Simple Exclusion Process Research Articles

Non-equilibrium phase transition in the heterogenous TASEPs with binding and unbinding processes among the adjacent subsystems

Abstract Non-equilibrium phase transition is always one of the most important issue for complexity science, since it can reveal physical mechanisms of abundant physical phenomena. Exploring applications of non-equilibrium phase transitions in basic paradigm models of complexity science is vital for better comprehending essences of real physical processes. Among these processes, totally asymmetric simple exclusion process (TASEP) stands out owing to important theoretical significances and practical value, whose importance is regarded as being equivalent to Ising model. In this paper, a heterogeneous interacting particle system constituted by three-lane TASEPs with binding and unbinding processes affected by interacting energies among adjacent subsystems is proposed. Nonlinear equations about all particle configuration states of boundaries and bulk are established. We find and analyze numerical local densities and currents by performing simple and cluster mean-field analyses. Nine species of phases including homogeneous phases and mixed ones are discovered. Specifically, as for mixed phases, densities of middle channel are much larger and smaller than those of neighboring ones when specific interacting energy is positive and negative, respectively. All triple points are found to move to upper left corner of phase space with increasing interacting energy. Current phase diagrams mapped into density phase ones are explored, which reveal currents in homogenous phases rely on just one critical governing parameter (namely, injecting rate or escaping one) while those in mixed phases are controlled by the coupling effects of these two critical governing rates. Theoretical results from mean-field analyses are confirmed by simulations, which yield to fine coincidences. This work will improve understanding of non-equilibrium phase transition mechanisms in such basic paradigm models and stochastic dynamics in corresponding critical phenomena to a certain extent, especially cluster effects and related dynamic processes of self-driven particles in such systems in the area of complex system science and statistical physics at mesoscopic scale.

Random Walk on the Simple Symmetric Exclusion Process

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density rho in [0, 1] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities rho except for at most two values rho _-, rho _+ in [0, 1]. The asymptotic speed we obtain in our LLN is a monotone function of rho . Also, rho _- and rho _+ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

TASEP fluctuations with soft-shock initial data

We consider the totally asymmetric simple exclusion process with soft-shock initial particle density, which is a step function increasing in the direction of flow and the step size chosen small to admit KPZ scaling. The initial configuration is deterministic and the dynamics create a shock.

Spectral gaps of open TASEP in the maximal current phase

We study spectral gaps of the one-dimensional totally asymmetric simple exclusion process (TASEP) with open boundaries in the maximal current phase. Earlier results for the model with periodic boundaries suggest that the gaps contributing to the universal KPZ regime may be understood as points on an infinite genus Riemann surface built from a parametric representation of the cumulant generating function of the current. We perform explicit analytic continuations from the known large deviations of the current for open TASEP, and confirm the results for the gaps by an exact Bethe ansatz calculation, with additional checks using high precision extrapolation numerics.

Integral Formulas of ASEP and q-TAZRP on a Ring

In this paper, we obtain the transition probability formulas for the Asymmetric Simple Exclusion Process (ASEP) and the $q$-deformed Totally Asymmetric Zero Range Process ($q$-TAZRP) on the ring by applying the coordinate Bethe ansatz. We also compute the distribution function for a tagged particle with general initial condition.

Current statistics in the q-boson zero range process

We obtain exact formulas of the first two cumulants of particle current in the q-boson zero range process on a ring via exact perturbative solution of the TQ-equation. The result is represented as an infinite sum of double contour integrals. We perform the asymptotic analysis of the large system size limit N → ∞ of the expressions obtained. For |q| ≠ 1 the leading terms of the second cumulant reproduce the N 3/2 scaling expected for models in the Kardar–Parisi–Zhang universality class. The scaling corresponds to the crossover between the Kardar–Parisi–Zhang and Edwards–Wilkinson universality classes. Under this scaling the sum converges to an integral, resulting in the crossover scaling function derived previously for the asymmetric simple exclusion process and conjectured to be universal.

