32 research outputs found
Connected Operators for the Totally Asymmetric Exclusion Process
We fully elucidate the structure of the hierarchy of the connected operators
that commute with the Markov matrix of the Totally Asymmetric Exclusion Process
(TASEP). We prove for the connected operators a combinatorial formula that was
conjectured in a previous work. Our derivation is purely algebraic and relies
on the algebra generated by the local jump operators involved in the TASEP.
Keywords: Non-Equilibrium Statistical Mechanics, ASEP, Exact Results,
Algebraic Bethe Ansatz.Comment: 10 page
Family of Commuting Operators for the Totally Asymmetric Exclusion Process
The algebraic structure underlying the totally asymmetric exclusion process
is studied by using the Bethe Ansatz technique. From the properties of the
algebra generated by the local jump operators, we explicitly construct the
hierarchy of operators (called generalized hamiltonians) that commute with the
Markov operator. The transfer matrix, which is the generating function of these
operators, is shown to represent a discrete Markov process with long-range
jumps. We give a general combinatorial formula for the connected hamiltonians
obtained by taking the logarithm of the transfer matrix. This formula is proved
using a symbolic calculation program for the first ten connected operators.
Keywords: ASEP, Algebraic Bethe Ansatz.
Pacs numbers: 02.30.Ik, 02.50.-r, 75.10.Pq.Comment: 26 pages, 1 figure; v2: published version with minor changes, revised
title, 4 refs adde
Derivation of a Matrix Product Representation for the Asymmetric Exclusion Process from Algebraic Bethe Ansatz
We derive, using the algebraic Bethe Ansatz, a generalized Matrix Product
Ansatz for the asymmetric exclusion process (ASEP) on a one-dimensional
periodic lattice. In this Matrix Product Ansatz, the components of the
eigenvectors of the ASEP Markov matrix can be expressed as traces of products
of non-commuting operators. We derive the relations between the operators
involved and show that they generate a quadratic algebra. Our construction
provides explicit finite dimensional representations for the generators of this
algebra.Comment: 16 page
Spectral gap of the totally asymmetric exclusion process at arbitrary filling
We calculate the spectral gap of the Markov matrix of the totally asymmetric
simple exclusion process (TASEP) on a ring of L sites with N particles. Our
derivation is simple and self-contained and extends a previous calculation that
was valid only for half-filling. We use a special property of the Bethe
equations for TASEP to reformulate them as a one-body problem. Our method is
closely related to the one used to derive exact large deviation functions of
the TASEP
Hidden symmetries in the asymmetric exclusion process
We present a spectral study of the evolution matrix of the totally asymmetric
exclusion process on a ring at half filling. The natural symmetries
(translation, charge conjugation combined with reflection) predict only two
fold degeneracies. However, we have found that degeneracies of higher order
also exist and, as the system size increases, higher and higher orders appear.
These degeneracies become generic in the limit of very large systems. This
behaviour can be explained by the Bethe Ansatz and suggests the presence of
hidden symmetries in the model.
Keywords: ASEP, Markov matrix, symmetries, spectral degeneracies, Bethe
Ansatz.Comment: 16 page
Generalization of the matrix product ansatz for integrable chains
We present a general formulation of the matrix product ansatz for exactly
integrable chains on periodic lattices. This new formulation extends the matrix
product ansatz present on our previous articles (F. C. Alcaraz and M. J. Lazo
J. Phys. A: Math. Gen. 37 (2004) L1-L7 and J. Phys. A: Math. Gen. 37 (2004)
4149-4182.)Comment: 5 pages. to appear in J. Phys. A: Math. Ge
Asymmetric exclusion model with several kinds of impurities
We formulate a new integrable asymmetric exclusion process with
kinds of impurities and with hierarchically ordered dynamics.
The model we proposed displays the full spectrum of the simple asymmetric
exclusion model plus new levels. The first excited state belongs to these new
levels and displays unusual scaling exponents. We conjecture that, while the
simple asymmetric exclusion process without impurities belongs to the KPZ
universality class with dynamical exponent 3/2, our model has a scaling
exponent . In order to check the conjecture, we solve numerically the
Bethe equation with N=3 and N=4 for the totally asymmetric diffusion and found
the dynamical exponents 7/2 and 9/2 in these cases.Comment: to appear in JSTA
The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
The asymmetric simple exclusion process (ASEP) plays the role of a paradigm
in non-equilibrium statistical mechanics. We review exact results for the ASEP
obtained by Bethe ansatz and put emphasis on the algebraic properties of this
model. The Bethe equations for the eigenvalues of the Markov matrix of the ASEP
are derived from the algebraic Bethe ansatz. Using these equations we explain
how to calculate the spectral gap of the model and how global spectral
properties such as the existence of multiplets can be predicted. An extension
of the Bethe ansatz leads to an analytic expression for the large deviation
function of the current in the ASEP that satisfies the Gallavotti-Cohen
relation. Finally, we describe some variants of the ASEP that are also solvable
by Bethe ansatz.
Keywords: ASEP, integrable models, Bethe ansatz, large deviations.Comment: 24 pages, 5 figures, published in the "special issue on recent
advances in low-dimensional quantum field theories", P. Dorey, G. Dunne and
J. Feinberg editor
Spectral Degeneracies in the Totally Asymmetric Exclusion Process
We study the spectrum of the Markov matrix of the totally asymmetric
exclusion process (TASEP) on a one-dimensional periodic lattice at ARBITRARY
filling. Although the system does not possess obvious symmetries except
translation invariance, the spectrum presents many multiplets with degeneracies
of high order. This behaviour is explained by a hidden symmetry property of the
Bethe Ansatz. Combinatorial formulae for the orders of degeneracy and the
corresponding number of multiplets are derived and compared with numerical
results obtained from exact diagonalisation of small size systems. This
unexpected structure of the TASEP spectrum suggests the existence of an
underlying large invariance group.
Keywords: ASEP, Markov matrix, Bethe Ansatz, Symmetries.Comment: 19 pages, 1 figur
Some Exact Results for the Exclusion Process
The asymmetric simple exclusion process (ASEP) is a paradigm for
non-equilibrium physics that appears as a building block to model various
low-dimensional transport phenomena, ranging from intracellular traffic to
quantum dots. We review some recent results obtained for the system on a
periodic ring by using the Bethe Ansatz. We show that this method allows to
derive analytically many properties of the dynamics of the model such as the
spectral gap and the generating function of the current. We also discuss the
solution of a generalized exclusion process with -species of particles and
explain how a geometric construction inspired from queuing theory sheds light
on the Matrix Product Representation technique that has been very fruitful to
derive exact results for the ASEP.Comment: 21 pages; Proceedings of STATPHYS24 (Cairns, Australia, July 2010