper we consider thereforeonly zeroth- and first-order terms in some simple disorder configurations. Insight thatthis approach gives us for general configurations of disorder is discussed in section 5.
4. Examples
The TASEP on a ring with a slow site placed at site L has only one blocked configuration C P with respect to hopping rate r <
1, the onein which all particles are immediately behind the slow site. It follows then that theequation (38) is trivially solved for any k ≥
0, i.e. f k ( C P ) = const. ≡ k ≥ f ( C ) = 0 for any C / ∈ B = { C P } , we can write f ( C ) using thedelta Kronecker function, f ( C ) = δ C,C P . To calculate the small- r expansion of thecurrent j L ( r ), we can choose j L = h rτ L ( C )[1 − τ ( C )] i , so that j L ( r ) = a b r − b − a b r + O( r ) , (41)where a k and b k are given by a k = X C τ L ( C )[1 − τ ( C )] f k ( C ) , b k = X C f k ( C ) , k ≥ . isordered exclusion process in the low-current regime a = b = 1 and therefore j L ( r ) = r + O( r ). Tocalculate the second-order term in r , we have to find f ( C ), i.e. λ ( C, C P ) because of f ( C ) = X C ′ λ ( C, C ′ ) f ( C ′ ) = X C ′ λ ( C, C ′ ) δ C ′ ,C P = λ ( C, C P ) . The construction of λ ( C, C ′ ), as explained in section 3.2, tells us to look forconfigurations C such that the blocked state C P is reached by moving particlesbackwards from C , provided the slow edge is crossed only by entering C P . It iseasy to see that any such C must have a particle displaced from the queue (figure 4).Thus the configurations giving non-zero first-order terms are particle-hole excitationsof the blocked state C P . Figure 4.
Configurations that have non-zero first-order term f ( C ) = 0 haveeither (a) one particle outside the queue and one hole inside the queue or (b) oneparticle taken out of the queue and added to its end. To calculate a and b , it proves useful to define a set of configurations J L , J L ≡ { C | τ L ( C ) = 1 , τ ( C ) = 0 } . In other words, J L is simply the set of all configurations such that rτ L ( C )[1 − τ L ( C )] =0. Using this definition, second-order term in (41) can be rewritten as c = − b − a b = − X C / ∈J L λ ( C, C P ) . This expression slightly simplifies the calculation of c , as we must calculate λ ( C, C P )only for C / ∈ J L , and not for all C having non-zero f ( C ). To calculate the matrixelement λ ( C, C P ) for C / ∈ J L , we can use the expression (31) derived in the previoussection. All possible C / ∈ J L having non-zero f ( C ) will either have a particle at site k = 1 , . . . , L − N − k = L , or particles both at k = 1 and L witha hole at site k = L − N + 1 , . . . , L −
1. Let’s first consider configurations
C / ∈ J L such that the particle outside the queue is placed at k = 1 , . . . , L − N − L . If k = L − N − A ( C ′ ) = 2 for any C ′ visited in going from C to C P . The matrixelement λ ( C, C P ) for such C is therefore given by λ ( C, C P ) = (cid:18) (cid:19) k , k = 1 , . . . , L − N − , k = L. If k = L − N , then A ( C ) = 1 and the rest of the configurations in going from C to C P have A ( C ′ ) = 2. This gives λ ( C, C P ) = (cid:18) (cid:19) L − N − , k = L − N, k = L. isordered exclusion process in the low-current regime k = 1 and k = L − N + 1 , . . . , L − C P , A is always 2 and therefore λ ( C, C P ) is given by λ ( C, C P ) = (cid:18) (cid:19) L − k +1 , k = 1 , k = L − N + 1 , . . . , L − . If k = 1 and k = L − N , then A ( C ) = 1 and so λ ( C, C P ) is given by λ ( C, C P ) = (cid:18) (cid:19) N , k = 1 , k = L − N. Summing all four contributions gives c = − L − N − X k =1 (cid:18) (cid:19) k − (cid:18) (cid:19) L − N − − L − X k = L − N +1 (cid:18) (cid:19) L − k +1 − (cid:18) (cid:19) N == − " − (cid:0) (cid:1) L − N − − − (cid:18) (cid:19) L − N − − " − (cid:0) (cid:1) N +1 − − − (cid:18) (cid:19) N = − . Our method thus gives us j L ( r ) up to O( r ) j L ( r ) = r − r + O( r ) , r ≪ , which was first calculated by Janowsky and Lebowitz [26]. Unfortunately, this is asfar as we can go without much effort. To calculate the next-order terms, we wouldhave to explore paths starting from configurations having either two particles outsidethe queue and a hole inside the queue, or one particle outside the queue and two holesinside the queue. However, tracking movement of two particles or two holes is nolonger trivial because of the exclusion, and therefore it becomes increasingly difficult,albeit possible, to calculate λ ( C, C ′ ). Open boundary conditions.
