Asymmetric Simple Exclusion Process Research Articles

Inferring the parameters of a Markov process from snapshots of the steady state

We seek to infer the parameters of an ergodic Markov process from samples taken independently from the steady state. Our focus is on non-equilibrium processes, where the steady state is not described by the Boltzmann measure, but is generally unknown and hard to compute, which prevents the application of established equilibrium inference methods. We propose a quantity we call propagator likelihood, which takes on the role of the likelihood in equilibrium processes. This propagator likelihood is based on fictitious transitions between those configurations of the system which occur in the samples. The propagator likelihood can be derived by minimising the relative entropy between the empirical distribution and a distribution generated by propagating the empirical distribution forward in time. Maximising the propagator likelihood leads to an efficient reconstruction of the parameters of the underlying model in different systems, both with discrete configurations and with continuous configurations. We apply the method to non-equilibrium models from statistical physics and theoretical biology, including the asymmetric simple exclusion process (ASEP), the kinetic Ising model, and replicator dynamics.

Dynamic ASEP, Duality, and Continuous q−1-Hermite Polynomials

Abstract We demonstrate a Markov duality between the dynamic asymmetric simple exclusion process (ASEP) and the standard ASEP. We then apply this to step initial data, as well as a half-stationary initial data (which we introduce). While investigating the duality for half-stationary initial data, we uncover and utilize connections to the continuous q−1-Hermite polynomials. Finally, we introduce a family of stationary initial data which are related to the indeterminate moment problem associated with these q−1-Hermite polynomials.

On the TASEP with Second Class Particles

In this paper we study some conditional probabilities for the totally asymmetric simple exclusion processes (TASEP) with second class particles. To be more specific, we consider a finite system with one first class particle and $N-1$ second class particles, and we assume that the first class particle is initially at the leftmost position. In this case, we find the probability that the first class particle is at $x$ and it is still the leftmost particle at time $t$. In particular, we show that this probability is expressed by the determinant of an $N\times N$ matrix of contour integrals if the initial positions of particles satisfy the step initial condition. The resulting formula is very similar to a known formula in the (usual) TASEP with the step initial condition which was used for asymptotics by Nagao and Sasamoto [Nuclear Phys. B 699 (2004), 487-502].

Current fluctuations of the stationary ASEP and six-vertex model

Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$, and show that they are of order $T^{1 / 3}$ along a characteristic line. Upon scaling by $T^{1 / 3}$, we establish that these fluctuations converge to the long-time height fluctuations of the stationary KPZ equation, that is, to the Baik-Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn in 2005 (who established the same result for the TASEP), and the work of Balazs and Seppalainen in 2010 (who established the $T^{1 / 3}$ scaling for the general ASEP). Second, we introduce a class of translation-invariant Gibbs measures that characterizes a one-parameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point) in the free energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size $T$ and show that they too are of order $T^{1 / 3}$ along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995 stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity. Upon scaling the height fluctuations by $T^{1 / 3}$, we again recover the Baik-Rains distribution in the large $T$ limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992.

TASEP hydrodynamics using microscopic characteristics

The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. In the way we show laws of large numbers for tagged particles, fluxes and second class particles, and simplify existing proofs in the shock cases. The presentation is self contained.

Quantitative estimates for the flux of TASEP with dilute site disorder

We prove that the flux function of the totally asymmetric simple exclusion process (TASEP) with site disorder exhibits a flat segment for sufficiently dilute disorder. For high dilution, we obtain an accurate description of the flux. The result is established under a decay assumption of the maximum current in finite boxes, which is implied in particular by a sufficiently slow power tail assumption on the disorder distribution near its minimum. To circumvent the absence of explicit invariant measures, we use an original renormalization procedure and some ideas inspired by homogenization.

Transition to Shocks in TASEP and Decoupling of Last Passage Times

We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes.

