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

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