Now let’s consider open boundaries case with slow siteplaced at site k . Here we can expand the current j L ( α, β, r ) in any of the hoppingrates α , β or r . The simplest case to consider is when none of the two remaininghopping rates are equal to the one that we are expanding in. In that case B has onlyone configuration: an empty chain if expanding around α = 0, a fully occupied chainif expanding around β = 0 and a semi-full chain with particles behind the slow site ifexpanding around r = 0. When expanding j L around r = 0, the calculation is similarto the one for the periodic case and in fact gives the same result, j L ( α, β, r ) = r − r + O( r ) , r ≪ α, β. That j L does not depend on α nor β in the small- r limit was recognized long timeago by Janowsky and Lebowitz [26] by studying α = β case. Note also that j L in thesmall r limit does not depend on the position k of the slow site either. isordered exclusion process in the low-current regime α = 0 or β = 0, the calculation is even simpler. There is only one blockedconfiguration with respect to α , and that is an empty chain. This gives us immediately a = b = 1, i.e. c = 1. The expression for c reads c = a b − c b b = − X C / ∈J f ( C ) , where J = { C | τ ( C ) = 1 } . There is only one configuration C ∈ J that has f ( C ) = 0, and that is the configuration with a particle at site 1, which has f ( C ) = 1.A similar calculation can be made for the expansion around β = 0. For the first twocoefficient, the final result is thus the same as for the pure TASEP, j L ( α, β, r ) = α − α + O( α ) , α ≪ β, rj L ( α, β, r ) = β − β + O( β ) , β ≪ α, r Again we see that the coefficients are pure numbers and do not depend on otherhopping rates, nor on the position of the slow site.When α or β is equal to r we immediately notice that | B | (the number of elementsin B ) is greater than 1, and therefore solving (35) cannot be avoided. It is the samedifficulty that we are going to encounter when dealing with more than one slow sitein the following sections. Let’s consider α = r ≪ β case first. Blocked configurationscan be described as having a queue behind the slow site and an empty segment infront of it. Compared to the r ≪ α, β case, the queue is now no longer of size k (i.e.occupying the whole segment behind the slow site), but can be of any size 0 , . . . , k giving | B | = k + 1. Let’s denote configurations belonging to B with C m , where m = 0 , . . . , k is the size of the queue behind the slow site placed at k . The equationsfor f ( C m ) are easily generated using (21) giving f ( C ) = f ( C )2 f ( C m ) = f ( C m − ) + f ( C ( m + 1)) , m = 1 , . . . , k − f ( C k ) = f ( C k − ) . According to our interpretation using partial exclusion process, this corresponds tohaving a single site with capacity k which exchanges particles with two reservoirs atrates α = β = 1. The solution to the system above is simply f ( C m ) = const.. If wechoose g ( C ) = rτ k [1 − τ k +1 ], then a = k and b = k +1 so that c = a /b = k/ ( k +1).A much more involved calculation is required to get the second-order terms. Here westate the final result leaving the details of this calculation to Appendix B j L ( α, β, r ) = kk + 1 r − ( k − k + 8)4( k + 1) r + O( r ) , α = r ≪ β. (42)How well truncating the expression (42) at second-order approximates j L ( r ) ispresented in figure 5, where (42) is compared to j L ( r ) obtained from Monte Carlosimulations on a lattice of L = 1000 sites for β = 1 and (a) k = 3 and (b) k = 10.Using the particle-hole symmetry τ i ↔ − τ L − i +1 , α ↔ β , k ↔ L − k , we canalso get the expansion for β = r ≪ α which reads j L ( α, β, r ) = L − kL − k + 1 r − ( L − k − L − k ) + 8]4( L − k + 1) r + O( r ) , β = r ≪ α. (43) isordered exclusion process in the low-current regime r j L (r , α , β ) sim.expr. (42)(a) r j L (r , α , β ) sim.expr. (42)(b) Figure 5.
