Asymmetric Simple Exclusion Process Research Articles

Quasi-static limit for the asymmetric simple exclusion

We study the one-dimensional asymmetric simple exclusion process on the lattice $\{1, \dots,N\}$ with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale $N^{-a}$ with $a>0$. We prove that at the time scale $N^{1+a}$ the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic boundary rates. We consider two different types of boundary rates: the "Liggett boundaries" that correspond to the projection of the infinite dynamics, and the reversible boundaries, that correspond to the contact with particle reservoirs in equilibrium. The proof is based on the control of the Lax boundary entropy--entropy flux pairs and a coupling argument.

Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift

We study a reaction-diffusion-convection problem with non-linear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this non-linearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder’s fixed point theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.

Totally asymmetric simple exclusion process with particle annihilation

The cargo transport in biological cells often happens under a crowded environment. Past experiments have revealed that cargoes have the ability to self-assemble by associating or dissociating multiple motor proteins, which can impede the forward motion of cargoes on biopolymeric tracks. Motivated by these processes, we study a totally asymmetric simple exclusion process with possibilities of annihilation of particles. The model consists of a one-dimensional track on which two species of particles, one carrying cargoes and the other representing free motor proteins, hop obeying the exclusion principle. Further, the cargo carrying particle can annihilate the other species of particles occupying the forward site at a rate r a . The annihilation process causing particle non-conservation leads to a nonlinear coupling between the two species of particles. We show that this system undergoes boundary induced phase transitions in the state. Using the method of boundary-layer analysis, we find mean-field solutions for the average particle distribution profile across the lattice in the steady state. Analyzing these solutions and the phase portrait of the boundary-layer differential equation, we predict the phase diagram, which consists of a low-density, a high-density and a shock phase. We find that the shapes of the density profiles are affected differently in different phases by the annihilation process. The shapes of the density profiles in different phases agree qualitatively with results from numerical simulations.

Schubert Polynomials, the Inhomogeneous TASEP, and Evil-Avoiding Permutations

Abstract Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\ldots ,n\}$; the possible arrangements of particles in sites thus correspond to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on $S_n$, in which two adjacent particles of weights $i<j$ swap places at rate $x_i - y_{n+1-j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise, nothing happens.) When $y_i=0$ for all $i$, the stationary distribution was conjecturally linked to Schubert polynomials [18], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues [4, 5]. In the case of general $y_i$, Cantini [7] showed that $n$ of the $n!$ states have probabilities proportional to double Schubert polynomials. In this paper, we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns $2413, 4132, 4213,$ and $3214$. We show that there are $\frac {(2+\sqrt {2})^{n-1}+(2-\sqrt {2})^{n-1}}{2}$ evil-avoiding permutations in $S_n$, and for each evil-avoiding permutation $w$, we give an explicit formula for the steady state probability $\psi _w$ as a product of double Schubert polynomials. (Conjecturally, all other probabilities are proportional to a positive sum of at least two Schubert polynomials.) When $y_i=0$ for all $i$, we give multiline queue formulas for the $\textbf {z}$-deformed steady state probabilities and use this to prove the monomial factor conjecture from [18]. Finally, we show that the Schubert polynomials arising in our formulas are flagged Schur functions, and we give a bijection in this case between multiline queues and semistandard Young tableaux.

Totally asymmetric simple exclusion process with a single defect coupling on-ramp

Totally asymmetric simple exclusion process with a single defect coupling on-ramp

Absorbing time asymptotics in the oriented swap process

The oriented swap process is a natural directed random walk on the symmetric group that can be interpreted as a multispecies version of the totally asymmetric simple exclusion process (TASEP) on a finite interval. An open problem from a 2009 paper of Angel, Holroyd, and Romik asks for the limiting distribution of the absorbing time of the process as the number of particles goes to infinity. We resolve this question by proving that this random variable satisfies GOE Tracy–Widom asymptotics. As a central ingredient of our proof, we reexamine a distributional identity relating the behavior of the oriented swap process to last passage percolation, conjectured in a recent paper of Bisi, Cunden, Gibbons, and Romik. We use a shift-invariance principle for multispecies TASEPs, obtained by exploiting recent results of Borodin, Gorin, and Wheeler for the stochastic colored six-vertex model, to prove a weakened form of the Bisi et al. conjectural identity, that is nonetheless sufficient for proving the asymptotic result for the absorbing time.

Two-species totally asymmetric simple exclusion process model: From a simple description to intermittency and traveling traffic jams.

We extend the paradigmatic and versatile totally asymmetric simple exclusion process (TASEP) for stochastic 1D transport to allow for two different particle species, each having specific entry and exit rates. We offer a complete mean-field analysis, including a phase diagram, by mapping this model onto an effective one-species TASEP. Stochastic simulations confirm the results, but indicate deviations when the particle species have very different exit rates. We illustrate that this is due to a phenomenon of intermittency, and formulate a refined "intermittent" mean-field theory for this regime. We discuss how nonstationary effects may further enrich the phenomenology.

