arXiv:1010.5741v2 [cond-mat.stat-mech] 18 Nov 2010
Dynamical Transition in the Open-boundary Totally Asymmetric Exclusion Process A. Proeme, R. A. Blythe and M. R. Evans SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom E-mail:
[email protected],
[email protected],
[email protected] Abstract. We revisit the totally asymmetric simple exclusion process with open boundaries (TASEP), focussing on the recent discovery by de Gier and Essler that the model has a dynamical transition along a nontrivial line in the phase diagram. This line coincides neither with any change in the steady-state properties of the TASEP, nor the corresponding line predicted by domain wall theory. We provide numerical evidence that the TASEP indeed has a dynamical transition along the de Gier–Essler line, finding that the most convincing evidence was obtained from Density Matrix Renormalisation Group (DMRG) calculations. By contrast, we find that the dynamical transition is rather hard to see in direct Monte Carlo simulations of the TASEP. We furthermore discuss in general terms scenarios that admit a distinction between static and dynamic phase behaviour. Date: October 26, 2010
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1. Introduction As a rare example of a class of exactly-solvable, nonequilibrium interacting particle systems, asymmetric simple exclusion processes of various types have found favour with researchers in statistical mechanics, mathematical physics and probability theory [1–4]. These systems comprise hard-core particles hopping in a preferred direction on a onedimensional lattice and have been used to describe systems as diverse as traffic flow [5], the dynamics of ribosomes [6] and molecular motors [7, 8], the flow of hydrocarbons through a zeolite pore [9], the growth of a fungal filament [10], and in queueing theory [11, 12]. The physical interest lies mainly in the rich phase behaviour that arises as a direct consequence of being driven away from equilibrium. Of particular interest are the open boundary cases. Here the system can be thought of as being placed between two boundary reservoirs, generally of different densities. The two reservoirs enforce a steady current across the lattice, and therewith a nonequilibrium steady state. It was first argued by Krug [13] that as the boundary densities are varied nonequilibrium phase transitions may occur in which steady state bulk quantities— such as the mean current or density—exhibit nonanalyticities. Such phase transitions were seen explicitly in first a mean-field approximation [6, 14] and then in the exact solution [15, 16] of the totally asymmetric exclusion process with open boundaries, hereafter abbreviated as TASEP, and of related processes [4]. Since then arguments have been developed to predict the phase diagram of more general one-dimensional driven diffusive systems [17–19]. Our interest in this work is in the distinction between two phase diagrams for the TASEP: one of which characterises the steady-state behaviour, and the other the dynamics. The static phase diagram [15,16] is shown in Fig. 1(a) where the left boundary density is α and the right is 1 − β. There are three possible phases in the steady state: a low density (LD) phase where the bulk density is controlled by the left boundary and is equal to α; a high density (HD) phase where the bulk density is controlled by the right boundary and is equal to 1 − β; a maximal current (MC) phase where the bulk density is 21 . The high and low density phases are further divided into subphases (e.g. LD1, LDII) according to the functional form of the spatial decay of the density profile to the bulk value near the non-controlling boundary. In these subphases the lengthscale over which the decay is observable remains finite as the system size is increased. This change at the subphase boundaries in the form of the decay is thus not a true phase transition in the thermodynamic sense. More recently, de Gier and Essler have performed an exact analysis of the ASEP’s dynamics [20–22] and derived the the longest relaxation times of the system by calculating the gap in the spectrum using the Bethe ansatz. The analysis builds on directly related work on the XXZ spin chain with nondiagonal boundary fields [23, 24]. The dynamical phase diagram they obtain, in which phases are associated with different functional forms of the longest relaxation time, is illustrated in figure 1(b). The same phases (high density, low density and maximal current) are found as in the static phase
Dynamical Transition in the TASEP
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1
1
LDII
MC j=
j = α(1 − α) ρ=α β
ρ=
LDII′
1 4 1 2
β
0.5
j = α(1 − α) ρ=α
LDI
0.5
LDI′
HDII
HDI
MC
HDII′ HDI′
j = β(1 − β)
j = β(1 − β)
ρ=1−β
ρ=1−β 0
0 0
0.5
α
(a)
1
0
0.5
1
α
(b)
Figure 1. (a) Static and (b) dynamic phase diagrams for the TASEP. Solid lines indicate thermodynamic phase transitions at which the current and bulk density are nonanalytic. Dotted lines indicate subphase boundaries. In the static case, the density profile a finite distance from the boundary is nonanalytic. In the dynamic case, the longest relaxation time exhibits a nonanalyticity.
