Dynamical Transitions in a One-Dimensional Katz–Lebowitz–Spohn Model

Abstract

Dynamical transitions, already found in the high-and low-density phases of the Totally Asymmetric Simple Exclusion Process and a couple of its generalizations, are singularities in the rate of relaxation towards the Non-Equilibrium Stationary State (NESS), which do not correspond to any transition in the NESS itself. We investigate dynamical transitions in the one-dimensional Katz-Lebowitz-Spohn model, a further generalization of the Totally Asymmetric Simple Exclusion Process where the hopping rate depends on the occupation state of the 2 nodes adjacent to the nodes affected by the hop. Following previous work, we choose Glauber rates and bulk-adapted boundary conditions. In particular, we consider a value of the repulsion which parameterizes the Glauber rates such that the fundamental diagram of the model exhibits 2 maxima and a minimum, and the NESS phase diagram is especially rich. We provide evidence, based on pair approximation, domain wall theory and exact finite size results, that dynamical transitions also occur in the one-dimensional Katz-Lebowitz-Spohn model, and discuss 2 new phenomena which are peculiar to this model. Entropy 2019, 21, 1028 2 of 18 current principles. It was observed [11][12][13] that, in order for this theory to be applicable to models with certain additional interactions, a specific choice of the coupling between the system and the boundary reservoirs must be made.More recently, on the basis of exact results [14-16] on relaxation towards the NESS, a dynamical transition has been identified in the TASEP, that is a singularity in the relaxation rate which does not correspond to any NESS phase transition. The existence of this dynamical transition has also been observed in numerical investigations based on the density matrix renormalization group [17], although a physical interpretation has long been lacking. In the last few years, two of us contributed to showing [18][19][20] that this dynamical transition can be at least qualitatively predicted by suitable mean-field-like approaches and is also exhibited by certain models (for which exact results are not available) which generalize the TASEP by including additional processes or interactions, namely the TASEP with Langmuir kinetics [21,22] and the Antal-Schütz (AS) model [11].The picture which seems to emerge is that, in those NESS phases where the bulk density is determined by one of the boundary reservoirs-that is, the so-called low-density (LD) and high-density (HD) phases-two phases can be identified on the basis of the behaviour of the relaxation rate. In one phase, dubbed slow in Reference [19] and typically located near a coexistence line (where the rate vanishes) in the NESS phase diagram, the relaxation rate depends on both reservoir densities. In the other phase, dubbed fast in Reference [19] and located away from coexistence lines, the relaxation rate depends only on the bulk density, that is, on just one reservoir density. A natural interpretation is that in the fast phase the slowest relaxation mode is a bulk one, while ...

Simple exclusion processes with local resetting

“…Regarding methods, based on the agreement between MF and KMC found in [10] and in our work, it would be interesting to rigorously assess whether, and to what extent, MF results can become exact in the thermodynamic limit. It would also be worth investigating the relaxation towards the stationary state, in which dynamical transitions without a static counterpart have been found in (T)ASEP with OBCs [18][19][20], also (at least with approximate methods) in the presence of Langmuir kinetics [21,22] or additional interactions between particles [23,24]. Finally, we hope that these theoretical results can stimulate progress in the experimental studies, which as far as we know have been so far limited to resetting in single-particle systems [25,26].…”

Unbalanced Langmuir kinetics affects TASEP dynamical transitions: mean-field theory

“…We summarize all the analytical results in a dynamical phase diagram (figure 5), partitioning the HD phase into regions (subphases), whose border lines coincide with the singularities described above. We first distinguish two main subphases, denoted as HD-s ("slow") and HD-f ("fast"), which respectively correspond to the regions where λ min (∞) depends on α or not (the terms "slow" and "fast" to denote these subphases have been previously used in [19,21]). In particular, in the fast phase λ min (∞) is not only independent of α but it also takes the highest admissible value at given β, namely…”

“…In the first place, we have observed that, depending on the model parameters, the singularity associated with the dynamical transition may also appear as a discontinuity in the first derivative of the relaxation rate with respect to either the injection or extraction rate. This is a novel feature, since both in the pure TASEP (which is exactly solvable) [16,17,18] and in some variants (including the TASEP with balanced Langmuir kinetics) [19,20,21] the relaxation rate only exhibits discontinuities in the second derivative (in such cases we speak of second-order-like dynamical transitions). Furthermore, we have found that, when the relaxation rate exhibits first-order singularities, the whole spectrum of the mean-field relaxation matrix exhibits a non-trivial behaviour, being characterized (at finite size) by avoided crossings.…”

“…The transition separates a "normal" dynamical regime, where the relaxation rate depends on a given control parameter (the injection/extraction rate, according to the region of the phase diagram) from a "saturated" one, in which the relaxation rate reaches a maximum and remains constant thereafter. Mostly relying on mean-field-like techniques, some recent works [19,20,21] provide evidence that similar dynamical transitions should occur as well in different generalizations of the TASEP, with Langmuir kinetics or with local particle interactions. For the pure TASEP, the same kind of approximations [22] predict a dynamical transition line in good qualitative agreement with the exact one, slightly improving previous approaches based either on the domain-wall theory [23,24,25] or on the viscous Burgers equation [26].…”

Simple Exclusion Processes with Local Resetting

“…Regarding methods, based on the agreement between MF and KMC found in [10] and in our work, it would be interesting to rigorously assess whether, and to what extent, MF results can become exact in the thermodynamic limit. It would also be worth investigating the relaxation towards the stationary state, in which dynamical transitions without a static counterpart have been found in (T)ASEP with OBCs [18][19][20], also (at least with approximate methods) in the presence of Langmuir kinetics [21,22] or additional interactions between particles [23,24]. Finally, we hope that these theoretical results can stimulate progress in the experimental studies, which as far as we know have been so far limited to resetting in single-particle systems [25,26].…”