Exact solution of asymmetric diffusion with N classes of particles of arbitrary size and hierarchical order

Alcaraz, Francisco C.; Bariev, R. Z.

doi:10.1590/s0103-97332000000400004

Cited by 34 publications

(80 citation statements)

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“…Otherwise said, each α or β in a factor W is a composition with the later T i . The W -product is determined by these (ordered) i, and we denote the sequence of them by I = {i 1 …”

Section: A General Formulamentioning

confidence: 99%

“…1 Each particle waits exponential time, then with probability p it moves one step to the right if the site is unoccupied, otherwise it stays put; and with probability q = 1 − p it moves one step to the left if the site is unoccupied, otherwise it stays put. Each particle does this independently of the other particles.…”

Section: Introductionmentioning

confidence: 99%

“…2 Thus, if a particle of species s tries to move to a neighboring site occupied by a particle of species s it is blocked if s ≤ s , but if s > s the particles interchange positions. Second-class particles were introduced by Liggett [13] and subsequently developed and generalized by several authors [1,4,5,[7][8][9]20].…”

Section: Introductionmentioning

confidence: 99%

“…But for multispecies ASEP we must show that the identities define A π σ consistently. If, as in [1], the multispecies ASEP is formulated as a nested Bethe Ansatz problem, one is led to show that the Yang-Baxter equations [6,21] are satisfied. That the model has nontrivial solutions to the Yang-Baxter equations can be traced back to early work of Perk and Schultz on multistate vertex models [16].…”

Section: Introductionmentioning

confidence: 99%

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The asymmetric simple exclusion process with an open boundary

Tracy

Widom

2013

Journal of Mathematical Physics

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In previous work the authors, using the Bethe Ansatz, found for the N -particle asymmetric simple exclusion process on the integers a formula-a sum of multiple integrals-for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species having priority over those of a lower species. Here the integrands in the multiple integrals are defined by a system of relations whose consistency requires verifying that the Yang-Baxter equations hold.

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“…Otherwise said, each α or β in a factor W is a composition with the later T i . The W -product is determined by these (ordered) i, and we denote the sequence of them by I = {i 1 …”

Section: A General Formulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The asymmetric simple exclusion process with an open boundary

Tracy

Widom

2013

Journal of Mathematical Physics

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“…Motivated by this fact we decided to verify if we can solve the above quantum chains directly though a matrix product ansatz, without considering any time dependence as in the case of the dynamical matrix ansatz. Surprisingly, we were able to rederive all the results previously obtained though the Bethe ansatz for the asymmetric diffusion problem with one species of particles [15] or more [18,19]. Moreover, our derivation turns out to be quite simple and it is not difficult to extend it to many other quantum Hamiltonians related or not to stochastic particle dynamics [41].…”

Section: Introductionmentioning

confidence: 60%

The exact solution of the asymmetric exclusion problem with particles of asrbitrary size: matrix product ansatz

Alcaraz¹,

Lazo²

2003

Braz. J. Phys.

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The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product ansatz. Due to the similarity of the master equation and the Schrödinger equation at imaginary times the solution of these problems reduces to the diagonalization of a one dimensional quantum Hamiltonian. Initially, we present the solution of the problem when an arbitrary mixture of molecules, each of then having an arbitrary size (s = 0, 1, 2, . . .) in units of lattice spacing, diffuses asymmetrically on the lattice. The solution of the more general problem where we have the diffusion of particles belonging to N distinct classes of particles (c = 1, . . . , N ), with hierarchical order and arbitrary sizes, is also presented. Our matrix product ansatz asserts that the amplitudes of an arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed by a product of matrices. The algebraic properties of the matrices defining the ansatz depend on the particular associated Hamiltonian. The absence of contradictions in the algebraic relations defining the algebra ensures the exact integrability of the model. In the case of particles distributed in N > 2 classes, the associativity of this algebra implies the Yang-Baxter relations of the exact integrable model. I IntroductionThe representation of interacting stochastic particle dynamics in terms of quantum spin systems produced interesting and fruitful interchanges between the areas of equilibrium and nonequilibrium statistical mechanics. The connection between these areas follows from the similarity between the master equation describing the time-fluctuations on the nonequilibrium stochastic problem and the quantum fluctuations of the equilibrium quantum spin chains [1]- [20].Unlike the area of nonequilibrium interacting systems, where very few models are fully solvable, there exists a huge family of quantum chains appearing in equilibrium problems that are exactly integrable. The machinery that allows the exact solutions of these quantum chains comes from the Bethe ansatz in its several formulations (see [21]-[24] for reviews). The above mentioned mathematical connection between equilibrium and nonequilibrium revealed that some quantum chains related to interacting stochastic problems are exactly solvable through the Bethe ansatz. The simplest example is the problem of asymmetric diffusion of hard-core particles on the one dimensional lattice (see [16,17,20] for reviews). The time fluctuations of this last model are governed by a time evolution operator that coincides with the exact integrable anisotropic Heisenberg chain, or the so called, XXZ quantum chain, in its ferromagnetically ordered regime. A generalization of this stochastic problem where exact integrability is also known [25]-[27] is the case where there are N (N = 1, 2, ...) classes of particles hierarchically ordered and diffusing asymmetrically on the lattice. The quantum chain related to this problem is known in the literature as the anisotropic Sutherland model [28...

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On the Asymmetric Simple Exclusion Process with Multiple Species

Tracy

Widom

2012

J Stat Phys

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In previous work the authors, using the Bethe Ansatz, found for the N-particle asymmetric simple exclusion process on the integers a formula-a sum of multiple integrals-for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species having priority over those of a lower species. Here the integrands in the multiple integrals are defined by a system of relations whose consistency requires verifying that the Yang-Baxter equations hold.

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Exact solution of asymmetric diffusion with N classes of particles of arbitrary size and hierarchical order

Abstract: c from holes (empty sites). We generalize and solve exactly this model by considering the molecules in each distinct class c = 1, 2, ..., N with sizes s c (s c = 0, 1, 2, ...), in units of the lattice spacing. The solution is derived via a Bethe ansatz of nested type.]]>

Cited by 34 publications

References 2 publications

The asymmetric simple exclusion process with an open boundary

The asymmetric simple exclusion process with an open boundary

The exact solution of the asymmetric exclusion problem with particles of asrbitrary size: matrix product ansatz

On the Asymmetric Simple Exclusion Process with Multiple Species

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