Convergence of exclusion processes and the KPZ equation to the KPZ fixed point

We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the Kardar-Parisi-Zhang (KPZ) equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of Wu [Tightness and local fluctuation estimates for the KPZ line ensemble, 2021], Dimitrov and Matetski [Ann. Probab. 49 (2021), pp. 2477–2529], the KPZ line ensemble converges to the Airy line ensemble. For the KPZ equation we are able to start from a continuous function plus a finite collection of narrow wedges. For nearest neighbour exclusions, we can take (discretizations) of continuous functions with | h ( x ) | ≤ C ( 1 + | x | ) |h(x)|\le C(1+\sqrt {|x|}) for some C > 0 C>0 , or one narrow wedge. For non-nearest neighbour exclusions, we are restricted at the present time to a class of (random) initial data, dense in continuous functions in the topology of uniform convergence on compacts. The method is by comparison of the transition probabilities of finite range exclusion processes and the totally asymmetric simple exclusion processes using energy estimates.

From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity

We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of a representation of a steady-state TASEP configurations in terms of (1 + 1)D heaps of pieces (HP) and a theorem of X Viennot which projects the partition function of (1 + 1)D HP onto that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.

Existence and uniqueness of solution of the differential equation describing the TASEP-LK coupled transport process

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C∞[0,1] steady-state solution by the method of upper and lower solution, and the uniqueness in both W1,2(0,1) and L∞(0,1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C([0,1]×[0,+∞))∩C2,1([0,1]×(0,+∞)), which, for any continuous initial value, converges to the steady-state solution in C[0,1] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species.

Particle creation and annihilation in an exclusion process on networks

To mimic the complex transport-like collective phenomena in a man-made or natural system, we study an open network junction model of totally asymmetric simple exclusion process with bulk particle attachment and detachment. The stationary system properties such as particle density, phase transitions, and phase diagrams are derived theoretically utilising the mean field approach. The steady-state phases have been categorized into various sub-classes based upon the phase transitions occurring across the junction. It is found that the number of steady-state phases depends on the number of incoming and outgoing segments at the junction. Further, an increase in the particle non-conserving rates significantly affects the topology of the phase diagram, and the number of stationary phases changes in a non-monotonic way. For both the case of equal and unequal incoming and outgoing segments, the critical values of non-conserving rates at which the topology of the phase diagram changes are identified. The theoretical results are validated using extensive Monte Carlo simulations.

Slow nucleosome dynamics set the transcriptional speed limit and induce RNA polymerase II traffic jams and bursts.

Nucleosomes are recognized as key regulators of transcription. However, the relationship between slow nucleosome unwrapping dynamics and bulk transcriptional properties has not been thoroughly explored. Here, an agent-based model that we call the dynamic defect Totally Asymmetric Simple Exclusion Process (ddTASEP) was constructed to investigate the effects of nucleosome-induced pausing on transcriptional dynamics. Pausing due to slow nucleosome dynamics induced RNAPII convoy formation, which would cooperatively prevent nucleosome rebinding leading to bursts of transcription. The mean first passage time (MFPT) and the variance of first passage time (VFPT) were analytically expressed in terms of the nucleosome rate constants, allowing for the direct quantification of the effects of nucleosome-induced pausing on pioneering polymerase dynamics. The mean first passage elongation rate γ(hc, ho) is inversely proportional to the MFPT and can be considered to be a new axis of the ddTASEP phase diagram, orthogonal to the classical αβ-plane (where α and β are the initiation and termination rates). Subsequently, we showed that, for β = 1, there is a novel jamming transition in the αγ-plane that separates the ddTASEP dynamics into initiation-limited and nucleosome pausing-limited regions. We propose analytical estimates for the RNAPII density ρ, average elongation rate v, and transcription flux J and verified them numerically. We demonstrate that the intra-burst RNAPII waiting times tin follow the time-headway distribution of a max flux TASEP and that the average inter-burst interval [Formula: see text] correlates with the index of dispersion De. In the limit γ→0, the average burst size reaches a maximum set by the closing rate hc. When α≪1, the burst sizes are geometrically distributed, allowing large bursts even while the average burst size [Formula: see text] is small. Last, preliminary results on the relative effects of static and dynamic defects are presented to show that dynamic defects can induce equal or greater pausing than static bottle necks.

Combinatorics of a disordered two-species ASEP on a torus

We define a new disordered asymmetric simple exclusion process (ASEP) with two species of particles, first-class particles labelled • and second-class particles labelled □, on a two-dimensional toroidal lattice. The dynamics is controlled by particles labelled •, which only move horizontally, with forward and backward hopping rates pi and qi respectively if the • is on row i. The motion of particles labelled □ depends on the relative position of these with respect to •’s, and can be both horizontal and vertical. We show that the stationary weight of any configuration is proportional to a monomial in the pi’s and qi’s. Our process projects to the disordered ASEP on a ring, and so explains combinatorially the stationary distribution of the latter first derived by Evans (Europhysics Letters, 1996). We compute the partition function, as well as densities and currents of •’s and □’s in the stationary state. We observe a novel mechanism we call the Scott Russell phenomenon: the current of □’s in the vertical direction is the same as that of •’s in the horizontal direction.