diagram, however, a hitherto unexpected dynamical transition line which subdivides the low density and high density phases is now apparent. This line which we shall refer to as the de Gier–Essler (dGE) line replaces the subphase boundaries at α = 1/2, β < 1/2 β = 1/2, α < 1/2 in the static phase diagram and, for example, subdivides the low density region into LDI′ , LDII′. Although there is nothing to rule out the prediction of dGE of a dynamical transition at a distinct location to any static transition, the result came as something of a surprise. This is because the static phase diagram had been successfully interpreted in terms of an effective, dynamical theory thought to be relevant for late-time dynamics, referred to as domain wall theory (DWT). We shall review DWT more fully in Section 2.4. For the moment we note that DWT correctly predicts the static subphase boundary and the associated change in the decay of the density profile, but does not predict the dGE line. Therefore, it was previously thought that a dynamical transition also occurred at the subphase boundary and initial calculations of relaxation times appeared to be consistent with this [25, 26]. Our primary goal in this work is to establish numerically that a dynamical transition does indeed occur along the dGE line rather than at the static subphase boundary. We used two distinct approaches. First, we carried out direct Monte Carlo simulations of the TASEP dynamics and identified the dominant transient in three time-dependent observables. We find that these simulations are consistent with a dynamical transition at the dGE line but are not sufficiently accurate to rule out the scenario of the dynamical transition occurring at the subphase boundary.
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We instead turn to a Density Matrix Renormalisation Group (DMRG) approach [27] to acquire convincing evidence that a dynamical transition occurs at the dGE line. The DMRG is an approximate means to obtain the lowest-lying energies and eigenstates of model Hamiltonians with short-range interactions. Although originally developed in the context of the Hubbard and Heisenberg models [28] (see also [29] for a comprehensive review), it has also been applied to nonequilibrium processes such as reaction-diffusion models [30]. In particular, it has also been used to investigate the spectrum of the TASEP [26]. However, the dynamical transition was not evident from the data presented in that work—perhaps because it had not been predicted at that time. By revisiting this approach, we obtain estimates of the longest relaxation time that allow us to rule out the domain wall theory prediction for the dynamical transition. The paper is organised as follows. We begin in Section 2 by recalling the definition of the TASEP, and by reviewing in more detail the static and dynamic phase diagrams discussed above. Then, in Section 3, we present our Monte Carlo simulation data, followed by the DMRG results in Section 4. In Section 5 we return to more general questions about the distinction between the static and dynamic phase diagram with particular reference to different theoretical approaches to the TASEP. We conclude in Section 6 with some open questions. 2. Model definition and phase diagrams Although we have alluded to the basic properties of the totally asymmetric simple exclusion process (TASEP) in the introduction, in the interest of a self-contained presentation we provide in this section a precise model definition and full details of the static and dynamic phase diagrams. 2.1. Model definition The TASEP describes the biased diffusion of particles on a one-dimensional lattice with L sites. No more than one particle can occupy a given site, and overtaking is prohibited. The stochastic dynamics are expressed in terms of transition rates, for example a particle on a site i in the bulk hops to the right as a Poisson process with rate 1, but only if site i + 1 is unoccupied: At the left boundary only particle influx takes place, with rate α, and at the right boundary only particle outflux takes place, with rate β, as shown in Fig. 2. The corresponding reservoir densities are α at the left and 1 − β at the right. 2.2. Static phase diagram As noted in the introduction, the TASEP static phase diagram can be determined from, for example, exact expressions for the current and density profile in the steady state [15,16]. The current takes three functional forms according to whether it is limited by a slow insertion rate (LD: α < β, α < 12 ), by a slow exit rate (HD: β < α, β < 21 ) or by the exclusion interaction in the bulk (MC: α > 12 , β > 21 ). This behaviour of
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Figure 2. The dynamics of the totally asymmetric simple exclusion process (TASEP) with open boundaries. Arrows show moves that may take place, and the labels indicate the corresponding stochastic rates.
