Asymmetric Simple Exclusion Process Research Articles

From single-file diffusion to two-dimensional cage diffusion and generalization of the totally asymmetric simple exclusion process to higher dimensions

A two-dimensional constrained diffusion model is presented and characterized by numerical simulations. The model generalizes the one-dimensional single-file diffusion model by considering a cage diffusion constraint induced by neighboring particles, which is a more stringent condition than volume exclusion. Using numerical simulations we characterize the diffusion process and we particularly show that asymmetric transition probabilities lead to the two-dimensional Kardar-Parisi-Zhang universality class. Therefore, this very simple model effectively generalizes the one-dimensional totally asymmetric simple exclusion process to higher dimensions.

Dynamical phase transitions in totally asymmetric simple exclusion processes with two types of particles under periodically driven boundary conditions.

Driven diffusive systems have provided simple models for nonequilibrium systems with nontrivial structures. Steady-state behavior of these systems with constant boundary conditions have been studied extensively. Comparatively less work has been carried out on the responses of these systems to time-dependent parameters. We report the modifications to the probability density function of a two-particle exclusion model in response to a periodically changing perturbation to its boundary conditions. The changes in the shape of the distribution as a function of the frequency of the perturbation contains considerable structure. A dynamical phase transition in which the system response changes abruptly as a function of perturbation frequency was observed. We interpret this structure to be a consequence of the existence of a typical time scale associated with the dynamics of density shock profiles within the system.

Towards the dynamics of Coulomb blockade in quantum dot via the totally asymmetric simple exclusion process with a single site

Towards the dynamics of Coulomb blockade in quantum dot via the totally asymmetric simple exclusion process with a single site

Entropy production and large deviation function for systems with microscopically irreversible transitions

We obtain the large deviation function for entropy production of the medium and its distribution function for the two-site totally asymmetric simple exclusion process (TASEP) and the three-state unicyclic network. Since such systems are described through microscopic irreversible transitions, we obtain time-dependent transition rates by sampling the states of these systems at a regular short time interval τ. These transition rates are used to derive the large deviation function for the entropy production in the nonequilibrium steady state and its asymptotic distribution function. The shapes of the large deviation function and the distribution function depend on the value of the mean entropy production rate which has a non-trivial dependence on the particle injection and withdrawal rates in the case of TASEP. Further, it is argued that in case of a TASEP, the distribution function tends to be like a Poisson distribution for smaller values of particle injection and withdrawal rates.

Totally asymmetric simple exclusion process simulations of molecular motor transport on random networks with asymmetric exit rates.

Using the totally asymmetric simple-exclusion-process and mean-field transport theory, we investigate the transport in closed random networks with simple crossing topology-two incoming, two outgoing segments, as a model for molecular motor motion along biopolymer networks. Inspired by in vitro observation of molecular motor motion, we model the motor behavior at the intersections by introducing different exit rates for the two outgoing segments. Our simulations of this simple network reveal surprisingly rich behavior of the transport current with respect to the global density and exit rate ratio. For asymmetric exit rates, we find a broad current plateau at intermediate motor densities resulting from the competition of two subnetwork populations. This current plateau leads to stabilization of transport properties within such networks.

Ribosome Flow Model on a Ring.

The asymmetric simple exclusion process (ASEP) is an important model from statistical physics describing particles that hop randomly from one site to the next along an ordered lattice of sites, but only if the next site is empty. ASEP has been used to model and analyze numerous multiagent systems with local interactions including the flow of ribosomes along the mRNA strand. In ASEP with periodic boundary conditions a particle that hops from the last site returns to the first one. The mean field approximation of this model is referred to as the ribosome flow model on a ring (RFMR). The RFMR may be used to model both synthetic and endogenous gene expression regimes. We analyze the RFMR using the theory of monotone dynamical systems. We show that it admits a continuum of equilibrium points and that every trajectory converges to an equilibrium point. Furthermore, we show that it entrains to periodic transition rates between the sites. We describe the implications of the analysis results to understanding and engineering cyclic mRNA translation in-vitro and in-vivo.

Current fluctuations and large deviations for periodic TASEP on the relaxation scale

.The one-dimensional totally asymmetric simple exclusion process (TASEP) with N particles on a periodic lattice of L sites is an interacting particle system with hopping rates breaking detailed balance. The total time-integrated current of particles Q between time 0 and time T is studied for this model in the thermodynamic limit with finite density of particles . The current Q takes at leading order a deterministic value which follows from the hydrodynamic evolution of the macroscopic density profile by the inviscid Burgers’ equation. Using asymptotics of Bethe ansatz formulas for eigenvalues and eigenvectors, an exact expression for the probability distribution of the fluctuations of Q is derived on the relaxation time scale for an evolution conditioned on simple initial and final states. For flat initial and final states, a large deviation function expressed simply in terms of the Airy function is obtained at small rescaled time T/L3/2.