Current j L ( r, α, β ) as a function of α = r obtained by Monte Carlosimulations (——) on a lattice of L = 1000 sites with β = 1, compared to theexpression (42) truncated at the second-order (– – –) for (a) k = 3 and (b) k = 10. In the most complicated case when α = β = r , which corresponds to the partialexclusion process with two sites having capacities k and L − k , unfortunately we werenot able to find even f ( C ), i.e. to solve (21) for general k . This already clearlydemonstrates that severe difficulties are to be expected whenever | B | is not small. We next consider two slow sites placed at k = L − d and k = L on a ring of L sites and N particles. We will further assume that d < N < L − d ‡ , which means the number of particles in the segment i = L − d +1 , . . . , L can take values 0 , . . . , d . The number of blocked configurations is then | B | = d +1. Thecorresponding partial exclusion process consists of two sites with periodic boundaryconditions, which has a simple steady state with all f ( C ) = const.. Choosingagain j L = h rτ k ( C )[1 − τ k +1 ( C )] i gives immediately a = d and b = d + 1, i.e. c = d/ ( d + 1), as in (7). The calculation of the second-order term is very similar tothe α = r case with a single slow site. The final result for j L ( r ) up to O( r ) is j L ( r ) = dd + 1 r − ( d − d + 4)2( d + 1) r + O( r ) . (44)Notice that as d → ∞ , j ∞ ( r ) = r − r / r ), as in the TASEP with a single slowsite. A comparison of (44), truncated at the second-order, with j L ( r ) obtained fromMonte Carlo simulations ( L = 1000, N = 500) is presented in figure 6 for (a) d = 3and (b) d = 10. Open boundary conditions.
For r ≪ α, β , the result is the same as in (44), j L ( r ) = dd + 1 r − ( d − d + 4)2( d + 1) r + O( r ) , r ≪ α, β. (45)Other cases, i.e. when α or β are equal to r , are more difficult to deal with as theanalytical solution to (21) is generally not known. ‡ Other values of N can be explored as well, but we are here mainly interested in large N and small d . isordered exclusion process in the low-current regime r j L (r) sim.expr. (44)(a) r j L (r) sim.expr. (44)(b) Figure 6.
Current j L ( r ) as a function of r obtained by Monte Carlo simulations(——) on a ring of L = 1000 sites and N = 500 particles, compared to theexpression (44) truncated at the second-order (– – –) for (a) d = 3 and (b) d = 10. Finally, we mention the case of all slow sites being clustered in a bottleneck , which waspreviously studied in [19]. Here the set B is 2 l § , where l is the number of slow sites.Using the previously developed mapping to the partial exclusion process in the limit r →
0, the periodic boundaries case becomes equivalent to the pure TASEP with onelarge but finite reservoir. If we further assume that l < N < L − l so that the finitereservoir is never empty nor fully occupied, f ( C ) is given by the exact solution ofthe pure TASEP of size l − α = β = 1. Using the knownsolution of the pure TASEP with open boundaries [5,6], the current j L ( r ) up to O( r )reads j L ( r ) = l + 14 l − r + O( r ) . (46)The same result applies to the open boundaries case for r ≪ α, β , which wasconjectured k in [19]. Unfortunately, due to large | B | we were not able to find thesecond-order term for general l . (The l = 2 case is already covered by the previousexample when the distance between two slow sites is 1.)
