Asymmetric Simple Exclusion Process Research Articles

Re-appearance of phases in the phase diagram of asymmetrically coupled two-lane exclusion process

Inspired by the finite resources in transport systems, we propose a two-lane totally asymmetric simple exclusion process model with attachment, detachment, and a limited supply of particles. We use a generalized mean-field theory in collaboration with a finite difference scheme and boundary layer analysis to evaluate the significance of limited resources in our proposed system. The outcomes are verified using Monte Carlo simulations. The steady-state behavior of the system is determined using phase diagrams, density profiles, and phase transitions. The study reports a novel phenomenon in which a phase occupies two different regions far from each other in the phase diagrams for an intermediate value of the total number of particles in the system. We also observe unique single and multiple re-entrance transitions, double shock, and varied new mixed phases, leading to bulk-induced phase transitions as exciting findings in the present study.

Tradeoff of generalization error in unsupervised learning

Abstract Finding the optimal model complexity that minimizes the generalization error (GE) is a key issue of machine learning. For the conventional supervised learning, this task typically involves the bias-variance tradeoff: lowering the bias by making the model more complex entails an increase in the variance. Meanwhile, little has been studied about whether the same tradeoff exists for unsupervised learning. In this study, we propose that unsupervised learning generally exhibits a two-component tradeoff of the GE, namely the model error (ME) and the data error (DE)—using a more complex model reduces the ME at the cost of the DE, with the DE playing a more significant role for a smaller training dataset. This is corroborated by training the restricted Boltzmann machine to generate the configurations of the two-dimensional Ising model at a given temperature and the totally asymmetric simple exclusion process with given entry and exit rates. Our results also indicate that the optimal model tends to be more complex when the data to be learned are more complex.

A reverse duality for the ASEP with open boundaries

We prove a duality between the asymmetric simple exclusion process (ASEP) with non-conservative open boundary conditions and an asymmetric exclusion process with particle-dependent hopping rates and conservative reflecting boundaries. This is a reverse duality in the sense that the duality function relates the measures of the dual processes rather than expectations. Specifically, for a certain parameter manifold of the boundary parameters of the open ASEP this duality expresses the time evolution of a family of shock product measures with N microscopic shocks in terms of the time evolution of N particles in the dual process. The reverse duality also elucidates some so far poorly understood properties of the stationary matrix product measures of the open ASEP given by finite-dimensional matrices.

Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder.

We investigate the current properties in the totally asymmetric simple exclusion process (TASEP) on a quenched random energy landscape. In low- and high-density regimes, the properties are characterized by single-particle dynamics. In the intermediate one, the current becomes constant and is maximized. Based on the renewal theory, we derive accurate results for the maximum current. The maximum current significantly depends on a disorder realization, i.e., non-self-averaging (SA). We demonstrate that the disorder average of the maximum current decreases with the system size, and the sample-to-sample fluctuations of the maximum current exceed those of current in the low- and high-density regimes. We find a significant difference between single-particle dynamics and the TASEP. In particular, the non-SA behavior of the maximum current is always observed, whereas the transition from non-SA to SA for current in single-particle dynamics exists.

Sample-to-sample fluctuations of transport coefficients in the totally asymmetric simple exclusion process with quenched disorder.

We consider the totally asymmetric simple exclusion processes on quenched random energy landscapes. We show that the current and the diffusion coefficient differ from those for homogeneous environments. Using the mean-field approximation, we analytically obtain the site density when the particle density is low or high. As a result, the current and the diffusion coefficient are described by the dilute limit of particles or holes, respectively. However, in the intermediate regime, due to the many-body effect, the current and the diffusion coefficient differ from those for single-particle dynamics. The current is almost constant and becomes the maximal value in the intermediate regime. Moreover, the diffusion coefficient decreases with the particle density in the intermediate regime. We obtain analytical expressions for the maximal current and the diffusion coefficient based on the renewal theory. The deepest energy depth plays a central role in determining the maximal current and the diffusion coefficient. As a result, the maximal current and the diffusion coefficient depend crucially on the disorder, i.e., non-self-averaging. Based on the extreme value theory, we find that sample-to-sample fluctuations of the maximal current and diffusion coefficient are characterized by the Weibull distribution. We show that the disorder averages of the maximal current and the diffusion coefficient converge to zero as the system size is increased and quantify the degree of the non-self-averaging effect for the maximal current and the diffusion coefficient.