Phase diagrams of three-lane asymmetrically coupled exclusion process with Langmuir kinetics

This letter studies a fully asymmetrically coupled three-lane totally asymmetric simple exclusion process with Langmuir kinetics under open boundary conditions. Phase diagrams and density profiles for different kinetic rates are obtained using a mean-field analysis along with a singular perturbation technique and are found to be in good agreement with Monte Carlo simulation results. Some mixed phases are observed in the middle lane resulting into bulk-induced phase transitions. We have found that a number of steady-state phases firstly increases then decreases with respect to an increase in lane changing rate. Critical values of the lane changing rate are identified at which the appearance or disappearance of certain phases is observed. We have identified the jumping effect in the position of shock in the middle lane with respect to an increase in the lane changing rate.

Tracy–Widom asymptotics for $q$-TASEP

We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time t are of order t^{1/3} and asymptotically distributed as the GUE Tracy-Widom distribution, which confirms the KPZ scaling theory conjecture.

Stochastic Higher Spin Vertex Models on the Line

We introduce a four-parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain Markov dualities. Using this, for the systems started in step initial data we write down nested contour integral formulas for moments and Fredholm determinant formulas for Laplace-type transforms. Taking various choices or limits of parameters, this family degenerates to many of the known exactly solvable models in the Kardar-Parisi-Zhang universality class, as well as leads to many new examples of such models. In particular, ASEP, the stochastic six-vertex model, q-TASEP and various directed polymer models all arise in this manner. Our systems are constructed from stochastic versions of the R-matrix related to the six-vertex model. One of the key tools used here is the fusion of R-matrices and we provide a probabilistic proof of this procedure.

Learning curve constraint cell automaton model for the lean production of CFRP airframe components

A new model is proposed for the lean production of carbon fiber reinforced plastic (CFRP) airframes. Our method improves the production rate by determining the ideal human-capital balance and inventory density on the factory line. The proposed model is derived as a two-step process: First, an analytical solution for the learning rate shift with a human-capital ratio is obtained by merging the Wright learning curve model into the Cobb-Douglas production function. Second, the solution is factored by an asymmetric simple exclusion process (ASEP) cell automaton model to assess whether the inventory density negates the theoretical learning effect. Recent moves toward lean production mean that aerospace CFRPs have a limited shelf life, minimizing buffer periods under metastable and stable production in the time discrete ASEP model. The shift from metastable to stable changes the production rate. Combined with the fact that ASEP is known to drastically reduce throughput if the production steps are not harmonized, the shipment probability p at each step becomes less than 1. Therefore, the human learning effect, which can alter the shipment rate, must be controlled so that p=1 at each production step. This paper describes the analytical aspects of the apparent learning rate to determine adequate values for the human and capital resources, and thus harmonize the learning rates of the production steps. The analytical model shows that factory planning dominates the production rate of CFRP aerospace components. The model is applied to Boeing 787 production data, and it is found that a reduction in inventory density could improve the apparent delivery rate up to the maximum of the human potential.

Quantum Algebra Symmetry of the ASEP with Second-Class Particles

We consider a two-component asymmetric simple exclusion process (ASEP) on a finite lattice with reflecting boundary conditions. For this process, which is equivalent to the ASEP with second-class particles, we construct the representation matrices of the quantum algebra $U_q[\mathfrak{gl}(3)]$ that commute with the generator. As a byproduct we prove reversibility and obtain in explicit form the reversible measure. A review of the algebraic techniques used in the proofs is given.

Multispecies totally asymmetric zero range process: I. Multiline process and combinatorialR

We introduce an $n$-species totally asymmetric zero range process ($n$-TAZRP) on one-dimensional periodic lattice with $L$ sites. It is a continuous time Markov process in which $n$ species of particles hop to the adjacent site only in one direction under the condition that smaller species ones have the priority to do so. Also introduced is an $n$-line process, a companion stochastic system having the uniform steady state from which the $n$-TAZRP is derived as the image by a certain projection $\pi$. We construct the $\pi$ by a combinatorial $R$ of the quantum affine algebra $U_q(\hat{sl}_L)$ and establish a matrix product formula of the steady state probability of the $n$-TAZRP in terms of corner transfer matrices of a $q=0$-oscillator valued vertex model. These results parallel the recent reformulation of the $n$-species totally asymmetric simple exclusion process ($n$-TASEP) by the authors, demonstrating that $n$-TAZRP and $n$-TASEP are the canonical sister models associated with the symmetric and the antisymmetric tensor representations of $U_q(\hat{sl}_L)$ at $q=0$, respectively.

On the Blockage Problem and the Non-analyticity of the Current for Parallel TASEP on a Ring

The Totally Asymmetric Simple Exclusion Process (TASEP) is an important example of a particle system driven by an irreversible Markov chain. In this paper we give a simple yet rigorous derivation of the chain stationary measure in the case of parallel updating rule. In this parallel framework we then consider the blockage problem (aka slow bond problem). We find the exact expression of the current for an arbitrary blockage intensity $\varepsilon$ in the case of the so-called rule-184 cellular automaton, i.e. a parallel TASEP where at each step all particles free-to-move are actually moved. Finally, we investigate through numerical experiments the conjecture that for parallel updates other than rule-184 the current may be non-analytic in the blockage intensity around the value $\varepsilon = 0$.

