Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra

We study the partially asymmetric exclusion process with open boundaries. W e generalize the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and c urrents valid for all values of the asymmetry parameter q. Due to the relat ionship between the matrix algebra and the q-deformed quantum harmonic osci llator algebra we find that q Hermite polynomials, along with their orthogo nality properties and generating functions, are of great utility. We employ two distinct sets of q-Hermite polynomials, one for q < 1 and the other fo r q > 1. It turns out that these correspond to two distinct regimes: the pr eviously studied case of forward bias (q < 1) and the regime of reverse bia s (q > 1) where the boundaries support a current opposite indirection to th e bulk bias. For the forward bias case we confirm the previously proposed p hase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size.