Disordered exclusion process revisited: some exact results in the low-current regime

We study the steady state of the totally asymmetric simple exclusion process with inhomogeneous hopping rates associated with sites (site-wise disorder). Using the fact that the non-normalized steady-state weights, which solve the master equation, are polynomials in all the hopping rates, we propose a general method for calculating their first few lowest coefficients exactly. In the case of binary disorder where all slow sites share the same hopping rate r < 1, we apply this method to calculate the steady-state current up to the quadratic term in r for some particular disorder configurations. For the most general (non-binary) disorder, we show that in the low-current regime the current is determined solely by the current-minimizing subset of equal hopping rates, provided all other slow rates are much larger. Our approach can be readily applied to any other driven diffusive system with unidirectional hopping, if one can identify a hopping rate such that the current vanishes when this rate is set to zero.