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arXiv:nlin/0201064 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite-size scaling in the steady state of the fully asymmetric exclusion process

J.G. Brankov

Published Mar 28, 2002• 11 citations
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Abstract

Finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition are found in the steady state of the one-dimensional fully asymmetric simple-exclusion process with open boundaries and discrete-time dynamics. The corresponding finite-size scaling variables are identified as the ratio of the chain length to the localization length of the relevant domain wall.

DOI
10.1103/physreve.65.046111

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Finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition are found in the steady state of the one-dimensional fully asymmetric simple-exclusion process with open boundaries and discrete-time dynamics. The corresponding finite-size scaling variables are identified as the ratio of the chain length to the localization length of the relevant domain wall.
The applicability of the concepts of finite-size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one-dimensional totally asymmetric simple-exclusion process with open boundaries. In the thermodynamic limit there are boundary-induced transitions both of the first and second order between steady-state phases of the model. We derive finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition under different stochastic dynamics and compare them to establish the existence of universal functions. Next we study numerically the finite-size behavior of the Lee-Yang zeros of the normalization factor for the different steady-state probabilities.
An exact and rigorous calculation of the current and density profile in the steady state of the one-dimensional fully asymmetric simple-exclusion process with open boundaries and forward-ordered sequential dynamics is presented. The method is based on a matrix product representation of the steady-state probability distribution. The main idea is to choose a suitable representation in which the scalar products describing the current and local density profile for a chain of arbitrary finite size depend only on the elements in a finite number of rows and columns. This makes possible the use of a truncated finite-dimensional representation of the matrices and vectors involved. After performing the calculations, we lift the truncation by letting its dimensionality go to infinity. In this limit the results become exact for any size of the chain. By rescaling one of the infinite-dimensional matrix representations found in the work of Derrida et al. [J. Phys. A 26, 1493 (1993)] for their algebra, we obtain a symmetric ``propagator'' matrix. Its truncated version is diagonalized by orthogonal transformation for easy calculation of the relevant scalar products. An interpretation of the phase transitions between the different phases is given in terms of eigenvalue splitting from a bounded quasicontinuous spectrum. A precise description of the local density profiles is given for all values of the parameters. It is shown that the leading-order asymptotic form of the position-dependent terms in the local density changes within the low- and high-density phases, signaling the presence of a higher-order transition.
We study the fluctuations of the current J ( t ) of the totally asymmetric exclusion process with open boundaries. Using a density matrix renormalization group approach, we calculate the cumulant generating function of the current. This function can be interpreted as a free energy for an ensemble in which histories are weighted by exp ( − sJ ( t )). We show that in this ensemble the model has a first-order spacetime phase transition at s = 0. We numerically determine the finite size scaling of the cumulant generating function near this phase transition, both in the non-equilibrium steady state and for large times.
We continue our studies [J. Stat. Phys. 71, 485 (1993)] of a one-dimensional anisotropic exclusion process with parallel dynamics describing particles moving to the right on a chain of L sites. Instead of considering periodic boundary conditions with a defect as in our studies, we study open boundary conditions with injection of particles with rate \ensuremath{\alpha} at the origin and absorption of particles with rate \ensuremath{\beta} at the boundary. We construct the steady state and compute the density profile as a function of \ensuremath{\alpha} and \ensuremath{\beta}. In the large-L limit we find a high-density phase (\ensuremath{\alpha}>\ensuremath{\beta}) and a low-density phase (\ensuremath{\alpha}). In both phases, the density distribution along the chain approaches its respective constant bulk value exponentially on a length scale \ensuremath{\xi}. They are separated by a phase-transition line where \ensuremath{\xi} diverges and where the density increases linearly with the distance from the origin. Furthermore, we present exact expressions for all equal-time n-point density correlation functions and for the time-dependent two-point function in the steady state. We compare our results with predictions from local dynamical scaling and discuss some conjectures for other exclusion models.
We analyse the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process on a finite lattice and with the most general open boundary conditions. We extend results obtained recently for totally asymmetric diffusion (de Gier J and Essler F H L 2006 J. Stat. Mech. P12011) to the case of partial asymmetry. We determine the finite-size scaling of the spectral gap, which characterizes the approach to stationarity at late times, in the low- and high-density regimes and on the coexistence line. We observe boundary-induced crossovers and discuss possible interpretations of our results in terms of effective domain wall theories.
We analyse the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process on a finite lattice and with the most general open boundary conditions. We extend results obtained recently for totally asymmetric diffusion (de Gier J and Essler F H L 2006 J. Stat. Mech. P12011) to the case of partial asymmetry. We determine the finite-size scaling of the spectral gap, which characterizes the approach to stationarity at late times, in the low- and high-density regimes and on the coexistence line. We observe boundary-induced crossovers and discuss possible interpretations of our results in terms of effective domain wall theories.