Exact solution of a 1D asymmetric exclusion model using a matrix formulation

Several recent works have shown that the onedimensional fully asymmetric exclusion model, which describes a ystem of panicles hopping in a preferred direction with hard core interactions, can be solved exactb in the case of open boundaries. Here we present a new approach based on representing the weights of each mnfiguration in the steady state as a product of noncommuting matrices. Wth this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisQ ve’y simple algebraic NI= We obtain several explicit toms for these noncommuting matrices which are, in the general case. infinite-dimensional. Our approach allows exam expresions to be derived for the current and density profiles. Finally we discuss h’efly two possible generalizations of our results: the pmblem of panially asymmetric exclusion and the case of a mixture of two kinds of panicles.