Exact solution of a 1D asymmetric exclusion model using a matrix formulation

Derrida, Bernard; Evans, Martin R.; Hakim, Vincent; Pasquier, Vincent

doi:10.1088/0305-4470/26/7/011

Cited by 1,217 publications

(2,168 citation statements)

References 28 publications

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“…The mean-field approach predicts phase transitions in the steady state as parameters controlling the rate of insertion and extraction of particles at the boundaries are varied [31]. The existence of these phase transitions is confirmed through an exact solution of the ASEP [29][30][31], achieved using a powerful matrix product approach [32,34] which has subsequently been used to solve many generalisations of the ASEP. The details of the matrix product method are not necessary for the following-suffice to say that one ends up calculating a normalisation proportional to (23) through a product of matrices, often of infinite dimension.…”

Section: Iv1 Driven Diffusive Systemsmentioning

confidence: 99%

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

2003

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We present a pedagogical account of the Lee-Yang theory of equilibrium phase transitions and review recent advances in applying this theory to nonequilibrium systems. Through both general considerations and explicit studies of specific models, we show that the Lee-Yang approach can be used to locate and classify phase transitions in nonequilibrium steady states. I IntroductionIn this work we seek a mathematical understanding of phase transitions in the steady state of stochastic many-body systems. Systems at equilibrium with their environment provide examples of such steady states, and the mechanisms underlying equilibrium phase transitions are long known and understood. Experimentally, one can distinguish between two types of transition: the first-order transition at which there is phase coexistence, e.g. between a high-density solid and a low-density fluid, and the continuous transition at which fluctuations and correlations grow to such an extent as to be macroscopically observable.From a thermodynamical perspective, one can understand first-order transitions by associating with each phase a free energy. For a given set of external parameters, the phase 'chosen' by the system is that with the lowest free energy, and so a phase transition occurs when the free energies of two (or more) phases are equal. The sudden changes in macroscopically measurable quantities that take place at first-order transitions are described mathematically as discontinuities in the first derivative of the free energy. Discontinuities in higher derivatives relate to continuous (higherorder) phase transitions.The tools of equilibrium statistical physics allow onein principle at least-to express the free energy solely in terms of microscopic interactions. More specifically, the free energy is given by the logarithm of a partition function, a quantity that normalises the steady-state probability distribution of microscopic configurations. Initially it was not universally accepted that this approach could faithfully describe phase transitions, in particular the first-order solidfluid transition [1]. In order to show that the statistical mechanical approach can reproduce the correct discontinuities in the free energy at a first-order transition, Lee and Yang [2] introduced a description of phase transitions concerning zeros of the partition function when generalised to the complex plane of an intensive thermodynamic quantity. Initially, the theory was couched in terms of zeros in the complex fugacity plane which is appropriate for fluids in the grand canonical (fluctuating particle number) ensemble. However, by mapping a lattice gas onto the Ising model [3], the theory was found to hold equally well for a magnet in a fixed magnetic field. Later [4-6] it became clear that the distribution of zeros in the complex temperature plane can reveal information about phase transitions in the canonical ensemble.In section II we present a brief, self-contained discussion of the Lee-Yang theory of equilibrium phase transitions which relates nonanalytic ...

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Section: Iv1 Driven Diffusive Systemsmentioning

confidence: 99%

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

2003

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“…Construction of such companion operatorsX α is a sufficient task to prove (1.1) as is well known. See for example [4]. In our setting of the n-TAZRP, the X α andX α are linear operators on the space of "internal degrees of freedom" F ⊗n(n−1)/2 as depicted in the corner transfer matrix type diagram (2.7).…”

Section: Introductionmentioning

confidence: 99%

Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

2015

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We consider a three-dimensional (3D) lattice model associated with the intertwiner of the quantized coordinate ring A q (sl 3 ), and introduce a family of layer to layer transfer matrices on m × n square lattice. By using the tetrahedron equation we derive their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n, they lead to a new proof of the steady state probability of the n-species totally asymmetric zero range process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.

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“…A particle is randomly chosen at a time step, and moves toward smaller coordinate values from (x ′ , y ′ ) to (x, y), where x < x ′ and y < y ′ , until reaching the boundary at x = 1 or y = 1. We emphasize that this type of CA with splitting differs from the asymmetric simple exclusion process (ASEP) [26] and the contact (or voter) model [25], since there are no spatial exclusions and no interaction between any particles. This representation of random walks with splitting is inspirational for deriving a combinatorial analytic form of the distribution of areas.…”

Section: Combinatorial Analysismentioning

confidence: 96%

Combinatorial and approximative analyses in a spatially random division process

Hayashi

Komaki

Ide

et al. 2013

Physica A: Statistical Mechanics and its Applications

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For a spatial characteristic, there exist commonly fat-tail frequency distributions of fragment-size and -mass of glass, areas enclosed by city roads, and pore size/volume in random packings. In order to give a new analytical approach for the distributions, we consider a simple model which constructs a fractal-like hierarchical network based on random divisions of rectangles. The stochastic process makes a Markov chain and corresponds to directional random walks with splitting into four particles. We derive a combinatorial analytical form and its continuous approximation for the distribution of rectangle areas, and numerically show a good fitting with the actual distribution in the averaging behavior of the divisions.

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Exact solution of a 1D asymmetric exclusion model using a matrix formulation

Cited by 1,217 publications

References 28 publications

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

Combinatorial and approximative analyses in a spatially random division process

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