EXACT SOLUTION OF A 1D ASYMMETRIC EXCLUSION MODEL USING A MATRIX FORMULATION

Several recent works have shown that the one-dimensional fully asymmetric exclusion model, which describes a system of particles hopping in a preferred direction with hard core interactions, can be solved exactly in the case of open boundaries. Here we present a new approach based on representing the weights of each configuration in the steady state as a product of non-commuting matrices. With this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules. We obtain several explicit forms for these non-commuting matrices which are, in the general case, infinite-dimensional. Our approach allows exact expressions to be derived for the current and density profiles. Finally we discuss briefly two possible generalizations of our results: the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles.