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(Color online) Schematic diagram of our model with T A (blue dash-dotted line) and T B (black dashed line) marked.
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Generic inhomogeneous steady states in an asymmetric exclusion process on a ring with a pair of point bottlenecks are studied. We show that, due to an underlying universal feature, measurements of coarse-grained steady-state densities in this model resolve the bottleneck structures only partially. Unexpectedly, it displays localization-delocalizati...
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... 1D model consists of a ring having 2N sites, with two bottlenecks (point defects) of reduced hopping rates q 1 ,q 2 < 1, from i = 1 to 2N and i = N (1 − ) + 1 to N (1 − ), || 1, respectively. The hopping rate elsewhere is unity (see Fig. 1). Site labels i run clockwise from i = 1, whereas particles move counterclockwise. When one of q 1 , q 2 , or || is set to unity, our model is physically identical to that of Ref. [8]. It is convenient to use a continuum labeling in the thermodynamic limit (TL): N → ∞, x = i/2N , and 0 < x < 1. The bottleneck positions are then at x = ...
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... the BLs, which may form close to a defect]. Thus, in the bulk, ρ should be a constant [15]. Therefore, in the ID phase, ρ can be piecewise continuous without any spatial variation, with the possibility of an LDW in the system. The system can be viewed as a combination of two TASEP channels T A [0 x (1 − )/2], marked as a blue dash-dotted line in Fig. 1, and T B [(1 − )/2 x 1] (black dashed line in Fig. 1), joined at x = 0 and x = (1 − )/2, respectively [16]; see Fig. 1. Channels T A and T B are generally of unequal length. We ...
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... the bulk, ρ should be a constant [15]. Therefore, in the ID phase, ρ can be piecewise continuous without any spatial variation, with the possibility of an LDW in the system. The system can be viewed as a combination of two TASEP channels T A [0 x (1 − )/2], marked as a blue dash-dotted line in Fig. 1, and T B [(1 − )/2 x 1] (black dashed line in Fig. 1), joined at x = 0 and x = (1 − )/2, respectively [16]; see Fig. 1. Channels T A and T B are generally of unequal length. We ...
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... ρ can be piecewise continuous without any spatial variation, with the possibility of an LDW in the system. The system can be viewed as a combination of two TASEP channels T A [0 x (1 − )/2], marked as a blue dash-dotted line in Fig. 1, and T B [(1 − )/2 x 1] (black dashed line in Fig. 1), joined at x = 0 and x = (1 − )/2, respectively [16]; see Fig. 1. Channels T A and T B are generally of unequal length. We ...
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... equivalent quantitative disagreements between MFT and MCS results for density profiles]. We have used various system sizes in our MCS studies, ranging from 2N = 500 to 2000, all of which agree with each other within the numerical accuracies of our MCS studies, ruling out any significant system size effects. Notice that in the region AOC (AOB) of Fig. 10, both q 1 and q 2 satisfy the ID phase condition, but q 2 (q 1 ) is screened by q 1 (q 2 ). Now with the current J m = ρ m (1 − ρ m ) (m = A,B) for channels T m in the bulk, J m is clearly continuous across the phase boundaries in Fig. 10 since the density ρ m changes continuously across the phase boundaries. This is reminiscent of a ...
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... of our MCS studies, ruling out any significant system size effects. Notice that in the region AOC (AOB) of Fig. 10, both q 1 and q 2 satisfy the ID phase condition, but q 2 (q 1 ) is screened by q 1 (q 2 ). Now with the current J m = ρ m (1 − ρ m ) (m = A,B) for channels T m in the bulk, J m is clearly continuous across the phase boundaries in Fig. 10 since the density ρ m changes continuously across the phase boundaries. This is reminiscent of a second-order phase transition between the LD and ID phases (and hence between the HD and ID phases by using the particle-hole symmetry in the model). Equivalently, considering the LDW position as an order parameter, the phase boundaries in ...
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... in Fig. 10 since the density ρ m changes continuously across the phase boundaries. This is reminiscent of a second-order phase transition between the LD and ID phases (and hence between the HD and ID phases by using the particle-hole symmetry in the model). Equivalently, considering the LDW position as an order parameter, the phase boundaries in Fig. 10 are second order in nature, with an order parameter exponent 1. This may be obtained as follows: We assume q 1 < q 2 (thus q 2 is irrelevant). Then, to obtain the behavior of x w A near the LD-ID phase transition, we use Eq. (7) and set q 1 = q c − δq, δq > 0, (19) with q c = n/(1 − n) at the threshold of the ID phase for a given n. ...