Dynamics of fluctuations in quantum simple exclusion processes

We consider the dynamics of fluctuations in the quantum asymmetric simple exclusion process (Q-ASEP) with periodic boundary conditions. The Q-ASEP describes a chain of spinless fermions with random hoppings that are induced by a Markovian environment. We show that fluctuations of the fermionic degrees of freedom obey evolution equations of Lindblad type, and derive the corresponding Lindbladians. We identify the underlying algebraic structure by mapping them to non-Hermitian spin chains and demonstrate that the operator space fragments into exponentially many (in system size) sectors that are invariant under time evolution. At the level of quadratic fluctuations we consider the Lindbladian on the sectors that determine the late time dynamics for the particular case of the quantum symmetric simple exclusion process (Q-SSEP). We show that the corresponding blocks in some cases correspond to known Yang-Baxter integrable models and investigate the level-spacing statistics in others. We carry out a detailed analysis of the steady states and slow modes that govern the late time behaviour and show that the dynamics of fluctuations of observables is described in terms of closed sets of coupled linear differential-difference equations. The behaviour of the solutions to these equations is essentially diffusive but with relevant deviations, that at sufficiently late times and large distances can be described in terms of a continuum scaling limit which we construct. We numerically check the validity of this scaling limit over a significant range of time and space scales. These results are then applied to the study of operator spreading at large scales, focusing on out-of-time ordered correlators and operator entanglement.

Two-lane totally asymmetric simple exclusion process with extended Langmuir kinetics.

Multilane totally asymmetric simple exclusion processes with interactions between the lanes have recently been investigated actively. This paper proposes a two-lane model with extended Langmuir kinetics on a periodic lattice. Both bidirectional and unidirectional flows are investigated. In our model, the hopping, attachment, and detachment rates vary depending on the state of the corresponding site in the other lane. We obtain a theoretical expression for the global density of the system in the steady state from three kinds of mean-field analyses [(1×1)-, (2×1)-, and (2×2)-cluster cases]. We verify that the (2×2)-cluster mean-field analysis reproduces the differences between the two directional flows and approximates well the results of computer simulations for some cases. We observe that (2×1)-cluster mean-field analyses are already good approximations of the simulation results for unidirectional flows; on the other hand, the accuracy of the approximations much improves by (2×2)-cluster one for bidirectional flows. We explain the phenomena in a qualitative manner by a simple analysis of correlations. We expect these findings to give informative suggestions for actual traffic systems.

Harold Widom’s work in random matrix theory

This is a survey of Harold Widom’s work in random matrices. We start with his pioneering papers on the sine-kernel determinant, continue with his and Craig Tracy’s groundbreaking results concerning the distribution functions of random matrix theory, touch on the remarkable universality of the Tracy–Widom distributions in mathematics and physics, and close with Tracy and Widom’s remarkable work on the asymmetric simple exclusion process.

Multi-point distribution of discrete time periodic TASEP

We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. For a class of initial conditions satisfying certain technical assumptions, we are able to derive large-time, large-period limit of the joint distribution, under the relaxation time scale t=O(L^{3/2}) when the height fluctuations are critically affected by the finite geometry. The assumptions are verified for the step and flat initial conditions. As a corollary we obtain the multi-point distribution of discrete time TASEP on the whole integer lattice {mathbb {Z}} by taking the period L large enough so that the finite-time distribution is not affected by the boundary. The large time limit for the multi-time distribution of discrete time TASEP on {mathbb {Z}} is then obtained for the step initial condition.

Cutoff profile of ASEP on a segment

This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length N. Our main result is that for particle densities in (0, 1), the total-variation cutoff window of ASEP is N^{1/3} and the cutoff profile is 1-F_{mathrm {GUE}}, where F_{mathrm {GUE}} is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labbé and Lacoin. Our proof combines coupling arguments, the result of Tracy–Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities coming from interpreting the multi-species ASEP as a random walk on a Hecke algebra.

Upper-tail large deviation principle for the ASEP

We consider the asymmetric simple exclusion process (ASEP) on Z started from step initial data and obtain the exact Lyapunov exponents for H0(t), the integrated current of ASEP. As a corollary, we derive an explicit formula for the upper-tail large deviation rate function for −H0(t). Our result matches with the rate function for the integrated current of the totally asymmetric simple exclusion process (TASEP) obtained in [40].

Emergent chirality and current generation

We investigate the phenomenon of producing vibration-induced rotational motion in a cylinder filled with achiral rods and being vibrated by an external drive. The arrangement of the rods develops chirality as a result of the interaction of gravity and steric hindrance, which is responsible for the induced motion. A two-rod arrangement is sufficient to generate a persistent motion. The average angular velocity $\langle \omega \rangle$ of the rods at long times varies non-monotonically with the packing fraction $\phi$ for a given drive strength $\Gamma$. Though the precise nature of the variation of $\langle \omega \rangle$ with $\phi$ depends on the details of the interaction between individual rods, its general characteristics hold true for rods of various materials and geometries. A stochastic model based on the asymmetric simple exclusion process helps in understanding the key features of our experiments.