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arXiv:1601.02651 · Physical review. E

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

Jun Sato, Katsuhiro Nishinari

Published Apr 14, 2016• 9 citations
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Abstract

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.

DOI
10.1103/physreve.93.042113

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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.
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.
We study the long time asymptotics of the relaxation dynamics of the totally asymmetric simple exclusion process on a ring.Evaluating the asymptotic amplitudes of the local currents by the algebraic Bethe ansatz method, we find that the relaxation times starting from the step and alternating initial conditions are governed by different eigenvalues of the Markov matrix.In both cases, the scaling exponents of the leading asymptotic amplitudes with respect to the total number of sites are found to be -1.We also study the asymptotics of correlation functions such as the emptiness formation probability.
The one-dimensional totally asymmetric simple exclusion process, a Markov process describing classical hard-core particles hopping in the same direction, is considered on a periodic lattice of L sites. The relaxation to the nonequilibrium steady state, which occurs on the time scale t∼L^{3/2} for large L, is studied for the half-filled system with N=L/2 particles. Using large L asymptotics of Bethe ansatz formulas for the eigenstates, exact expressions depending explicitly on the rescaled time t/L^{3/2} are obtained for the average and two-point function of the local density, and for the current fluctuations for simple (stationary, flat and step) initial conditions, relating previous results for the infinite system to stationary large deviations. The final formulas have a nice interpretation in terms of a functional integral with the action of a scalar field in a linear potential.
The exact stationary state of an asymmetric exclusion process with fully parallel dynamics is obtained using the matrix product ansatz. We give a simple derivation for the deterministic case by a physical interpretation of the dimension of the matrices. We prove the stationarity via a cancellation mechanism, and by making use of an explicit representation of the matrix algebra we easily find closed expressions for the correlation functions in the general probabilistic case. Asymptotic expressions, obtained by making use of earlier results, allow us to derive the exact phase diagram.
We apply the matrix-product ansatz to study the totally asymmetric simple exclusion process on a ring with a generalized discrete-time dynamics depending on two hopping probabilities, ppp and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}$. The model contains as special cases the TASEP with parallel update, when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}=0$, and with sequential backward-ordered update, when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}=p$. We construct a quadratic algebra and its two-dimensional matrix-product representation to obtain exact finite-size expressions for the partition function, the current of particles, and the two-point correlation function. Our main new result is the derivation of the finite-size pair correlation function. Its behavior is analyzed in different regimes of effective attraction and repulsion between the particles, depending on whether $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}>p$ or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}<p$. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}\ensuremath{\rightarrow}1$, when the stationary configurations contain just one cluster.