5. Further applications
The approach developed in section 3 is general and can be applied to any unidirectional driven diffusive system in which one can identified blocked configurations with respectto one of its hopping rates. (Here the unidirectional hopping is necessary to relate λ ( C, C ′ ) to weighted backwards paths in the configuration space.) The success of thisapproach will mostly depend on our ability to solve (21) and (35). If B = { C P } , theseare trivially solved and the main problem is to find λ ( C, C ′ ). For | B | >
1, we maytry to solve (21) analytically for small values of | B | or on a computer for larger valuesusing the analogy with the partial exclusion process.As a further application that goes beyond binary disorder discussed so far, herewe mention some results for the slow hopping rates that are not necessary all equal. § In the periodic boundaries case that is true provided l < N , which is the case we consider. k Although the notion that the bottleneck behaves as a small TASEP within a big one is not surprisingand new, it is unclear to us whether the authors of [19] were actually aware that this picture is exactin the limit r → isordered exclusion process in the low-current regime et al [2, 3], whointroduced the TASEP to model the process of translation in protein biosynthesis. Ina simplified description of translation, a ribosome binds to mRNA and moves along itcodon by codon translating the mRNA sequence into sequence of specific amino acids.At each step, the corresponding amino acid is transported to the ribosome by tRNA.The availability (abundance) of tRNA is thus believed to be responsible for the timescale on which the ribosome moves along the mRNA. Codons with lower concentrationsof corresponding tRNA will locally suppress ribosome motion across them, acting thusas slow sites. An important question, explored extensively in [33], is how are proteinproduction rates correlated with specific sequences of codons. Translated into theTASEP, the question is to determine the limiting factor for the current with respectto the strength and the positions of slow sites. Here we discuss some immediate resultsthat stem from our approach applied to the non-binary disorder. We will not includeanother important ingredient for modelling translation - the fact that ribosomes bindto approximately 12 codon sites - as it become technically difficult to do it in ourapproach due to the exclusion.For two slow sites with hopping rates r = r our approach readily gives j L ( r ) = r ∗ − r ∗ + O( r ∗ ) , r ∗ = min { r , r } ≪ α, β (47)This result can be easily generalized to arbitrary number M of slow sites providedthey all have different hopping rates j L ( r ) = r ∗ − r ∗ + O( r ∗ ) , r ∗ = min { r i | i = 1 , . . . , M } ≪ α, βr i = r j , ∀ i, j, i = j If two slow sites however share the same hopping rate r , then (45) applies providedall other slow hopping rates (including α and β ) are mutually different and not equalto r . What this tells us generally is that the low-current regime of the TASEP withsitewise disorder depends only on the current-minimizing subset of slow sites with equalhopping rates, regardless of other slow sites .Here we make a modest attempt to test this idea by simulating the TASEP with15 randomly distributed slow sites of which 10 have rates r (type 1) and 5 haverates r > r (type 2) on a lattice of 1000 sites with α = β = 1. Figure 7 comparescurrent j L ( r , r , α, β ) as a function of r for two values of r , one with r = 0 . r = 1 (only type 1 present). Our data shows nosignificant difference between these two currents for small r < r = 0 .
3. Notice also agap between the current with type 2 slow sites only (– – –) and the point r = r = 0 . E. Coli [34]) and compare it to the approximate theory of estimatingcurrents for the disordered TASEP developed in [33]. isordered exclusion process in the low-current regime j L (r , r , α , β ) r = 0.3r = 1.0r = 1.0, r = 0.3 Figure 7.