the current leads to the identification of the thermodynamic phases separated by solid lines in Fig. 1(a). Along these lines there are nonanalyticities in both the current and the density far from either boundary. The relevant expressions for these quantities are shown on Fig. 1(a). The density near one of the boundaries is nonanalytic along the lines α = 12 and β = 21 in the HD and LD phases respectively. These subphases are shown dotted in Fig. 1(a). Inspection of the functional form of the density profile reveals that these are not true thermodynamic phase transitions, in the sense that the deviation from the bulk value extends only a finite distance into the bulk, and thus contributes only subextensively to the nonequilibrium analogue of a free energy (see e.g., [31] for the definition of such a quantity). To be more explicit, consider for example the behaviour near the right boundary in the low-density phase. Here the bulk density is ρ = α. The mean occupancy of the lattice site positioned a distance j from the right boundary approaches in the thermodynamic limit L → ∞ the form [4, 15, 16] ρL−j ∼
α+
α(1−α) j β(1−β) j cII (α, β) [4α(1−α)] j 3/2
α + cI (β)
α0 cn e
P
φn (x) ′ φ0 (x)
(λn −λ0 )t φ (x) n n≥0 cn e
.
(B.8)
The final term, involving the ratio of sums, vanishes as t → ∞, as all λn with n > 0 exceed λ0 . Thus the stationary profile is "
1 φ′ (x) ρ(x) = 1+ 0 2 φ0 (x)
#
.
(B.9)
We can define the Burgers gap, εB via the asymptotic rate of the exponential decay of the density profile to its stationary form, viz, d ln[ρ(x, t) − ρ(x)] = λ1 − λ0 . (B.10) εB = lim t→∞ dt The eigenfunctions given by (B.6) themselves take the form φ(x) = Aeγx + Be−γx
(B.11)
whence λ = 12 γ 2 . The allowed values of γ are quantised due to the boundary conditions (B.4) and (B.5). We first look for solutions with λ > 0, i.e., real γ. Imposing the boundary conditions leads to the pair of equations γ(A − B) γL
γ(Ae
−γL
− Be
= (2α − 1)(A + B) γx
) = (1 − 2β)(Ae
(B.12) −γx
+ Be
).
(B.13)
These equations are consistent only if tanh(γL) =
2(1 − α − β)γ . (1 − 2α)(1 − 2β) + γ 2
(B.14)
In the limit L → ∞, the left-hand side approaches ±1. The resulting equation has a solution γ = 1 − 2α if α < 12 and a solution γ = 1 − 2β if β < 21 . Solutions with a negative λ have purely imaginary γ = ik (k real), and by following through the same procedure, one finds that the allowed values of k are the roots of tan(kL) =
2(α + β − 1)k . k 2 − (2α − 1)(2β − 1)
(B.15)
In the large-L limit, we find that k ∼ 1/L, and hence the associated eigenvalues vanish as 1/L2 in the thermodynamic limit. We can now state the L → ∞ behaviour of the Burgers gap ε as α and β are varied. When both α and β are smaller than 12 , there are two isolated positive eigenvalues 2 (2β−1)2 , 2 . In this region, then, the gap behaves as λ = (2α−1) 2 |(2α − 1)2 − (2β − 1)2 | = −2|α(1 − α) − β(1 − β)| . (B.16) 2 Like the exact gap (4), this vanishes along the HD-LD coexistence line α = β < 12 . However, unlike the exact gap, the mean-field gap has a nonanalyticity along this line. εB = −
Dynamical Transition in the TASEP
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If α or β is increased above 21 , one of the two solutions with positive λ ceases to exist, and the second-largest eigenvalue λ1 = 0 in the thermodynamic limit. Hence then, the size of the gap is simply equal to the value of the largest eigenvalue. That is, within the low-density phase α < 12 , β > 12 , (1 − 2α)2 2 and within the high-density phase α > 12 , β < εB = −
(B.17) 1 2
(1 − 2β)2 . (B.18) 2 The Burgers gap thus exhibits nonanalyticities along the coexistence line α = β < 21 , and along the lines α < 21 , β = 12 and α = 21 , β < 21 . Thus the dynamic phase diagram for the Burgers equation appears the same as the static phase diagram, Fig. 1(a), except that the subphase boundaries have become dynamical transition lines. εB = −
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