Asymmetric Simple Exclusion Process with open boundaries and Koornwinder polynomials

TL;DR

This paper studies the steady state of an asymmetric exclusion process with open boundaries, revealing connections to Koornwinder and Macdonald polynomials, and computes key physical quantities like current and density.

Contribution

It introduces a spectral parameter deformation of the process, linking steady state probabilities to non-symmetric Koornwinder polynomials and identifying the partition function with a Macdonald-Koornwinder polynomial.

Findings

Partition function equals a symmetric Macdonald-Koornwinder polynomial.

Derived explicit formulas for steady current.

Calculated average density of first class particles.

Abstract

In this paper we analyze the steady state of the Asymmetric Simple Exclusion process with open boundaries and second class particles by deforming it through the introduction of spectral parameters. The (unnormalized) probabilities of the particle configurations get promoted to Laurent polynomials in the spectral parameters and are constructed in terms of non-symmetric Koornwinder polynomials. In particular we show that the partition function coincides with a symmetric Macdonald-Koornwinder polynomial. As an outcome we compute the steady current and the average density of first class particles.