Flagellar length control in biflagellate eukaryotes: time-of-flight, shared pool, train traffic and cooperative phenomena

Flagella of eukaryotic cells are transient long cylindrical protrusions. The proteins needed to form and maintain flagella are synthesized in the cell body and transported to the distal tips. What ‘rulers’ or ‘timers’ a specific type of cells use to strike a balance between the outward and inward transport of materials so as to maintain a particular length of its flagella in the steady state is one of the open questions in cellular self-organization. Even more curious is how the two flagella of biflagellates, like Chlamydomonas reinhardtii, communicate through their base to coordinate their lengths. In this paper we develop a stochastic model for flagellar length control based on a time-of-flight (ToF) mechanism. This ToF mechanism decides whether or not structural proteins are to be loaded onto an intraflagellar transport (IFT) train just before it begins its motorized journey from the base to the tip of the flagellum. Because of the ongoing turnover, the structural proteins released from the flagellar tip are transported back to the cell body also by IFT trains. We represent the traffic of IFT trains as a totally asymmetric simple exclusion process (TASEP). The ToF mechanism for each flagellum, together with the TASEP-based description of the IFT trains, combined with a scenario of sharing of a common pool of flagellar structural proteins in biflagellates, can account for all key features of experimentally known phenomena. These include ciliogenesis, resorption, deflagellation as well as regeneration after selective amputation of one of the two flagella. We also show that the experimental observations of Ishikawa and Marshall are consistent with the ToF mechanism of length control if the effects of the mutual exclusion of the IFT trains captured by the TASEP are taken into account. Moreover, we make new predictions on the flagellar length fluctuations and the role of the common pool.

Modelling cytoskeletal transport by clusters of non-processive molecular motors with limited binding sites.

Molecular motors are responsible for intracellular transport of a variety of biological cargo. We consider the collective behaviour of a finite number of motors attached on a cargo. We extend previous analytical work on processive motors to the case of non-processive motors, which stochastically bind on and off cytoskeletal filaments with a limited number of binding sites available. Physically, motors attached to a cargo cannot bind anywhere along the filaments, so the number of accessible binding sites on the filament should be limited. Thus, we analytically study the distribution and the velocity of a cluster of non-processive motors with limited number of binding sites. To validate our analytical results and to go beyond the level of detail possible analytically, we perform Monte Carlo latticed based stochastic simulations. In particular, in our simulations, we include sequence preservation of motors performing stepping and binding obeying a simple exclusion process. We find that limiting the number of binding sites reduces the probability of non-processive motors binding but has a relatively small effect on force–velocity relations. Our analytical and stochastic simulation results compare well to published data from in vitro and in vivo experiments.

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

In a previous study we developed a mean-field theory of dynamical transitions for the totally-asymmetric simple-exclusion process with open boundaries and Langmuir kinetics, in the so-called balanced regime, characterized by equal binding and unbinding rates. Here we show that simply including the possibility of unbalanced rates gives rise to an unexpectedly richer dynamical phase diagram. In particular, the current work predicts an unusual type of dynamical transition, which exhibits certain similarities with first-order phase transitions of equilibrium systems. We also point out that different types of dynamical transition are accompanied by different structural changes in the (mean-field) relaxation spectrum.

Stochastic Process Emerged from Lattice Fermion Systems by Repeated Measurements and Long-Time Limit

It is known that, in quantum theory, measurements may suppress Hamiltonian dynamics of a system. A famous example is the ‘Quantum Zeno Effect’. This is the phenomena that, if one performs the measurements M times asking whether the system is in the same state as the one at the initial time until the fixed measurement time t, then survival probability tends to 1 by taking the limit M→∞. This is the case for fixed measurement time t. It is known that, if one takes measurement time infinite at appropriate scaling, the ‘Quantum Zeno Effect’ does not occur and the effect of Hamiltonian dynamics emerges. In the present paper, we consider the long time repeated measurements and the dynamics of quantum many body systems in the scaling where the effect of measurements and dynamics are balanced. We show that the stochastic process, called the symmetric simple exclusion process (SSEP), is obtained from the repeated and long time measurements of configuration of particles in finite lattice fermion systems. The emerging stochastic process is independent of potential and interaction of the underlying Hamiltonian of the system.

When diffusion faces drift: Consequences of exclusion processes for bi-directional pedestrian flows

Stochastic particle-based models are useful tools for describing the collective movement of large crowds of pedestrians in crowded confined environments. Using descriptions based on the simple exclusion process, two populations of particles, mimicking pedestrians walking in a built environment, enter a room from two opposite sides. One population is passive — being unaware of the local environment; particles belonging to this group perform a symmetric random walk. The other population has information on the local geometry in the sense that as soon as particles enter a visibility zone, a drift activates them. Their self-propulsion leads them towards the exit. This second type of species is referred here as active. The assumed crowdedness corresponds to a near-jammed scenario. The main question we ask in this paper is: Can we induce modifications of the dynamics of the active particles to improve the outgoing current of the passive particles? To address this question, we compute occupation number profiles and currents for both populations in selected parameter ranges. Besides observing the more classical faster-is-slower effect, new features appear as prominent like the non-monotonicity of currents, self-induced phase separation within the active population, as well as acceleration of passive particles for large-drift regimes of active particles.