Current j L ( r , r , α, β ) obtained by Monte Carlo simulations on alattice of L = 1000 sites with 15 slow sites of two types, 10 slow sites with rates r (type 1) and 5 slow sites with rates r (type 2), plotted as a function of r forfixed r = 0 . r = 1 ( ◦ ). Setting r = 1 in the latter case meansthat only type 1 slow sites are present. Broken line (– – –) corresponds to r = 1and r = 0 . α = β = 1. Slow sites of type 1were placed at i = 27 , , , , , , , , ,
888 and those of type 2at i = 156 , , , ,
6. Conclusion
The matrix-product ansatz and mean-field approximation are both powerful ana-lytical approaches for studying driven diffusive systems, the ASEP in particular.Unfortunately, in some cases such as site-wise disorder, it is not known how toapply the matrix-product ansatz and mean-field approximation is mostly reduced tonumerical studies. In this article we showed how to access some exact steady-stateproperties of the TASEP with site-wise disorder in the low-current regime, i.e. whenone of the hopping rates is small.Our approach is based on a simple fact, proved in Appendix A, that the steady-state (non-normalized) weights are polynomials in hopping rates. Using this fact thesteady-state average of any physical observable (e.g. current) can be expanded inone of the small hopping rates with coefficients that obey a specific set of equations.While our approach is not restricted particularly to the TASEP, it requires that we canidentify what we call blocked configurations (configurations that the system freezesinto if one of the hopping rates is set to zero). In that case the zeroth-order coefficientsare non-zero only for blocked configurations, which drastically reduces the numberof unknowns. For the TASEP with binary disorder where all slow sites share thesame hopping rate r <
1, we show that the zeroth-order coefficients in the small r expansion are in fact steady-state weights of another (but rarely studied) processcalled partial exclusion process in which more than one particle per site is allowed.This mapping, which is exact in the limit r →
0, is made by replacing all slow siteswith boxes of capacities equal to the distances between neighbouring slow sites. Asimple interpretation of this result is that the limit r → r and 1, so that any particle thatjumps across a slow site immediately joins a queue in the front. It is remarkable (anddiscouraging at the same time) that the TASEP with binary disorder even in this isordered exclusion process in the low-current regime r expansion of the steady-statecurrent for particular disorder configurations previously studied in [19, 26]. As thenumber of blocked configurations increases it becomes increasingly difficult to followthis programme analytically. However, because of the mapping to the partial exclusionprocess the reduction of the unknowns is still huge and allows us potentially to use acomputer instead, even for large lattices.Our approach can readily be applied to non-binary disorder with more than onetype of slow rates, which is relevant for modelling protein synthesis. Remarkably,the lowest-order coefficients we get by expanding in one of the slow hopping ratesdo not depend on the other hopping rates. In other words, the low-current regime ofthe TASEP with site-wise disorder depends only on the current-minimizing set of slowsites with equal hopping rates, regardless of other slow sites. Once this subset is found(which may be a hard problem in itself), one can work with binary disorder only andgo from there using either the approach developed here or using e.g. phenomenologicalapproach that looks for the largest cluster of slow sites [27, 28].Finally, we mention that our approach can be applied to any other driven diffusivesystem with particles hopping unidirectionally, provided we can identify blockedconfigurations with respect to one of the model’s hopping rates. In the networklanguage, unidirectional hopping is necessary to avoid loops that may exist even ifone of the hopping rates is equal to zero. For example, our approach does not workfor Langmuir kinetics [20] or a multi-lane exclusion process where particles can changelanes [22]. However, it could be useful for studying more complex lattice geometrieswithout resorting to mean-field approximation or even to understand why in someinstances the mean-field approximation is satisfactory. Acknowledgments
This work has been funded in part by EPSRC under grant number EP/J007404/1.Early part of this work has been supported by the Croatian Ministry of Science,Education and Sports under grant number 035-0000000-3187.