Counterflow-induced clustering: Exact results.

We analyze the cluster formation in a nonergodic stochastic system as a result of counterflow, with the aid of an exactly solvable model. To illustrate the clustering, a two species asymmetric simple exclusion process with impurities on a periodic lattice is considered, where the impurity can activate flips between the two nonconserved species. Exact analytical results, supported by Monte Carlo simulations, show two distinct phases, free-flowing phase and clustering phase. The clustering phase is characterized by constant density and vanishing current of the nonconserved species, whereas the free-flowing phase is identified with nonmonotonic density and nonmonotonic finite current of the same. The n-point spatial correlation between n consecutive vacancies grows with increasing n in the clustering phase, indicating the formation of two macroscopic clusters in this phase, one of the vacancies and the other consisting of all the particles. We define a rearrangement parameter that permutes the ordering of particles in the initial configuration, keeping all the input parameters fixed. This rearrangement parameter reveals the significant effect of nonergodicity on the onset of clustering. For a special choice of the microscopic dynamics, we connect the present model to a system of run-and-tumble particles used to model active matter, where the two species having opposite net bias manifest the two possible run directions of the run-and-tumble particles, and the impurities act as tumbling reagents that enable the tumbling process.

Orthogonal Polynomial Duality of a Two-Species Asymmetric Exclusion Process

We examine type D ASEP, a two-species interacting particle system which generalizes the usual asymmetric simple exclusion process. For certain cases of type D ASEP, the process does not give priority for one species over another, even though there is nontrivial interaction between the two species. For those specific cases, we prove that the type D ASEP is self-dual with respect to an independent product of q-Krawtchouk polynomials. The type D ASEP was originally constructed in [15], using the type D quantum groups $$\mathcal {U}_q(\mathfrak {so}_6)$$ and $$\mathcal {U}_q(\mathfrak {so}_8)$$ . That paper claimed that certain states needed to be “discarded” in order to ensure non-negativity. Here, we also provide a more efficient argument for the same claim.

Similarity revisited: shock random walks in the asymmetric simple exclusion process with open boundaries

A reverse duality between the asymmetric simple exclusion process (ASEP) with open boundary conditions and a biased random walk of a single particle is proved for a special manifold of boundary parameters of the ASEP. The duality function is given by the configuration probabilities of a family of Bernoulli shock measures with a microscopic shock at site x of the lattice. The boundary conditions of the dual random walk, which can be reflecting or absorbing, depend on the choice of the duality function. As a consequence of this duality, the full time-dependent distribution of the open ASEP starting from a fixed shock measure at x is given for any time t>0 by a convex combination of shock measures with shock at position y. The coefficients are the transition probabilities P(x, t|y, 0) at time t of the random walk starting at site y. It is also shown that this family of shock measures arises from a similarity transformation of the two-dimensional representation of the matrix algebra for the stationary matrix product measure of the open ASEP. This implies a further duality between random walks with different boundary conditions.

Asymmetric coupling induces two-directional reentrance transition in three-lane exclusion process.

Inspired by vehicular traffic phenomena, we study a three-lane open totally asymmetric simple exclusion process with both-sided lane switching in the companionship of Langmuir kinetics. We calculate the phase diagrams, density profiles, and phase transitions through mean-field theory and successfully validate these findings with Monte Carlo simulation results. It has been found that both the qualitative and quantitative topology of phase diagrams crucially rely on the ratio of lane-switching rates called coupling strength. The proposed model has various unique mixed phases, including a double shock resulting in bulk-induced phase transitions. The interplay between both-sided coupling, third lane, and Langmuir kinetics produces unusual features, including a back-and-forth phase transition, also called a reentrance transition, in two directions for relatively nominal values of coupling strength. The presence of reentrance transition and peculiar phase boundaries leads to a rare type of phase division in which one phase lies entirely within another region. Moreover, we scrutinize the shock dynamics by analyzing four different types of shock and finite-size effects.

