Exact diffusion constant for the one-dimensional partially asymmetric exclusion model

We calculate exactly the diffusion constant associated with the fluctuations of the current for the partial asymmetric exclusion model on a ring with an arbitrary number of particles and holes. We also give the diffusion constant of a tagged particle on that ring. Our approach extends, using the deformed harmonic oscillator algebra, a result already known for the fully asymmetric case. In the limit of weak asymmetry, we extract from our exact expression the crossover between the Edwards - Wilkinson and the Kardar - Parisi - Zhang equations in (1 + 1) dimensions.