From Stochastic Spin Chains to Quantum Kardar-Parisi-Zhang Dynamics.

We introduce the asymmetric extension of the quantum symmetric simple exclusion process which is a stochastic model of fermions on a lattice hopping with random amplitudes. In this setting, we analytically show that the time-integrated current of fermions defines a height field that exhibits quantum nonlinear stochastic Kardar-Parisi-Zhang dynamics. Similarly to classical simple exclusion processes, we further introduce the discrete Cole-Hopf (or Gärtner) transform of the height field that satisfies a quantum version of the stochastic heat equation. Finally, we investigate the limit of the height field theory in the continuum under the celebrated Kardar-Parisi-Zhang scaling and the regime of almost-commuting quantum noise.

Combinatorics of the 2-Species Exclusion Processes, Marked Laguerre Histories, and Partially Signed Permutations

Starting from the two-species partially asymmetric simple exclusion process, we study a subclass of signed permutations, the partially signed permutations, using the combinatorics of Laguerre histories. From this physical and bijective point of view, we obtain a natural descent statistic on partially signed permutations; as well as partially signed permutations patterns.

Stochastic modelling of collective motor protein transport through a crossing of microtubules

The cytoskeleton in eukaryotic cells plays several crucial roles. In terms of intracellular transport, motor proteins use the cytoskeletal filaments as a backbone along which they can actively transport biological cargos such as vesicles carrying biochemical reactants. Crossings between such filaments constitute a key element, as they may serve to alter the destination of such payload. Although motor proteins are known to display a rich behaviour at such crossings, the latter have so far only been modelled as simple branching points. Here we explore a model for a crossing between two microtubules which retains the individual tracks consisting of protofilaments, and we construct a schematic representation of the transport paths. We study collective transport exemplified by the Totally Asymmetric Simple Exclusion Process (TASEP), and provide a full analysis of the transport features and the associated phase diagram, by a generic mean-field approach which we confirm through particle-based stochastic simulations. In particular we show that transport through such a compound crossing cannot be approximated from a coarse-grained structure with a simple branching point. Instead, it gives rise to entirely new and counterintuitive features: the fundamental current-density relation for traffic flow is no longer a single-valued function, and it furthermore differs according to whether it is observed upstream or downstream from the crossing. We argue that these novel features may be directly relevant for interpreting experimental measurements.

KPZ Statistics of Second Class Particles in ASEP via Mixing

We consider the asymmetric simple exclusion process (ASEP) on {mathbb {Z}} with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from 0 to 1. We are interested in X(t), the position of the second class particle at time t. We show that, under the KPZ 1/3 scaling, X(t) is asymptotically distributed as the difference of two independent, {mathrm {GUE}}-distributed random variables. The key part of the proof is to show that X(t) equals, up to a negligible term, the difference of a random number of holes and particles, with the randomness built up by ASEP itself. This provides a KPZ analogue to the 1994 result of Ferrari and Fontes (Probab Theory Relat Fields 99:305–319, 1994), where this randomness comes from the initial data and leads to Gaussian limit laws.

Dynamics of ribosomes in mRNA translation under steady- and nonsteady-state conditions.

Recent advances in DNA sequencing and fluorescence imaging have made it possible to monitor the dynamics of ribosomes actively engaged in messenger RNA (mRNA) translation. Here, we model these experiments within the inhomogeneous totally asymmetric simple exclusion process (TASEP) using realistic kinetic parameters. In particular, we present analytic expressions to describe the following three cases: (a) translation of a newly transcribed mRNA, (b) translation in the steady state and, specifically, the dynamics of individual (tagged) ribosomes, and (c) runoff translation after inhibition of translation initiation. In cases (b) and (c) we develop an effective medium approximation to describe many-ribosome dynamics in terms of a single tagged ribosome in an effective medium. The predictions are in good agreement with stochastic simulations.