Mixing times for the simple exclusion process with open boundaries

We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some special cases even cutoff.

Distribution of a Second-Class Particle’s Position in the Two-Species ASEP with a Special Initial Configuration

In this paper, we consider the two-species asymmetric simple exclusion process (ASEP) consisting of $$N-1$$ first-class particles and one second-class particle. We assume that all particles are located at arbitrary positions but the second-class particle is the rightmost particle at time $$t=0$$ . We find the exact formula of the distribution of the second-class particle’s position at time t by directly using the transition probabilities of the two-species ASEP, which is a different approach from the coupling method Tracy and Widom used in [J Phys A 42:425002, 2009].

Stationary measures of the KPZ equation on an interval from Enaud–Derrida’s matrix product ansatz representation

The stationary measures of the Kardar–Parisi–Zhang equation on an interval have been computed recently. We present a rather direct derivation of this result by taking the weak asymmetry limit of the matrix product ansatz for the asymmetric simple exclusion process. We rely on the matrix product ansatz representation of Enaud and Derrida, which allows to express the steady-state in terms of re-weighted simple random walks. In the continuum limit, its measure becomes a path integral (or re-weighted Brownian motion) of the form encountered in Liouville quantum mechanics, recovering the recent formula.

Interplay of reservoirs in a bidirectional system.

Motivated by the interplay of multiple species in several real world transport processes, we propose a bidirectional totally asymmetric simple exclusion process with two finite particle reservoirs regulating the inflow of oppositely directed particles corresponding to two different species. The system's stationary characteristics, such as densities, currents, etc., are investigated using a theoretical framework based on mean-field approximation and are supported by extensive Monte Carlo simulations. The impact of individual species populations, quantified by filling factor, has been comprehensively analyzed considering both equal and unequal conditions. For the equal case, the system exhibits the spontaneous symmetry-breaking phenomena and admits both symmetric as well as asymmetric phases. Moreover, the phase diagram exhibits a different asymmetric phase and displays a nonmonotonic variation in the number of phases with respect to the filling factor. For unequal filling factors, the phase schema can display at most five phases including a phase that shows maximal current for one of the species.

Symmetry phases of asymmetric simple exclusion processes on two lanes with an intersection

This paper studies two-lane asymmetric simple exclusion processes (ASEPs) with an intersection. In the upstream segments of the intersection, one particle can move to the next site with rate 1 if the site is empty, and the other particle can move forward with rate p in the sites of downstream segments. The parameter p can represent the rate of slowing of motion, and the parameter is introduced to investigate spontaneous symmetry breaking (SSB) phenomenon. Extensive Monte Carlo simulations are carried out. It is shown that three symmetric phases exist and the SSB does not exist in the system. Simple mean field approach in which correlation of sites is ignored is firstly adopted to analyze the system, and the system is divided into four independent segments. It is found that the analytical results deviate from the simulation ones, especially when p is small. In addition, the inexsitence of SSB can only be explained qualitatively. Motivated by this, we carry out the cluster mean field analysis in which correlation of five sites is considered. It is shown that densities of the two upstream segments are equal, which demonstrates that the SSB does not exist. It is also shown that, as expected, the cluster mean field analysis performs much better than the simple mean field analysis.

Multi species asymmetric simple exclusion process with impurity activated flips

We obtain an exact matrix product steady state for a class of multi species asymmetric simple exclusion process with impurities, under periodic boundary condition. Alongside the usual hopping dynamics, an additional flip dynamics is activated only in the presence of impurities. Although the microscopic dynamics renders the system to be non-ergodic, exact analytical results for observables are obtained in steady states for a specific class of initial configurations. Interesting physical features including negative differential mobility and transition of correlations from negative to positive with changing vacancy density, have been observed. We discuss plausible connections of this exactly solvable model with multi lane asymmetric simple exclusion processes as well as enzymatic chemical reactions.