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... analogous mean-field phase diagram in the n-q 1 (with q 1 q 2 and = 0) plane is shown in Fig. 11. The LD, HD, and ID phases are shown; q 1 = n/(1 − n) gives the boundary between the LD and ID phases; q 1 = (1 − n)/n gives the boundary between the ID and HD phases. The upper limit of q 1 is confined up to q 2 since above this value q 1 gets screened by q 2 ; 0.4 < n < 0.6 gives the location of the DDWs. Not surprisingly, the ...
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... n/(1 − n) gives the boundary between the LD and ID phases; q 1 = (1 − n)/n gives the boundary between the ID and HD phases. The upper limit of q 1 is confined up to q 2 since above this value q 1 gets screened by q 2 ; 0.4 < n < 0.6 gives the location of the DDWs. Not surprisingly, the particle current is continuous across the phase boundaries in Fig. 11, similar to the continuity of the particle current across the phase boundaries in Fig. 10. This is consistent with the second-order phase transitions between the LD (or HD) and ID phases in the ...
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... between the ID and HD phases. The upper limit of q 1 is confined up to q 2 since above this value q 1 gets screened by q 2 ; 0.4 < n < 0.6 gives the location of the DDWs. Not surprisingly, the particle current is continuous across the phase boundaries in Fig. 11, similar to the continuity of the particle current across the phase boundaries in Fig. 10. This is consistent with the second-order phase transitions between the LD (or HD) and ID phases in the ...
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... the locations of the LDW and DDWs in the -q plane, where q = |q 1 − q 2 | with n = 1/2 (see Fig. 12). Evidently, the q = 0 line corresponds to the DDWs in the model, with the DDW spans shrinking as increases to 1, finally being fully confined at = 1. The rest of the box with q > 0 corresponds to LDWs in the ...
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... These results complement the recent studies on number conserving TASEP with quenched but nonrandom hopping rates [38][39][40], where two generic types of steady states were found, that revealed the existence of a type of universality. Similarly, in the present study too, the scaling of the density fluctuations are very sensitive to whether the system is away or close to half-filling. ...
We develop the hydrodynamic theory for number conserving asymmetric exclusion processes with short-range random quenched disordered hopping rates, which is a one-dimensional Kardar-Parisi-Zhang (KPZ) equation with quenched columnar disorder. We show that when the system is away from half-filling, the universal spatiotemporal scaling of the density fluctuations is indistinguishable from its pure counterpart, with the model belonging to the one-dimensional Kardar-Parisi-Zhang universality class. In contrast, close to half-filling, the quenched disorder is relevant, leading to a new universality class. We physically argue that the irrelevance of the quenched disorder when away from half-filling is a consequence of the averaging of the disorder by the propagating density fluctuations in the system. In contrast, close to half-filling the density fluctuations are overdamped, and as a result, are strongly influenced by the quenched disorder.
... Steady states of periodic TASEPs are known to be affected by quenched but non-random disorders, e.g., by localised "obstacles" or defects. This has been studied either with isolated point defects [12,13], or with extended defects [14][15][16]. The general outcome from these studies point towards the existence of two generic classes of steady states in which the steady state currents are either independent of or vary explicitly with the particle numbers. ...
... We highlight a technical point here. With (13), at halffilling ...
We develop the hydrodynamic theory for number conserving asymmetric exclusion processes with short-range random quenched disordered hopping rates. We show that when the system is away from half-filling, the spatio-temporal scaling of the density fluctuations is indistinguishable from its pure counterpart, with the associated scaling exponents belonging to the one-dimensional Kardar- Parisi-Zhang universality class. In contrast, close to half-filling, the scaling exponents take different values, indicating the relevance of the quenched disorder and the emergence of a new universality class.
... When there are multiple global minima, the abovementioned theory of LDW breaks down and very different physics emerges in the shock phase. For instance, consider [19,20], of equal length. Furthermore, from (3) and (4) we have ...
... The shock phase of the present model is analogous to the shocks in open TASEPs. To see this we note that our ring model can be viewed as an open TASEP with effective entry [α e = ρ − (x 0 )] and exit [β e = 1 − ρ + (x 0 )] rates that are joined at x 0 , the location of q min (assuming a single global minimum) [19][20][21]. From the condition of the shock phase, J in = J out = J max = q min /4 holds automatically. ...