Appendix A. A formal solution of the stationary master equation
Consider an ergodic continuous-time Markov jump process with transition rates W ( i → j ). Master equation for the steady-state probabilities P i is then given by0 = X j W ( j → i ) P j − X j W ( i → j ) P j , i = 1 , . . . , N , (A.1)where N is the total number of states. Equation (A.1) can be written in a morecompact form by introducing a matrix L ij , X j L ij P j = 0 , (A.2) L ij = W ( j → i ) − δ ij X k W ( i → k ) , (A.3) isordered exclusion process in the low-current regime L is a left stochastic matrix, which means that P i L ij = 0 for any j .An important property of a left stochastic matrix is that it has all of its cofactors C ij = ( − ij M ij independent of i . (Here M ij is defined to be the determinantof the submatrix of L obtained by removing i -th row and j -th column from L .)Since we assumed that the process is ergodic, the stationary state satisfying equation P j L ij P j = 0 is unique and non-trivial, which means that det L = 0. By using theLaplace expansion for det L and the aforementioned fact that C jk = C kk we get,det L = 0 = X k L ik C jk X k L ik C kk , (A.4)which means that P i must be proportional to C ii . Since C ii is the determinant ofa matrix whose matrix elements are linear combinations of the transition rates, weconclude that P i must be a polynomial in all the transitions rates present in (A.1). Appendix B. Second-order term c for α = r in the TASEP with a singleslow site The starting point is the expression for c c = a b − c b b = ( a − b ) c b + a c − b = − k ( k + 1) X C / ∈J k f ( C ) + 1( k + 1) X C ∈J k f ( C ) , where J k = { C | τ k ( C ) = 1 , τ k +1 ( C ) = 0 } . Summations in (B.1) can be furtherseparated into X C / ∈J k f ( C ) = X C / ∈J kC ∈ B f ( C ) + X C / ∈J kC / ∈ B f ( C ) (B.1) X C ∈J k f ( C ) = X C ∈J kC ∈ B f ( C ) + X C ∈J kC / ∈ B f ( C ) . (B.2)The first sum in (B.1) can be greatly simplified by that fact that there is only one C ∈ B for which g ( C ) = rτ k ( C )[1 − τ k +1 ( C )] = 0, and that is an empty lattice, X C / ∈J kC ∈ B f ( C ) = f ( C ) . To find f ( C ) for C ∈ B , we have to solve (35), which in this case reads f ( C ) = f ( C ) + h ( C )2 f ( C m ) = f ( C m − ) + f ( C m +1 ) + h ( C m ) , m = 1 , . . . , k − f ( C k ) = f ( C k − ) + h ( C k ) , (B.3)where h ( C m ) is given by (36). Luckily, this system admits closed expression for f ( C m ) which is f ( C m ) = f ( C ) − m X i =1 i · h ( C m − i ) , m = 1 , . . . , k. isordered exclusion process in the low-current regime X C ∈J kC ∈ B f ( C ) = k · f ( C ) − k X m =1 m X i =1 i · h ( C m − i )= k · f ( C ) − k − X i =0 ( k − i )( k − i + 1) h ( C i ) . Altogether, the expression for c can be written as c = 1( k + 1) X C ∈J kC / ∈ B f ( C ) − k X C / ∈J kC / ∈ B f ( C ) − k − X i =0 ( k − i )( k − i + 1) h ( C i ) . (B.4)Now, let’s go back to the expression (36) for h ( C ). For a given C m , the set S ( C m ) ∩ B consists of all configurations C having f ( C ) = 0 that can be reachedfrom C m by moving particles backwards