On the classical aspects of electrons tunnelling through a quantum dot via a driven lattice gas model in one dimension

A theoretical study of classical aspects, i.e.: density, current density, and average speed of electrons tunnelling through a quantum dot (QD) via a simple driven lattice gas model have been carried out. The main objective of this study is to determine some classical aspects of electron tunnelling through a QD using the driven lattice gas model. The study is conducted by considering a resemblance between the components of the QD with the components of the totally asymmetric simple exclusion process (TASEP) that consists of only a single site and open boundary conditions. The former consists of a source, an island, and a drain, which corresponds respectively to the left reservoir (i = 0), site i = 1, and the right reservoir (i = 2) of the latter. Explicit expressions of the density, current densities, and average speed for electrons tunnelling through the QD in the classical regime are obtained. At the steady state, the density of electrons tunnelling through the dot is 0.5 and the current density becomes v/2, where v is the speed of the electrons. Furthermore, the speed of the electrons may be obtained as functions of temperature and the difference between gate and source-drain potentials. For very low temperatures, the speed of electrons rapidly goes to zero pointing to the occurrence of Coulomb blockade.

Irreversible Local Markov Chains with Rapid Convergence towards Equilibrium.

We study the continuous one-dimensional hard-sphere model and present irreversible local Markov chains that mix on faster time scales than the reversible heat bath or Metropolis algorithms. The mixing time scales appear to fall into two distinct universality classes, both faster than for reversible local Markov chains. The event-chain algorithm, the infinitesimal limit of one of these Markov chains, belongs to the class presenting the fastest decay. For the lattice-gas limit of the hard-sphere model, reversible local Markov chains correspond to the symmetric simple exclusion process (SEP) with periodic boundary conditions. The two universality classes for irreversible Markov chains are realized by the totally asymmetric SEP (TASEP), and by a faster variant (lifted TASEP) that we propose here. We discuss how our irreversible hard-sphere Markov chains generalize to arbitrary repulsive pair interactions and carry over to higher dimensions through the concept of lifted Markov chains and the recently introduced factorized Metropolis acceptance rule.

A model of irreversible jam formation in dense traffic

We study an one-dimensional stochastic model of vehicular traffic on open segments of a single-lane road of finite size L. The vehicles obey a stochastic discrete-time dynamics which is a limiting case of the generalized Totally Asymmetric Simple Exclusion Process. This dynamics has been previously used by Bunzarova and Pesheva (2017) for an one-dimensional model of irreversible aggregation. The model was shown to have three stationary phases: a many-particle one, MP, a phase with completely filled configuration, CF, and a boundary perturbed MP+CF phase, depending on the values of the particle injection (α), ejection (β) and hopping (p) probabilities.Here we extend the results for the stationary properties of the MP+CF phase, by deriving exact expressions for the local density at the first site of the chain and the probability P(1) of a completely jammed configuration. The unusual phase transition, characterized by jumps in both the bulk density and the current (in the thermodynamic limit), as α crosses the boundary α=p from the MP to the CF phase, is explained by the finite-size behavior of P(1). By using a random walk theory, we find that, when α approaches from below the boundary α=p, three different regimes appear, as the size L→∞: (i) the lifetime of the gap between the rightmost clusters is of the order O(L) in the MP phase; (ii) small jams, separated by gaps with lifetime O(1), exist in the MP+CF phase close to the left chain boundary; and (iii) when β=p, the jams are divided by gaps with lifetime of the order O(L1∕2). These results are supported by extensive Monte Carlo calculations.

Blocks in the asymmetric simple exclusion process

In earlier work the authors obtained formulas for the probability in the asymmetric simple exclusion process that the $m$th particle from the left is at site $x$ at time $t$. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work these results are generalized to the case where the $m$th particle is the left-most one in a contiguous block of $L$ particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general $L$.

Dynamics of relaxation to a stationary state for interacting molecular motors

Motor proteins are active enzymatic molecules that drive a variety of biological processes, including transfer of genetic information, cellular transport, cell motility and muscle contraction. It is known that these biological molecular motors usually perform their cellular tasks by acting collectively, and there are interactions between individual motors that specify the overall collective behavior. One of the fundamental issues related to the collective dynamics of motor proteins is the question if they function at stationary-state conditions. To investigate this problem, we analyze a relaxation to the stationary state for the system of interacting molecular motors. Our approach utilizes a recently developed theoretical framework, which views the collective dynamics of motor proteins as a totally asymmetric simple exclusion process of interacting particles, where interactions are taken into account via a thermodynamically consistent approach. The dynamics of relaxation to the stationary state is analyzed using a domain-wall method that relies on a mean-field description, which takes into account some correlations. It is found that the system quickly relaxes for repulsive interactions, while attractive interactions always slow down reaching the stationary state. It is also predicted that for some range of parameters the fastest relaxation might be achieved for a weak repulsive interaction. Our theoretical predictions are tested with Monte Carlo computer simulations. The implications of our findings for biological systems are briefly discussed.