Cluster dynamics in the open-boundary heterogeneous ASEPs coupled with interacting energies

Cluster dynamics possess the promising studies in statistical physics and complexity science. While proposing physical models, applying analytical methods and exploring macroscopic properties and microscopic interactions are critical to investigate physical mechanisms of cluster dynamics. Among these physical models, totally asymmetric simple exclusion process is an important non-equilibrium statistical physics model, as it is essential enough to lattice models. Besides, it is also abbreviated as TASEP. Additionally, asymmetric simple exclusion processes, ASEPs, are derivatives, whose dynamics are based on TASEP with other nonlinear physical processes. Actually, heterogeneity of ASEPs is available to depict the real physical and engineering phenomena in a more reasonable manner. Different with previous researches, connected ASEPs are proposed here, which are constituted by two isolated ASEP coupled with a conjunction. As for each individual ASEP, its dynamics are dominated by interacting energies. Different with previous work, two completely different energies are introduced in system, which are used to describe the competition between subsystems. By analyzing complete transition processes among different particle configurations, mean-field dynamical equations are established. Phase diagrams under different energies are obtained, which yield to findings of coexistence phase of low densities, coexistence phase of high densities and coexistence phase of low and high densities. These local densities are qualitatively different. Hereafter, evolution regularities of phase diagrams obtained from local densities and currents are obtained to deeply understand physical mechanisms of complexity science. Expanded results and discussions about potential applications and fewer basic results are also presented. Expanded model with different Langmuir kinetics and internal interacting energies is established. Governing partially differential equations of the whole system are derived. Four steady stacking states of kinesin-II and OSM-3 on bovine brain tubulins affected by two different kinds of buffers are found. This work will help and promote the understanding of non-equilibrium statistical evolutions and nonlinear dynamic behaviors of such multibody system.

Traffic flow with multiple quenched disorders.

We study heterogeneous traffic dynamics by introducing quenched disorders in all the parameters of Newell's car-following model. Specifically, we consider randomness in the free-flow speed, the jam density, and the backward wave speed. The quenched disorders are modeled using beta distributions. It is observed that, at low densities, the average platoon size and the average speed of vehicles evolve as power laws in time as derived by Ben-Naim, Krapivsky, and Redner. No power-law behavior has been observed in the time evolution of the second moment of density and density distribution function, indicating no equivalence between the present system and the sticky gas. As opposed to a totally asymmetric simple exclusion process, we found no power-law behavior in the stationary gap distribution and the transition from the platoon forming phase to the laminar phase coincides with the free flow to congestion transition and is always of first order, independent of the quenched disorder in the free-flow speed. Using mean-field theory, we derived the gap distribution of vehicles and showed that the phase transition is always of first order, independent of the quenched disorder in the free-flow speed corroborating the simulation results. We also showed that the transition density is the reciprocal of the average gap of vehicles in the platoon in the thermodynamic limit.

Analysis of interactions in totally asymmetric exclusion process with site-dependent hopping rates: theory and simulations

Biological molecular motors are special enzymes that support biological processes such as intracellular transport, vesicle locomotion, RNA translation and many more. Experimental works suggest that the motor proteins interact among each other and moreover they experience a push by other motors during the intracellular transport. To incorporate these dynamics, we consider a variant of open one-dimensional totally asymmetric simple exclusion process with site-dependent hopping rates and interactions. The qualitative properties of our system do not depend on the hopping rate function. We utilize the simple mean-field, the cluster mean-field, the correlated cluster mean-field to theoretically calculate the stationary phase diagrams, density and the maximal particle current. The limitations of all the three theories are extensively discussed. It has been found that although the interactions do not change the number of phases in a phase diagram, it significantly changes the density profiles, the phase transition lines and the maximal particle current. The theoretical results obtained are supported by Monte Carlo simulations.

Eigenstates of triangularisable open XXX spin chains and closed-form solutions for the steady state of the open SSEP

In this article we study the relation between the eigenstates of open rational spin Heisenberg chains with different boundary conditions. The focus lies on the relation between the spin chain with diagonal boundary conditions and the spin chain with triangular boundary conditions as well as the class of spin chains that can be brought to such form by certain similarity transformations in the physical space. The boundary driven symmetric simple exclusion process (open SSEP) belongs to the latter. We derive a transformation that maps the eigenvectors of the diagonal spin chain to the eigenvectors of the triangular chain. This transformation yields an essential simplification for determining the states beyond half-filling. It allows to first determine the eigenstates of the diagonal chain through the Bethe ansatz on the fully excited reference state and subsequently map them to the triangular chain for which only the vacuum serves as a reference state. In particular the transformed reference state, i.e. the fully excited eigenstate of the triangular chain, is presented at any length of the chain. It can be mapped to the steady state of the open SSEP. This results in a concise closed-form expression for the probabilities of particle distributions and correlation functions in the steady state. Further, the complete set of eigenstates of the Markov generator is expressed in terms of the eigenstates of the diagonal open chain.