and hitting a slow edge in the last move only.Starting from C m (with m particles in the segment i = 1 , . . . , k by the definition), theresulting C will have either m − m + 1 particles in the same segment dependingon whether we crossed the left boundary or site k in the last move, respectively. Theresulting C will necessarily have a hole at site 1 in the former case and a particle-holepar at k, k + 1 in the latter case. Thus by defining J = { C | τ ( C ) = 0 } in the samespirit as we defined J k , we can write X C ∈ S C C / ∈ B f ( C ) = X C ∈J kC / ∈ B f ( C ) · { k X i =1 τ i ( C ) = 1 } X C ∈ S Cm ) C / ∈ B f ( C ) = X C ∈J C / ∈ B f ( C ) · { k X i =1 τ i ( C ) = m − } + X C ∈J kC / ∈ B f ( C ) · { k X i =1 τ i ( C ) = m + 1 } , m = 1 , . . . , k − X C ∈ S Ck − C / ∈ B f ( C ) = X C ∈J C / ∈ B f ( C ) · { k X i =1 τ i ( C ) = k − } X C ∈ S Ck ) C / ∈ B f ( C ) = X C ∈J C / ∈ B f ( C ) · { k X i =1 τ i ( C ) = k − } where { X = x } is an indicator function, { X = x } = 0 if X = and 1 if X = x .Similarly, V ( C m ) in (36) consists of all configurations C having f ( C ) = 0 andprecisely m particles in the segment i = 1 , . . . , k . We can further divide this set intosubsets depending on whether C belongs to J ∩ J k (for which A r ( C ) = 2), J butnot J k and vice versa (for which A r ( C ) = 1) or to none of these sets (for which A r ( C ) = 0). The second sum in (36) can be then written as isordered exclusion process in the low-current regime X C ∈ V ( C ) f ( C ) = X C ∈J C / ∈ B f ( C ) · { k X i =1 τ i ( C ) = 0 } X C ∈ V ( C m ) f ( C ) = X C ∈J C / ∈ B f ( C ) · { k X i =1 τ i ( C ) = m } + X C ∈J kC / ∈ B f ( C ) · { k X i =1 τ i ( C ) = m } , m = 1 , . . . , k − . To ease the notation let’s introduce a (0)1 ( m ) and a ( k )1 ( m ) defined as a (0)1 ( m ) = X C ∈J C / ∈ B f ( C ) · { k X i =1 τ i ( C ) = m } , m = 0 , . . . , k, (B.5) a ( k )1 ( m ) = X C ∈J kC / ∈ B f ( C ) · { k X i =1 τ i ( C ) = m } , m = 0 , . . . , k. (B.6)From here it is easy to see that a (0)1 ( k ) = 0 , a ( k )1 (0) = a ( k )1 ( k ) = 0 . We can now write h ( C m ) as h ( C ) = a ( k )1 (1) − a (0)1 (0) h ( C m ) = a ( k )1 ( m + 1) − a ( k )1 ( m ) + a (0)1 ( m − − a (0)1 ( m ) , m = 1 , . . . , k − ,h ( C k − ) = a (0)1 ( k − − a (0)1 ( k − − a ( k )1 ( k −
1) (B.7)Using definition for a ( k )1 ( m ) we can also write X C ∈J kC / ∈ B f ( C ) = k − X m =1 a ( k )1 ( m ) . (B.8)While each a (0)1 ( m ) and a ( k )1 ( m ) can be calculated explicitly using (31), we canfurther simplify calculation by showing that we must only calculate their difference a (0)1 ( m ) − a ( k )1 ( m ). Using (B.7), we can show after some algebra that12 k − X i =0 ( k − i )( k − i + 1) h ( C i ) = − ka (0)1 (0) + k − X m =1 a ( k )1 ( m )+ k − X m =1 ( k − m )[ a ( k )1 ( m ) − a (0)1 ( m )] . (B.9)Using (31) it is also straightforward to calculate the second sum in c . Depending onthe number of particles m in the segment i = 1 , . . . , k we can show that X C / ∈J kC / ∈ B f ( C ) = a (0)1 (0) + 52 ( k − . (B.10) isordered exclusion process in the low-current regime c = 0 , k = 1 c = 1( k + 1) ( − k ( k −
