Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries

Cantini, Luigi; Garbali, Alexandr; Gier, Jan de; Wheeler, Michael

doi:10.1088/1751-8113/49/44/444002

Cited by 23 publications

(52 citation statements)

References 67 publications

(109 reference statements)

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“…The coefficients of this power series in z are the sums of squared weights of the TASEP for increasing system size with β = 1, and match with those calculated in (17)(18)(19) using the matrix reduction relations. By the symmetry between α and β, we also have from (76) Q(z; 1, β), whereby α is fixed and β is variable.…”

Section: Coefficient Extraction Q(z; 1 1) Resultsmentioning

confidence: 59%

“…Phase transitions in this system are characterised by changes in the macroscopic forms of density profile and particle current, both of which are exactly calculable by the matrix product formalism. The matrix product formalism has also allowed quantities such as the moments of the current [13][14][15] to be computed and has been extended to solve open systems with many species of particle [16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

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Rényi entropy of the totally asymmetric exclusion process

Wood¹,

Blythe²,

Evans³

2017

J. Phys. A: Math. Theor.

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The Rényi entropy is a generalisation of the Shannon entropy that is sensitive to the fine details of a probability distribution. We present results for the Rényi entropy of the totally asymmetric exclusion process (TASEP). We calculate explicitly an entropy whereby the squares of configuration probabilities are summed, using the matrix product formalism to map the problem to one involving a six direction lattice walk in the upper quarter plane. We derive the generating function across the whole phase diagram, using an obstinate kernel method. This gives the leading behaviour of the Rényi entropy and corrections in all phases of the TASEP. The leading behaviour is given by the result for a Bernoulli measure and we conjecture that this holds for all Rényi entropies. Within the maximal current phase the correction to the leading behaviour is logarithmic in the system size. Finally, we remark upon a special property of equilibrium systems whereby discontinuities in the Rényi entropy arise away from phase transitions, which we refer to as secondary transitions. We find no such secondary transition for this nonequilibrium system, supporting the notion that these are specific to equilibrium cases.

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Section: Coefficient Extraction Q(z; 1 1) Resultsmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Rényi entropy of the totally asymmetric exclusion process

Wood¹,

Blythe²,

Evans³

2017

J. Phys. A: Math. Theor.

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“…The 2-species case has now been settled [5,13,15,33] but we are still lacking for a solution for the general N-species case. In particular it would be nice to construct the stationary state of the model pointed out in [10] in a matrix product form because of its connection with Koornwinder polynomials.…”

Section: Resultsmentioning

confidence: 99%

“…Specific N-species cases with open boundaries have been studied in [28] or with reflexive boundaries in [1] and a large class of integrable boundaries have been provided in [14]. Some connections of the stationary state with orthogonal polynomials have been revealed [9,10]. The phase diagram of such multi-species ASEP with open boundaries has been exactly computed in some specific cases [7].…”

Section: Introductionmentioning

confidence: 99%

Matrix product solution to multi-species ASEP with open boundaries

Finn¹,

Ragoucy²,

Vanicat³

2018

J. Stat. Mech.

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We study a class of multi-species ASEP with open boundaries. The boundaries are chosen in such a way that all species of particles interact non-trivially with the boundaries and are present in the stationary state. We give the exact expression of the stationary state in a matrix product form and we compute its normalisation. Densities and currents for the different species are then computed in term of this normalisation.

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“…The multi-species ASEP (m-ASEP) is also exactly solvable with periodic boundaries [27,28] and a variety of open boundaries [20,[29][30][31]. While a number of results is known for the exact solution of m-ASEP with periodic boundary conditions, not much is known for open boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

T-Q relations for the integrable two-species asymmetric simple exclusion process with open boundaries

Zhang¹,

Wen²,

Gier³

2019

J. Stat. Mech.

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We study the integrable two-species asymmetric simple exclusion process (ASEP) for two inequivalent types of open, non particle conserving boundary conditions. Employing the nested off-diagonal Bethe ansatz method, we construct for each case the corresponding homogeneous T -Q relations and obtain the Bethe ansatz equations. Numerical checks for small system sizes show completeness for some Bethe ansatz equations, and partial completeness for others.

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Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries

Cited by 23 publications

References 67 publications

Rényi entropy of the totally asymmetric exclusion process

Rényi entropy of the totally asymmetric exclusion process

Matrix product solution to multi-species ASEP with open boundaries

T-Q relations for the integrable two-species asymmetric simple exclusion process with open boundaries

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