A reverse duality for the ASEP with open boundaries

We study the stirring process with N − 1 species on a generic graph G = ( V , E ) with reservoirs. The multispecies stirring process generalizes the symmetric exclusion process, which is recovered in the case N = 2. We prove the existence of a dual process defined on an extended graph G ~ = ( V ~ , E ~ ) which includes additional sites in V ~ ∖ V where dual particles get absorbed in the long-time limit. We thus obtain a characterization of the non-equilibrium steady state of the boundary-driven system in terms of the absorption probabilities of dual particles. The process is integrable for the case of the one-dimensional chain with two reservoirs at the boundaries and with maximally one particle per site. We compute the absorption probabilities by relying on the underlying g l ( N ) symmetry and the matrix product ansatz. Thus one gets a closed-formula for (long-ranged) correlations and for the non-equilibrium stationary measure. Extensions beyond this integrable set-up are also discussed.

Section: The Modelmentioning

confidence: 99%

Duality for the multispecies stirring process with open boundaries

Casini,

Frassek,

Giardinà

2024

“…More recent results include [2] where a duality relation between an half-line open ASEP and a sub-Markov process where particles perform an asymmetric exclusion dynamics in the bulk and are killed at the boundary is proven. In [28,29] it is shown a reverse duality relation for an open ASEP with open boundary and a shock ASEP with reflecting boundary.…”

Section: Introductionmentioning

confidence: 99%

Duality for a boundary driven asymmetric model of energy transport

Carinci,

Casini,

Franceschini

2024

We study the Asymmetric Brownian Energy, a model of heat conduction deﬁned on the one-dimensional ﬁnite lattice with open boundaries. The system is shown to be dual to the Symmetric inclusion process with absorbing boundaries. The proof relies on a non-local map transformation procedure relating the model to its symmetric version. As an application, we show how the duality relation can be used to analytically compute suitable exponential moments with respect to the stationary measure.

“…However, unfortunately, the pathway outlined above has not yet been fully developed for the asymmetric case (XXZ-type models). For some recent interesting developments in this direction for the ASEP, we refer the reader to [16,17].…”

Section: Introductionmentioning

confidence: 99%

The steady state of the boundary-driven multiparticle asymmetric diffusion model

Frassek,

Szécsényi

2024

We consider the multiparticle asymmetric diﬀusion model (MADM) introduced by Sasamoto and Wadati with integrability preserving reservoirs at the boundaries. In contrast to the open asymmetric simple exclusion process (ASEP) the number of particles allowed per site is unbounded in the MADM. Taking inspiration from the stationary measure in the symmetric case, i.e. the rational limit, we ﬁrst obtain the length 1 solution and then show that the steady state can be expressed as an iterated product of Jackson q-integrals. In the proof of the stationarity condition, we observe a cancellation mechanism that closely resembles the one of the matrix product ansatz. To our knowledge, the occupation probabilities in the steady state of the boundary-driven MADM were not available before.