... That we obtain an LDW here as opposed to a DDW in an open TASEP under equivalent conditions is due to the strict particle number conservation here; see Refs. [7,18,19,21]. When there are two or more global minima of q(x), the system can be thought to consist of those many TASEPs joined together to form a ring; see Refs. ...
We study the nonequilibrium steady states in a unidirectional or driven single file motion (DSFM) of a collection of particles with hard-core repulsion in a closed system. For driven propulsion that is spatially smoothly varying with a few discontinuities, we show that the steady states are broadly classified into two classes, independently of any system detail: (i) when the steady state current depends explicitly on the conserved number density n, and (ii) when it is independent of n. This manifests itself in the universal topology of the phase diagrams and fundamental diagrams (i.e., the current versus density curves) for DSFM, which are determined solely by the interplay between two control parameters n and the minimum propulsion speed along the chain. Our theory can be tested in laboratory experiments on driven particles in a closed geometry.
We consider a nonequilibrium transition that leads to the formation of nonlinear steady-state structures due to the gas flow scattering on a partially penetrable obstacle. The resulting nonequilibrium steady state (NESS) corresponds to a two-domain gas structure attained at certain critical parameters. We use a simple mean-field model of the driven lattice gas with ring topology to demonstrate that this transition is accompanied by the emergence of local invariants related to a complex composed of the obstacle and its nearest gas surrounding, which we refer to as obstacle edges. These invariants are independent of the main system parameters and behave as local first integrals, at least qualitatively. As a result, the complex becomes insensitive to the noise of external driving field within the overcritical domain. The emerged invariants describe the conservation of the number of particles inside the obstacle and strong temporal synchronization or correlation of gas states at obstacle edges. Such synchronization guarantees the equality to zero of the total edge current at any time. The robustness against external drive fluctuations is shown to be accompanied by strong spatial localization of induced gas fluctuations near the domain wall separating the depleted and dense gas phases. Such a behavior can be associated with nonequilibrium protection effect and synchronization of edges. The transition rates between different NESSs are shown to be different. The relaxation rates from one NESS to another take complex and real values in the sub- and overcritical regimes, respectively. The mechanism of these transitions is governed by the generation of shock waves at the back side of the obstacle. In the subcritical regime, these solitary waves are generated sequentially many times, while only a single excitation is sufficient to rearrange the system state in the overcritical regime.
We propose and study a one-dimensional (1D) model consisting of two lanes with open boundaries. One of the lanes executes diffusive and the other lane driven unidirectional or asymmetric exclusion dynamics, which are mutually coupled through particle exchanges in the bulk. We elucidate the generic nonuniform steady states in this model. We show that in a parameter regime, where hopping along the TASEP lane, diffusion along the SEP lane, and the exchange of particles between the TASEP and SEP lanes compete, the SEP diffusivity D appears as a tuning parameter for both the SEP and TASEP densities for a given exchange rate in the nonequilibrium steady states of this model. Indeed, D can be tuned to achieve phase coexistence in the asymmetric exclusion dynamics together with spatially smoothly varying density in the diffusive dynamics in the steady state. We obtain phase diagrams of the model using mean field theories, and corroborate and complement the results with stochastic Monte Carlo simulations. This model reduces to an isolated open totally asymmetric exclusion process (TASEP) and an open TASEP with bulk particle nonconserving Langmuir kinetics (LK), respectively, in the limits of vanishing and diverging particle diffusivity in the lane executing diffusive dynamics. Thus, this model works as an overarching general model, connecting both pure TASEPs and TASEPs with LK in different asymptotic limits. We further define phases in the SEP and obtain phase diagrams and show their correspondence with the TASEP phases. In addition to its significance as a 1D driven, diffusive model, this model also serves as a simple reduced model for cell biological transport by molecular motors undergoing diffusive and directed motion inside eukaryotic cells.
The proposed study is motivated by the scenario of two-way vehicular traffic. We consider a totally asymmetric simple exclusion process in the presence of a finite reservoir along with the particle attachment, detachment, and lane-switching phenomena. The various system properties in terms of phase diagrams, density profiles, phase transitions, finite size effect, and shock position are analyzed, considering the available number of particles in the system and different values of coupling rate, by employing the generalized mean-field theory and the obtained results are detected to be a good match with the Monte Carlo simulation outcomes. It is discovered that the finite resources significantly affect the phase diagram for different coupling rate values, which leads to nonmonotonic changes in the number of phases in the phase plane for comparatively minor lane-changing rates and produces various exciting features. We calculate the critical value of the total number of particles in the system at which the multiple phases in the phase diagram appear or disappear. The competition between the limited particles, bidirectional motion, Langmuir kinetics, and particle lane-shifting behavior yields unanticipated and unique mixed phases, including the double shock phase, multiple reentrance and bulk-induced phase transitions, and phase segregation of the single shock phase.