A model of jam formation in congested traffic

We study a model of irreversible jam formation in congested vehicular traffic on an open segment of a single-lane road. The vehicles obey a stochastic discrete-time dynamics which is a limiting case of the generalized Totally Asymmetric Simple Exclusion Process. Its characteristic features are: (a) the existing clusters of jammed cars cannot break into parts; (b) when the leading vehicle of a cluster hops to the right, the whole cluster follows it deterministically, and (c) any two clusters of vehicles, occupying consecutive positions on the chain, may become nearest-neighbors and merge irreversibly into a single cluster. The above dynamics was used in a one-dimensional model of irreversible aggregation by Bunzarova and Pesheva [Phys. Rev. E 95, 052105 (2017)]. The model has three stationary non-equilibrium phases, depending on the probabilities of injection (α), ejection (β), and hopping (p) of particles: a many-particle one, MP, a completely jammed phase CF, and a mixed MP+CF phase. An exact expression for the stationary probability P(1) of a completely jammed configuration in the mixed MP+CF phase is obtained. The gap distribution between neighboring clusters of jammed cars at large lengths L of the road is studied. Three regimes of evolution of the width of a single gap are found: (i) growing gaps with length of the order O(L) when β > p; (ii) shrinking gaps with length of the order O(1) when β < p; and (iii) critical gaps at β = p, of the order O(L1/2). These results are supported by extensive Monte Carlo calculations.

Analytical and simulation studies of 2D asymmetric simple exclusion process

Two species of particles driven perpendicularly and interacting through hardcore exclusion on the 2D lattice is studied. Each particle has three moving directions without the back step. Under periodic boundary conditions, an intermediate phase has been found at which some particles could move along the border of jamming stripes. We have performed mean field analysis for the moving and intermediate phase respectively. The analytical results agree with the simulation results well. The empty site moves along the interface of jamming stripes when the system only has one empty site. The average movement of empty site in one Monte Carlo step (MCS) has been analyzed through the master equation. Under open boundary conditions, the system exhibits moving and jamming phases. The critical injection probability αc shows non-monotonically against the forward moving probability q. The analytical results of average velocity, the density and the flow rate against the injection probability in the moving phase also agree with simulation results well.

How to initialize a second class particle?

We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile, which in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with nonconcave, nonconvex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of Ferrari and Kipnis’ arguments (Ann. Inst. H. Poincare 31 (1995) 143–154) for the totally asymmetric simple exclusion process. The main novelty is the introduction of a signed coupling measure as initial data, which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible. We also reveal in full generality a very interesting invariance property of the one-site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution. Finally, we give a lower estimate on the probability of survival of a second class particle–antiparticle pair.

Shortcut with pool under parallel update rule on one-dimensional lattice totally asymmetric simple exclusion process

This paper studies one-dimensional totally asymmetric simple exclusion process (TASEP) with a pool-shortcut under parallel update rule. Here, the pool is a special area which is introduced into a TASEP on a single lane with shortcut with the probabilities of entering (p) and leaving (q). Different from the existed researches, the parameters of p and q are unequal. To simplify the calculations, the simplest case with parallel update is considered, namely the hopping probability in the bulk equals unity. The theoretical calculation results are compared to the densities acquired by Monte Carlo simulations, and they agree very well. The diagram includes five phases, namely LD/LD/LD, HD/HD/HD, LD/LD/HD, MC/LD/LD and MC/LD/HD phases. It is divided into the LD/LD/LD and HD/HD/HD two regions by the LD/LD/HD phase. Interestingly, the MC/LD/LD and MC/LD/HD phases become a straight line and a point, respectively. The influences of parameters p and q (p>q) on the phases are examined. It is found that the HD/HD/HD phase region shrinks, when the variable q reduces from 0.6 to 0.2 (p is fixed, p=0.8).