1) + k − X m =1 ( k − m ) h a (0)1 ( m ) − a ( k )1 ( m ) i) , k ≥ . The difference a (0)1 ( m ) − a ( k )1 ( m ) can be further written as a (0)1 ( m ) − a ( k )1 ( m ) = X C ∈J ,C / ∈J kC / ∈ B f ( C ) · { C | k X i =1 τ i ( C ) = m }− X C / ∈J ,C ∈J kC / ∈ B f ( C ) · { C | k X i =1 τ i ( C ) = m } . (B.11)Using (31) we can easily find that k − X m =1 X C ∈J ,C / ∈J kC / ∈ B ( k − m ) f ( C ) · { C | k X i =1 τ i ( C ) = m } = ( k − k − (cid:18) (cid:19) k − To compute the second sum in (B.11) we must first identify configurations
C / ∈ J and C ∈ J k that give f ( C ) = 0. Since sites 1 and k must be occupied and site k + 1must be empty, we have only one option for m = 2 , . . . , k − m = k −
1. For m = 2 , . . . , k −
2, the only way we can reach a blocked configurationby moving particle backwards is to start from a configuration that has a queue behindthe slow site and a particle at site 1. The blocked configuration with m − λ ( C, C m − ) = 1.We can thus write k − X m =1 X C / ∈J ,C ∈J kC / ∈ B f ( C ) · { C | k X i =1 τ i ( C ) = m } = k − X m =2 ( k − m ) · X C / ∈J ,C ∈J kC / ∈ B f ( C ) · { C | k X i =1 τ i ( C ) = k − } (B.12)For m = k − i = 1 , . . . , k , so inaddition to moving backwards a particle at site 1 we can reach a blocked configurationwith m = k particles by moving a particle from the segment i = k + 1 , . . . , L or fromthe right reservoir. Computing λ ( C, C k ) is then straightforward but more complicated,because we must count all the ways in which a particle and a hole can both move until C k is reached. Instead of calculating each λ ( C, C k ) individually and then make thesummation, we will use the following trick. Let’s denote with C k ,k configurationsthat have a hole placed at k = 2 , . . . , k − k = 2 , . . . , L − k +1,where both k and k are measured relative to the site k . Here k = L − k + 1 denotesa particle in the right reservoir, i.e. a configuration with no particles in the segment i = k + 1 , . . . , L . For a given k = 2 , . . . , k − f ( C k ,k ) read isordered exclusion process in the low-current regime f ( C k ,k ) = f ( C k − ,k ) + f ( C k ,k − ) , k = 2 , . . . , L − k − β ) f ( C k ,L − k ) = f ( C k − ,k ) + f ( C k ,L − k − ) f ( C k ,L − k +1 ) = f ( C k − ,L − k +1 ) + βf ( C k ,L − k )Summing all the equations for a fixed k = 2 . . . , k − L − k +1 X k =2 f ( C k ,k ) = L − k +1 X k =2 f ( C k − ,k ) + f ( k , . The advantage here is that we is no dependence on β . Solving this is as a recursionrelation in k we get L − k +1 X k =2 f ( C k ,k ) = L − k +1 X k =2 f ( C ,k ) + k X i =2 f ( C i, )Both sums on the r.h.s. are now simple to calculate using (31) because either particleor hole is always fixed. The final result is k − X k L − k +1 X k =2 f ( C k ,k ) = k −
52 + (cid:18) (cid:19) k − . (B.13)Inserting (B.13) in (B.12) and then in the expression for c we finally get c = − ( k − k + 8)4( k + 1) , k ≥ . (B.14) References [1] Marro J and Dickman D 1999
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