Phase coexistence and particle nonconservation in a closed asymmetric exclusion process with inhomogeneities.

We construct a one-dimensional totally asymmetric simple exclusion process (TASEP) on a ring with two segments having unequal hopping rates, coupled to particle nonconserving Langmuir kinetics (LK) characterized by equal attachment and detachment rates. In the steady state, in the limit of competing LK and TASEP, the model is always found in states of phase coexistence. We uncover a nonequilibrium phase transition between a three-phase and a two-phase coexistence in the faster segment, controlled by the underlying inhomogeneity configurations and LK. The model is always found to be half-filled on average in the steady state, regardless of the hopping rates and the attachment-detachment rate.

Sensitivity of mRNA Translation.

Using the dynamic mean-field approximation of the totally asymmetric simple exclusion process (TASEP), we investigate the effect of small changes in the initiation, elongation, and termination rates along the mRNA strand on the steady-state protein translation rate. We show that the sensitivity of mRNA translation is equal to the sensitivity of the maximal eigenvalue of a symmetric, nonnegative, tridiagonal, and irreducible matrix. This leads to new analytical results as well as efficient numerical schemes that are applicable for large-scale models. Our results show that in the usual endogenous case, when initiation is more rate-limiting than elongation, the sensitivity of the translation rate to small mutations rapidly increases towards the 5′ end of the ORF. When the initiation rate is high, as may be the case for highly expressed and/or heterologous optimized genes, the maximal sensitivity is with respect to the elongation rates at the middle of the mRNA strand. We also show that the maximal possible effect of a small increase/decrease in any of the rates along the mRNA is an increase/decrease of the same magnitude in the translation rate. These results are in agreement with previous molecular evolutionary and synthetic biology experimental studies.

Multispecies TASEP and combinatorial R*

We identify the algorithm for constructing steady states of the n-species totally asymmetric simple exclusion process (TASEP) on an L site periodic chain by Ferrari and Martin with a composition of combinatorial R for the quantum affine algebra in crystal base theory. Based on this connection and the factorized form of the R matrix derived recently from the tetrahedron equation, we establish a new matrix product formula for the steady state of the TASEP, which is expressed in terms of corner transfer matrices of the q-oscillator valued five-vertex model at q = 0.

Self-duality for the two-component asymmetric simple exclusion process

We study a two-component asymmetric simple exclusion process (ASEP) that is equivalent to the ASEP with second-class particles. We prove self-duality with respect to a family of duality functions which are shown to arise from the reversible measures of the process and the symmetry of the generator under the quantum algebra Uq[𝔤𝔩3]. We construct all invariant measures in explicit form and discuss some of their properties. We also prove a sum rule for the duality functions.

Bi-orthogonal polynomial sequences and the asymmetric simple exclusion process

We state the diffusion algebra equations of the stationary state of the three parameter (α, β and q) asymmetric simple exclusion Process as a linear functional acting on a tensor algebra. From we construct a pair of sequences, and , of monic polynomials which are bi-orthogonal, that is, they satisfy (where is a scalar). The uniqueness and existence of the pair of sequences arises from the determinant of the bi-moment matrix whose elements satisfy a pair of q-recurrence relations. The determinant is evaluated using an LDU-decomposition. If the linear functional is represented as an inner product, then the action of the polynomials Qn on the boundary vector generate a basis whose orthogonal dual vectors are given by the action of Pn on the dual boundary vector , that is . This basis gives the representation of the algebra which is associated with the Al-Salam–Chihara polynomials obtained by Sasamoto.

On the appearance of traffic jams in a long chain with a shortcut in the bulk

The Totally Asymmetric Simple Exclusion Process (TASEP) is studied on open long chains with a shunted section between two simple chain segments in the maximum current phase. The reference case, when the two branches are chosen with equal probability, is considered. The conditions for the occurrence of traffic jams and their properties are investigated both within the effective rates approximation and by extensive Monte Carlo simulations for arbitrary length of the shortcut. Our main results are: (1) For any length of the shortcut and any values of the external rates in the domain of the maximum current phase, there exists a position of the shortcut where the shunted segment is in a phase of coexistence with a completely delocalized domain wall; (2) The main features of the coexistence phase and the density profiles in the whole network are well described by the domain wall theory. Apart from the small inter-chain correlations, they depend only on the current through the shortcut; (3) The model displays unexpected features: (a) the current through the longer shunted segment is larger than the current through the shortcut, and (b) the delocalized domain wall in the coexistence phase of the long shunted segment induces similar behavior even in shortcuts containing a small number of sites; (4) From the viewpoint of vehicular traffic, most comfortable conditions for the drivers are provided when the shortcut is shifted downstream from the position of coexistence, when both the shunted segment and the shortcut exhibit low-density lamellar flow. Most unfavorable is the opposite case of upstream shifted shortcut, when both the shunted segment and the shortcut are in a high-density phase describing congested traffic of slowly moving cars. The above results are relevant also to phenomena like crowding of molecular motors moving along twisted protofilaments.