We study nonequilibrium steady states in totally asymmetric exclusion processes (TASEPs) with open boundary conditions having spatially inhomogeneous hopping rates. Considering smoothly varying hopping rates, we show that the steady states are in general classified by the steady state currents in direct analogy with open TASEPs having uniform hopping rates. We calculate the steady state bulk density profiles, which are now spatially nonuniform. We also obtain the phase diagrams in the plane of the control parameters, which, despite having phase boundaries that are in general curved lines, have the same topology as their counterparts for conventional open TASEPs, independent of the form of the hopping rate functions. This reveals a type of universality, not encountered in critical phenomena. Surprisingly and in contrast to the phase transitions in an open TASEP with uniform hopping, our studies on the phase transitions in the model reveal that all three transitions are first order in nature. We also demonstrate that this model admits delocalised domain walls (DDWs) on the phase boundaries, demarcating the generalised low and high density phases in this model. However, in contrast to the DDWs observed in an open TASEP with uniform hopping, the envelopes of the DDWs in the present model are generally curved lines.
Inspired from the vehicular traffic scenarios, including the companionship of two consecutive speed bumps placed within a sufficient distance on the road, we investigate a bidirectional two-lane symmetrically coupled totally asymmetric simple exclusion process with two bottlenecks in the presence of Langmuir kinetics. The steady-state system dynamics in terms of density profiles, phase diagrams, phase transitions, and shock dynamics are investigated thoroughly, exploiting hybrid mean-field theory with various strengths of the bottleneck and lane changing rates which match well with Monte Carlo simulation outcomes. It has been detected that the qualitative and quantitative topology of phase diagrams crucially depend on bottleneck and lane switching rate, emanating in monotonic as well as non-monotonic alterations in the number of steady-state phases. We observe that the effect of the bottlenecks is weakened for the increasing values of the lane changing rate. The proposed study provides many novel mixed phases resulting in bulk-induced phase transitions. The interplay between bottlenecks, lane switching, bidirectional movement, and Langmuir kinetics produces unique phenomena, including reentrance transition and phase division of mixed shock region for comparatively lower lane switching rate.
For a given inhomogeneous exclusion processes on N sites between two reservoirs, the trajectories probabilities allow to identify the relevant local empirical observables and to obtain the corresponding rate function at level 2.5. In order to close the hierarchy of the empirical dynamics that appear in the stationarity constraints, we consider the simplest approximation, namely the mean-field approximation for the empirical density of two consecutive sites, in direct correspondence with the previously studied mean-field approximation for the steady state. For a given inhomogeneous totally asymmetric model, this mean-field approximation yields the large deviations for the joint distribution of the empirical density profile and of the empirical current around the mean-field steady state; the further explicit contraction over the current allows to obtain the large deviations of the empirical density profile alone. For a given inhomogeneous asymmetric model, the local empirical observables also involve the empirical activities of the links and of the reservoirs; the further explicit contraction over these activities yields the large deviations for the joint distribution of the empirical density profile and of the empirical current. The consequences for the large deviations properties of time-additive space-local observables are also discussed in both cases.
We study the reservoir crowding effect by considering the nonequilibrium steady states of an asymmetric exclusion process (TASEP) coupled to a reservoir with fixed available resources and dynamically coupled entry and exit rate. We elucidate how the steady states are controlled by the interplay between the coupled entry and exit rates, both being dynamically controlled by the reservoir population, and the fixed total particle number in the system. The TASEP can be in the low-density, high-density, maximal current, and shock phases. We show that such a TASEP is different from an open TASEP for all values of available resources: here the TASEP can support only localized domain walls for any (finite) amount of resources that do not tend to delocalize even for large resources, a feature attributed to the form of the dynamic coupling between the entry and exit rates. Furthermore, in the limit of infinite resources, in contrast to an open TASEP, the TASEP can be found in its high-density phase only for any finite values of the control parameters, again as a consequence of the coupling between the entry and exit rates.
