One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach

Spectral boundary of the asymmetric simple exclusion process: Free fermions, Bethe ansatz, and random matrix theory

In nonequilibrium statistical mechanics, the asymmetric simple exclusion process (ASEP) serves as a paradigmatic example. We investigate the spectral characteristics of the ASEP, focusing on the spectral boundary of its generator matrix. We examine finite ASEP chains of length L, under periodic boundary conditions (PBCs) and open boundary conditions (OBCs). Notably, the spectral boundary exhibits L spikes for PBCs and L+1 spikes for OBCs. Treating the ASEP generator as an interacting non-Hermitian fermionic model, we extend the model to have tunable interaction. In the noninteracting case, the analytically computed many-body spectrum shows a spectral boundary with prominent spikes. For PBCs, we use the coordinate Bethe ansatz to interpolate between the noninteracting case to the ASEP limit and show that these spikes stem from clustering of Bethe roots. The robustness of the spikes in the spectral boundary is demonstrated by linking the ASEP generator to random matrices with trace correlations or, equivalently, random graphs with distinct cycle structures, both displaying similar spiked spectral boundaries.

Entropy Governed by the Absorbing State of Directed Percolation

We investigate the informational aspect of (1+1)-dimensional directed percolation, a canonical model of a nonequilibrium continuous transition to a phase dominated by a single special state called the "absorbing" state. Using a tensor network scheme, we numerically calculate the time evolution of state probability distribution of directed percolation. We find a universal relaxation of Rényi entropy at the absorbing phase transition point as well as a new singularity in the active phase, slightly but distinctly away from the absorbing transition point. At the new singular point, the second-order Rényi entropy has a clear cusp. There we also detect a singular behavior of "entanglement entropy," defined by regarding the probability distribution as a wave function. The entanglement entropy vanishes below the singular point and stays finite above. We confirm that the absorbing state, though its occurrence is exponentially rare in the active phase, is responsible for these phenomena. This interpretation provides us with a unified understanding of time evolution of the Rényi entropy at the critical point as well as in the active phase.

Relaxation dynamics of closed diffusive systems with infinitesimal Langmuir kinetics

We consider the asymmetric simple exclusion process with Langmuir kinetics in the closed boundary condition. We analytically obtain the exact stationary state and a series of excited states of the system in the limit where Langmuir kinetics is infinitesimally small. Based on this result, we propose an analytical formula for the time evolution of physical quantities of the system.

Matrix product solution of a left-permeable two-species asymmetric exclusion process

We study a two-species partially asymmetric exclusion process where the left boundary is permeable for the "slower" species but the right boundary is not. We find a matrix product solution for the stationary state and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero.

Relaxation dynamics of the asymmetric simple exclusion process with Langmuir kinetics on a ring

We consider the asymmetric simple exclusion process with Langmuir kinetics on a periodic lattice. We analytically obtain the exact time evolution of correlation functions with arbitrary length starting from the initial state with no particle in the system. The exact stationary state of this model has been known for the totally asymmetric case. We propose a basis transformation which simplifies the proof of the stationarity of this state and enables the generalization to the partially asymmetric case. Moreover, we construct low-energy excitations and obtain the exact relaxation time.

Exact current statistics of the asymmetric simple exclusion process with open boundaries

Nonequilibrium systems are often characterized by the transport of some quantity at a macroscopic scale, such as, for instance, a current of particles through a wire. The asymmetric simple exclusion process (ASEP) is a paradigm for nonequilibrium transport that is amenable to exact analytical solution. In the present work, we determine the full statistics of the current in the finite size open ASEP for all values of the parameters. Our exact analytical results are checked against numerical calculations using density matrix renormalization group techniques.

Exact relaxation dynamics in the totally asymmetric simple exclusion process

The relaxation dynamics of the one-dimensional totally asymmetric simple exclusion process on a ring is considered in the case of step initial condition. Analyzing the time evolution of the local particle densities and currents by the Bethe ansatz method, we examine their full relaxation dynamics. As a result, we observe peculiar behaviors, such as the emergence of a ripple in the density profile and the existence of the excessive particle currents. Moreover, by making a finite-size scaling analysis of the asymptotic amplitudes of the local densities and currents, we find the scaling exponents with respect to the total number of sites to be -3/2 and -1, respectively.