Modeling and controlling congestion caused by a bottleneck in an overcrowded aquarium

In this study, we modeled a crowded aquarium using a modified totally asymmetric simple exclusion process and propose a new ticketing system with optimized input rates. We performed a case study for Kaiyukan, a large-scale aquarium in Japan, which encounters heavy congestion primarily caused by a bottleneck in the visitor traffic flow. Kaiyukan has a simple structure and is distinct in that visitors walk through a fixed one-way aisle, as opposed to a typical museum or aquarium layout where visitors can move freely through open areas. We investigated basic congestion features caused by the bottleneck using theoretical analysis and numerical simulations, and developed a new indicator to estimate congestion. Using the model and indicator, we developed a novel ticketing system designed to reduce congestion while preserving the maximum flow. The proposed system includes a 30 min break and indicated a better performance than the conventional method of aiming to maintain a constant input rate. The results indicate that Kaiyukan can reduce the number of visitors waiting and maintain its current daily attendance by introducing our ticketing system.

Asymmetrically coupled two lane totally asymmetric simple exclusion process with extended Langmuir kinetics

The proposed study is motivated by the transport phenomena where the particle hopping, attachment, and detachment dynamics rely on the occupancy state of the corresponding site on another lane. To depict this kind of scenario, we consider an open two-lane Totally Asymmetric Simple Exclusion Process with the presence of asymmetrically coupled and extended Langmuir kinetics regulations. We have analyzed the steady-state properties of the system by calculating characteristics including phase diagrams, density profiles, phase transitions, and finite size effect. The theoretical results are captured by utilizing the framework of a mean-field theory, and it is recognized that numerical findings match closely with the Monte Carlo simulation outcomes.

Modelling of transport processes: Theory and simulations

The transport processes, being a non-equilibrium system, have been a point of interest for physicists since many years revealing and explaining several unexpected effects. Such systems are often dealt with an archetypal model, known as totally asymmetric simple exclusion process, with two different types of boundary conditions: open and periodic. Moreover, these models are analyzed in two varieties of dynamics, random sequential and parallel updates, even at the micro level which play an important role in the global dynamics of the system. On contrary to the random sequential rule, the parallel updates introduce correlations in the system. Using theoretical and numerical methods in the framework based on mean-field approaches, the system properties are analyzed in both transient and steady state.•Both the updating rules are realized using Monte Carlo simulations.•In simplest form, mean-field approach ignores all the correlations and the results coincide with the random sequential update.•Correlations are induced in the system due to parallel update, therefore, a cluster mean-field theory is also discussed to handle them.

Optimization of transition behaviors in a two-lane system

Transitions between two lanes often have a significant impact on various forms of road traffic. To address this problem, we have developed a two-lane asymmetric simple exclusion process model and two hypothetical traffic control strategies, to simulate a futuristic scenario where the timing and location of transitions between two lanes are highly controlled. Various scenarios were proposed to study the effectiveness of these control strategies. An optimized control strategy, whose parameters were determined through an optimization algorithm, is confirmed to effectively maximize the average traffic flow. Consequently, we may identify suitable road sections and the corresponding timings of transitions to resolve congestion in this model.

Totally asymmetric simple exclusion process with local resetting and open boundary conditions

We study a totally asymmetric simple exclusion process with open boundary conditions and local resetting at the injection node. We investigate the stationary state of the model, using both mean–field (MF) approximation and kinetic Monte Carlo simulations, and identify three regimes, depending on the way the resetting rate scales with the lattice size. The most interesting regime is the intermediate resetting one, as in the case of periodic boundary conditions. In this regime we find pure phases and phase separation phenomena, including a low–density/high–density phase separation, which was not possible with periodic boundary conditions. We discuss density profiles, characterizing bulk regions and boundary layers, and nearest–neighbor covariances, finding a remarkable agreement between MF and simulation results. The stationary state phase diagram is mapped out analytically at the MF level, but we conjecture that it may be exact in the thermodynamic limit. We also briefly discuss the large resetting regime, which exhibits an inverse characteristic length scale diverging logarithmically with the lattice size.