A matrix ansatz for the diffusion of an impurity in the asymmetric exclusion process

Boutillier, Cédric; François, Paul; Mallick, Kirone; Mallick, S.

doi:10.1088/0305-4470/35/46/301

Cited by 26 publications

(22 citation statements)

References 30 publications

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“…The matrix ansatz of section VI (which gives the weights of the microscopic configurations in the steady state) has been generalized to describe the steady state of several other systems [12,62,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124], with of course modified algebraic rules for the matrices the vectors W | and |V .…”

Section: The Matrix Approach For the Asymmetric Exclusion Processmentioning

confidence: 99%

Non-equilibrium steady states: fluctuations and large deviations of the density and of the current

Derrida¹

2007

J. Stat. Mech.

736

1,279

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These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow to calculate the fluctuations and large deviations of the density and of the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example non-Gaussian fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium. P non-equilibrium (C) = ? * Electronic address: bernard.derrida@lps.ens.fr

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Section: The Matrix Approach For the Asymmetric Exclusion Processmentioning

confidence: 99%

Non-equilibrium steady states: fluctuations and large deviations of the density and of the current

Derrida¹

2007

J. Stat. Mech.

736

1,279

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“…From this definition of A(t), and using (13) and the fact that Q is of degree n, we know the value of A(t) in t = 0, t = 1 and in the limit t → ∞ A(0) = x n , A(1) = e nγ and lim t→∞ A(t) = 1 (18) From (11), we also know that…”

Section: Bethe Ansatz For the Fluctuations Of The Currentmentioning

confidence: 99%

“…The fluctuations of the current have been studied for different variants of the ASEP: finite size open lattice [12], systems with second class particles [11,13], infinite lattice [14,15],... For the totally asymmetric exclusion process, in which the particles only hop in one direction (TASEP), on a finite size lattice with periodic boundary conditions, all the cumulants of the probability distribution of the current have been calculated by Bethe Ansatz, using a simplification of the Bethe Ansatz equations that occur only in this case. For the partially asymmetric exclusion process on a ring, a particular scaling limit of the probability distribution function of the current was calculated using a thermodynamic limit of the Bethe Ansatz equations [16].…”

Section: Introductionmentioning

confidence: 99%

Fluctuations and skewness of the current in the partially asymmetric exclusion process

Prolhac¹

2008

J. Phys. A: Math. Theor.

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We use functional Bethe Ansatz equations to calculate the cumulants of the total current in the partially asymmetric exclusion process. We recover known formulas for the first two cumulants (mean value of the current and diffusion constant) and obtain an explicit finite size formula for the third cumulant. The expression for the third cumulant takes a simple integral form in the limit where the asymmetry scales as the inverse of the square root of the size of the system, which corresponds to a natural separation between weak and strong asymmetry. PACS numbers: 05-40.-a; 05-60.-k

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“…Each product of D's and E's can be written as a sum of products of A's and B's which can be ordered using (12) by pushing all the A's to the right and all the B's to the left. One gets that way a sum of terms of the form B p A q , the matrix elements of which can be evaluated easily ( W |B p A q |V = W |V ) from (7) and (11).…”

Section: The Matrix Ansatz For the Ssepmentioning

confidence: 99%

“…The matrix ansatz of Section 3 (which gives the weights of the microscopic configurations in the steady state) has been generalized to describe the steady state of several other systems [1], [8], [11], [14], [18], [25], [30], [31], [32], [37], [40], [42], [43], [45], with of course modified algebraic rules for the matrices the vectors W | and |V .…”

Section: The Matrix Approach For the Asymmetric Exclusion Processmentioning

confidence: 99%

Matrix ansatz and large deviations of the density in exclusion processes

Derrida¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Exclusion processes describe a gas of particles on a lattice with hard core repulsion. When such a lattice gas is maintained in contact with two reservoirs at unequal densities, or driven by an external field, it exhibits a non-equilibrium steady state. In one dimension, a number of properties of this steady state can be calculated exactly using a matrix ansatz. This talk gives a short review on results obtained recently by this matrix ansatz approach.

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A matrix ansatz for the diffusion of an impurity in the asymmetric exclusion process

Cited by 26 publications

References 30 publications

Non-equilibrium steady states: fluctuations and large deviations of the density and of the current

Non-equilibrium steady states: fluctuations and large deviations of the density and of the current

Fluctuations and skewness of the current in the partially asymmetric exclusion process

Matrix ansatz and large deviations of the density in exclusion processes

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