Large deviation function for the current in the open asymmetric simple exclusion process

We consider the one-dimensional asymmetric exclusion process with particle injection and extraction at two boundaries. The model is known to exhibit four distinct phases in its stationary state. We analyze the current statistics at the first site in the low and high density phases. In the limit of infinite system size, we conjecture an exact expression for the current large deviation function.

Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials

We introduce some combinatorial objects called staircase tableaux, which have cardinality 4(n)n!, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and translation in protein synthesis. In its most general form, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all parameters general, in terms of staircase tableaux. Our second result is a formula for the moments of (the weight function of) Askey-Wilson polynomials, also in terms of staircase tableaux. Since the 1980s there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre); among these polynomials, the Askey-Wilson polynomials are the most important, because they are at the top of the hierarchy of classical orthogonal polynomials.

Ion transport through confined ion channels in the presence of immobile charges

We study charge transport in an ionic solution in a confined nanoscale geometry in the presence of an externally applied electric field and immobile background charges. For a range of parameters, the ion current shows nonmonotonic behavior as a function of the external ion concentration. For small applied electric field, the ion transport can be understood from simple analytic arguments, which are supported by Monte Carlo simulations. The results qualitatively explain measurements of ion current seen in a recent experiment on ion transport through a DNA-threaded nanopore [D. J. Bonthuis, Phys. Rev. Lett, 97, 128104 (2006)].

Queueing process with excluded-volume effect

We introduce an extension of the M/M/1 queueing process with a spatial structure and excluded-volume effect. The rule of particle hopping is the same as for the totally asymmetric simple exclusion process (TASEP). A stationary-state solution is constructed in a slightly arranged matrix product form of the open TASEP. We obtain the critical line that separates the parameter space depending on whether the model has the stationary state. We calculate the average length of the model and the number of particles and show the monotonicity of the probability of the length in the stationary state. We also consider a generalization of the model with backward hopping of particles allowed and an alternate joined system of the M/M/1 queueing process and the open TASEP.

Crossover from reptation to Rouse dynamics in the cage model

The two-dimensional cage model for polymer motion is discussed with an emphasis on the effect of sideways motions, which cross the barriers imposed by the lattice. Using the density matrix method as a solver of the master equation, the renewal time and the diffusion coefficient are calculated as a function of the strength of the barrier crossings. A strong crossover influence of the barrier crossings is found and it is analyzed in terms of effective exponents for a given chain length. The crossover scaling functions and the crossover scaling exponents are calculated.

Exact solution for a one-dimensional model for reptation

We discuss the exact solution for the properties of the recently introduced "necklace" model for reptation. The solution gives the drift velocity, diffusion constant, and renewal time for asymptotically long chains. Its properties are also related to a special case of the Rubinstein-Duke model in one dimension.

Bethe ansatz solution of the asymmetric exclusion process with open boundaries

We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. For totally asymmetric diffusion we calculate the spectral gap, which characterizes the approach to stationarity at large times. We observe boundary induced crossovers in and between massive, diffusive, and Kardar-Parisi-Zhang scaling regimes.

Exact probability function for bulk density and current in the asymmetric exclusion process

We examine the asymmetric simple exclusion process with open boundaries, a paradigm of driven diffusive systems, having a nonequilibrium steady-state transition. We provide a full derivation and expanded discussion and digression on results previously reported briefly in M. Depken and R. Stinchcombe, Phys. Rev. Lett. 93, 040602 (2004). In particular we derive an exact form for the joint probability function for the bulk density and current, both for finite systems, and also in the thermodynamic limit. The resulting distribution is non-Gaussian, and while the fluctuations in the current are continuous at the continuous phase transitions, the density fluctuations are discontinuous. The derivations are done by using the standard operator algebraic techniques and by introducing a modified version of the original operator algebra. As a by-product of these considerations we also arrive at a very simple way of calculating the normalization constant appearing in the standard treatment with the operator algebra. Like the partition function in equilibrium systems, this normalization constant is shown to completely characterize the fluctuations, albeit in a very different manner.