Role of Interactions and Correlations on Collective Dynamics of Molecular Motors Along Parallel Filaments

Cytoskeletal motors known as motor proteins are molecules that drive cellular transport along several parallel cytoskeletal filaments and support many biological processes. Experimental evidences suggest that they interact with the nearest molecules of their filament while performing mechanical work. To understand such mechanism theoretically, a new version of two-channel totally asymmetric simple exclusion process which incorporates interactions in a thermodynamically consistent way, is introduced. As the existing approaches for multi-channel systems deviate from analyzing the effect of inter and intra channel interactions, a new approach known as modified vertical cluster mean field is developed. The approach along with monte-carlo simulations successfully encounters some correlations and computes all complex dynamic properties of the system. Role of symmetry of interactions and inter-channel coupling is observed on triple points and particle maximal current. Surprisingly, for each coupling rate and most of the interaction splittings, there corresponds an optimal interaction strength for the maximal current which belongs to the case of weak repulsive interactions. Moreover, for a fixed interaction splitting, as coupling rate increases, the optimal strength decreases and tends towards the experimental predictions. Correlations are found to be short-range and weaker for repulsive and weak attractive interactions, while for stronger attractions they are long-range and stronger. All these findings are also discussed in the context of experimental observations.

Cluster approximations for the TASEP: stationary state and dynamical transition

We develop and test cluster approximations, which generalize simple mean--field by taking into account more and more local correlations, for the Totally Asymmetric Simple Exclusion Process with open boundaries. We consider in detail the pair and triplet approximations, discussing the improvements with respect to mean field in various steady state properties. Moreover, we analyze the recently discovered dynamical transition, describing how the spectrum of the relaxation matrix changes at the transition.

Height distribution tails in the Kardar–Parisi–Zhang equation with Brownian initial conditions

For stationary interface growth, governed by the Kardar–Parisi–Zhang (KPZ) equation in dimensions, typical fluctuations of the interface height at long times are described by the Baik–Rains distribution. Recently Chhita et al (2016 arXiv:1611.06690) used the totally asymmetric simple exclusion process (TASEP) to study the height fluctuations in systems of the KPZ universality class for Brownian interfaces with arbitrary diffusion constant. They showed that there is a one-parameter family of long-time distributions, parameterized by the diffusion constant of the initial random height profile. They also computed these distributions numerically by using Monte Carlo (MC) simulations. Here we address this problem analytically and focus on the distribution tails at short times. We determine the (stretched exponential) tails of the height distribution by applying the optimal fluctuation method (OFM) to the KPZ equation. We argue that, by analogy with other initial conditions, the ‘slow’ tail holds at arbitrary times and therefore provides a proper asymptotic to the family of long-time distributions studied in Chhita et al (2016 arXiv:1611.06690). We verify this hypothesis by performing large-scale MC simulations of a TASEP with a parallel-update rule. The ‘fast’ tail, predicted by the OFM, is also expected to hold at arbitrary times, at sufficiently large heights.

Ribosome flow model with extended objects

We study a deterministic mechanistic model for the flow of ribosomes along the mRNA molecule, called the ribosome flow model with extended objects(RFMEO). This model encapsulates many realistic features of translation including non-homogeneous transition rates along mRNA, the fact that every ribosome covers several codons, and the fact that ribosomes cannot overtake one another. The RFMEO is a mean-field approximation of an important model from statistical mechanics called the totally asymmetric simple exclusion process with extended objects (TASEPEO). We demonstrate that the RFMEO describes biophysical aspects of translation better than previous mean-field approximations, and that its predictions correlate well with those ofTASEPEO. However, unlikeTASEPEO, the RFMEO is amenable to rigorous analysis using tools from systems and control theory. We show that the ribosome density profile along the mRNA in the RFMEO converges to a unique steady-state density that depends on the length of the mRNA, the transition rates along it, and the number of codons covered by every ribosome, but not on the initial density of ribosomes along the mRNA. In particular, the protein production rate also converges to a unique steady state. Furthermore, if the transition rates along the mRNA are periodic with a common periodT then the ribosome density along the mRNA and the protein production rate converge to a unique periodic pattern with periodT, that is, the model entrains to periodic excitations in the transition rates. Analysis and simulations of the RFMEO demonstrate several counterintuitive results. For example, increasing the ribosome footprint may sometimes lead to an increase in the production rate. Also, for large values of the footprint the steady-state density along the mRNA may be quite complex (e.g. with quasi-periodic patterns) even for relatively simple (and non-periodic) transition rates along the mRNA. This implies that inferring the transition rates from the ribosome density may be non-trivial. We believe that the RFMEO could be useful for modelling, understanding and re-engineering translation as well as other important biological processes.