Exact results for a fully asymmetric exclusion process with sequential dynamics and open boundaries

Abstract

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 infinitedimensional matrix representations found in the work of Derrida et al. ͓J. Phys. A 26, 1493 ͑1993͔͒ 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.

Statistical physics of vehicular traffic and some related systems

“…A), e.g. random-sequential [167,168], ordered-sequential [166,169,170] and sublattice-parallel update [171][172][173]166]. One finds that the phase diagram has the same basic structure for all updates [166].…”

Nonequilibrium steady states of matrix-product form: a solver's guide

“…Finally the phase diagram for forward ordered updating can be obtained from the particle-hole symmetry mentioned above. Current and density profiles have been calculated in [179,180].…”

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

“…Next, the method of MPA was successfully applied for obtaining the steady-state properties in all the basic cases of true discrete-time dynamics: forward-(→) and backward-ordered (←) sequential [8], [9], [10], sublattice-parallel (s-) [11], [12], and, finally, fully parallel dynamics [13], [14].…”

“…The coexistence line between the low-and highdensity phases is given by (α = β, 0 ≤ β ≤ β c ); on crossing it the bulk density undergoes a finite jump. Here, the exact finite-size expressions for the current and local density in the steady state of FASEP with open boundaries and forward-ordered sequential update, derived in [10], are analysed within the framework of finite-size scaling (FSS) at continuous (for the current) and first-order (for the density) phase transitions. The appropriate scaling variables are identified and the corresponding scaling functions for the current and local desity are explicitly obtained.…”