Exact joint density-current probability function for the asymmetric exclusion process

We study the asymmetric simple exclusion process with open boundaries and derive the exact form of the joint probability function for the occupation number and the current through the system. We further consider the thermodynamic limit, showing that the resulting distribution is non-Gaussian and that the density fluctuations have a discontinuity at the continuous phase transition, while the current fluctuations are continuous. The derivations are performed by using the standard operator algebraic approach and by the introduction of new operators satisfying a modified version of the original algebra.

Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov-de Gennes random matrices

Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is the motion of radial coordinate of the three-dimensional Brownian motion represented in spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of Gaussian ensembles of Bogoliubov-de Gennes Hamiltonians of the mean-field theory of superconductivity, which have a particle-hole symmetry. We report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type C to type CI in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the noncolliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.

Power-law behavior in the power spectrum induced by Brownian motion of a domain wall

We show that Brownian motion of a one-dimensional domain wall in a large but finite system yields a omega(-3/2) power spectrum. This is successfully applied to the totally asymmetric simple exclusion process with open boundaries. An excellent agreement between our theory and numerical results is obtained in a frequency range where the domain wall motion dominates and the discreteness of the system is not effective.

Lee-Yang zeros and phase transitions in nonequilibrium steady states

We consider how the Lee-Yang description of phase transitions in terms of partition function zeros applies to nonequilibrium systems. Here, one does not have a partition function; instead we consider the zeros of a steady-state normalization factor in the complex plane of the transition rates. We obtain the exact distribution of zeros in the thermodynamic limit for a specific model, the boundary-driven asymmetric simple exclusion process. We show that the distributions of zeros at the first- and the second-order nonequilibrium phase transitions of this model follow the patterns known in the Lee-Yang equilibrium theory.

Criterion for phase separation in one-dimensional driven systems

A general criterion for the existence of phase separation in driven density-conserving one-dimensional systems is proposed. It is suggested that phase separation is related to the size dependence of the steady-state currents of domains in the system. A quantitative criterion for the existence of phase separation is conjectured using a correspondence made between driven diffusive models and zero-range processes. The criterion is verified in all cases where analytical results are available, and predictions for other models are provided.

Exact free energy functional for a driven diffusive open stationary nonequilibrium system

We obtain the exact probability exp[-LF([rho(x)])] of finding a macroscopic density profile rho(x) in the stationary nonequilibrium state of an open driven diffusive system, when the size of the system L-->infinity. F, which plays the role of a nonequilibrium free energy, has a very different structure from that found in the purely diffusive case. As there, F is nonlocal, but the shocks and dynamic phase transitions of the driven system are reflected in nonconvexity of F, in discontinuities in its second derivatives, and in non-Gaussian fluctuations in the steady state.

Free energy functional for nonequilibrium systems: an exactly solvable case

We consider the steady state of an open system in which there is a flux of matter between two reservoirs at different chemical potentials. For a large system of size N, the probability of any macroscopic density profile rho(x) is exp[-NF([rho])]; F thus generalizes to nonequilibrium systems the notion of free energy density for equilibrium systems. Our exact expression for F is a nonlocal functional of rho, which yields the macroscopically long range correlations in the nonequilibrium steady state previously predicted by fluctuating hydrodynamics and observed experimentally.

Stochastic ballistic annihilation and coalescence

We study a class of stochastic ballistic annihilation and coalescence models with a binary velocity distribution in one dimension. We obtain an exact solution for the density which reveals a universal phase diagram for the asymptotic density decay. By universal we mean that all models in the class are described by a single phase diagram spanned by two reduced parameters. The phase diagram reveals four regimes, two of which contain the previously studied cases of ballistic annihilation. The two new phases are a direct consequence of the stochasticity. The solution is obtained through a matrix product approach and builds on properties of a q-deformed harmonic oscillator algebra.

One-dimensional partially asymmetric simple exclusion process on a ring with a defect particle

The effect of a moving defect particle for the one-dimensional partially asymmetric simple exclusion process on a ring is considered. The current of the ordinary particles, the speed of the defect particle, and the density profile of the ordinary particles are calculated exactly. The phase diagram for the correlation length is identified. As a by-product, the average and the variance of the particle density of the one-dimensional partially asymmetric simple exclusion process with open boundaries are also computed.