Current Fluctuations in the exclusion process and Bethe Ansatz

We use the Bethe Ansatz to derive analytical expressions for the current statistics in the asymmetric exclusion process with both forward and backward jumps. The Bethe equations are highly coupled and this fact has impeded their use to derive exact results for finite systems. We overcome this technical difficulty by a reformulation of the Bethe equations into a one variable polynomial problem, akin to the functional Bethe Ansatz. The perturbative solution of this equation leads to the cumulants of the current. We calculate here the first two orders and derive exact formulae for the mean value of the current and its fluctuations.